Search: a007510 -id:a007510
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79, 97, 163, 173, 223, 277, 331, 367, 373, 383, 397, 439, 457, 547, 593, 607, 613, 673, 691, 709, 727, 733, 739, 757, 773, 853, 907, 929, 967, 997, 1013, 1069, 1087, 1103, 1123, 1129, 1171, 1181, 1213, 1223, 1237, 1249, 1307, 1373, 1423, 1433, 1447, 1493
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OFFSET
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1,1
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LINKS
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EXAMPLE
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79 is single prime, but not Chen prime, since 79 -2 = 77 = 7*11 is composite, and 79 + 2 = 81 = 3^4 is neither prime nor semiprime.
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MAPLE
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isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc: isA109611 := proc(n) if isprime(n) then isprime(n+2) or isA001358(n+2) ; else false; end if; end proc: isA007510 := proc(n) if isprime(n) then not isprime(n-2) and not isprime(n+2) ; else false; end if ; end proc: isA117244 := proc(n) isA007510(n) and not isA109611(n) ; end proc: for n from 1 to 4000 do if isA117244(n) then printf("%d, ", n) ; fi; end do ; # R. J. Mathar, Dec 09 2009
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MATHEMATICA
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Select[Range[1500], PrimeQ[#] && !PrimeQ[#-2] && PrimeOmega[#+2] > 2 &] (* Amiram Eldar, Oct 19 2021 *)
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PROG
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(PARI) isok(p) = isprime(p) && !isprime(p-2) && !isprime(p+2) && (bigomega(p+2) > 2); \\ Michel Marcus, Oct 19 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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2, 25, 62, 109, 162, 229, 308, 391, 480, 577, 690, 817, 948, 1105, 1268, 1435, 1608, 1819, 2042, 2275, 2526, 2783, 3046, 3323, 3616, 3923, 4240, 4571, 4908, 5261, 5620, 5987, 6360, 6739, 7122, 7511, 7908, 8309, 8718, 9157, 9600
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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Accumulate[Select[Prime[Range[100]], !PrimeQ[#-2]&&!PrimeQ[#+2]&]] (* Harvey P. Dale, Feb 08 2011 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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0, 7, 9, 11, 11, 13, 15, 15, 15, 15, 19, 19, 19, 23, 23, 23, 23, 29, 29, 31, 33, 33, 33, 35, 37, 37, 39, 39, 39, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 45, 45, 45, 45, 47, 47, 47, 47, 47, 47, 47, 49, 49, 49, 49, 51, 51, 51, 53, 53, 55, 57, 57, 59, 59, 59, 59, 59, 59, 59
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OFFSET
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1,2
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LINKS
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EXAMPLE
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a(2)=7 counts the numbers 3, 5, 7, 11, 13, 17, 19 below 23=A007510(2).
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MAPLE
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isA007510 := proc(n) RETURN(isprime(n) and not isprime(n-2) and not isprime(n+2)) ; end:
isA001097 := proc(n) RETURN(isprime(n) and (isprime(n-2) or isprime(n+2)) ) ; end:
A007510 := proc(n) local a; if n = 1 then 2; else for a from procname(n-1)+1 do if isA007510(a) then RETURN(a) ; fi; od: fi; end:
A162308 := proc(n) local a, k; a := 0 ; for k from 3 to A007510(n)-1 do if isA001097(k) then a := a+1; fi; od; a; end:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A064110
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Let s(n) = n-th single prime (cf. A007510). Sequence is defined by recurrence a(n+1) = s(a(n)), n = 0,1,2,..., a(0)=1.
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+20
0
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OFFSET
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0,2
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COMMENTS
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This is the "isolated prime Eratosthenes progression at base 1 (ipep(1))". The next ipep are: ipep(3) = 3, 37, 397, 4751, 64403, 1038629, 19661749,...; ipep(4) = 4, 47, 491, 5897, 81131, 1328167, 25467419,...; ipep(5) = 5, 53, 557, 6709, 93287, 1541191, 29778547,...; ...; ipep(22)= 22, 257, 2861, 37799, 589181, 10821757, 230452837,... ipep(24)= 24, 277, 3079, 40823, 640121, 11807167, 252480587,... and so on.
In the terminology of A007097 the name is "isolated_prime-th recurrence ..."
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REFERENCES
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"Isolated Primes", by Richard L. Francis, J. Rec. Math., 11 (1978), 17-22.
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LINKS
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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2, 2, 3, 3, 7, 4, 7, 5, 3, 6, 7, 7, 9, 8, 3, 8, 9, 9, 7, 1, 1, 3, 1, 2, 7, 1, 3, 1, 1, 5, 7, 1, 6, 3, 1, 6, 7, 1, 7, 3, 2, 1, 1, 2, 2, 3, 2, 3, 3, 2, 5, 1, 2, 5, 7, 2, 6, 3, 2, 7, 7, 2, 9, 3, 3, 0, 7, 3, 1, 7, 3, 3, 1, 3, 3, 7, 3, 5, 3, 3, 5, 9
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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With[{prs=Prime[Range[100]]}, Flatten[IntegerDigits/@Complement[prs, Flatten[Select[Partition[prs, 2, 1], Last[#]-First[#]==2&]]]]] (* Harvey P. Dale, Sep 20 2011 *)
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CROSSREFS
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KEYWORD
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easy,nonn,base,less
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AUTHOR
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STATUS
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approved
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1, 1, 2, 3, 4, 5, 6, 7, 2, 8, 9, 3, 10, 11, 4, 5, 12, 13, 6, 14, 15, 7, 8, 9, 10, 16, 17, 18, 19, 11, 12, 13, 20, 21, 22, 23, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 18, 19, 30, 31, 20, 32, 33, 21, 22, 23, 34, 35, 24, 36, 37, 25, 26, 38, 39, 27, 28, 29, 40, 41, 30, 31, 32, 33, 34
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A374591
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Even numbers that can be written as the sum of two isolated primes (A007510).
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+20
0
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4, 46, 60, 70, 74, 76, 84, 90, 94, 100, 102, 104, 106, 112, 114, 116, 120, 126, 130, 132, 134, 136, 142, 144, 146, 150, 154, 156, 158, 160, 162, 164, 166, 168, 172, 174, 176, 178, 180, 184, 186, 190, 192, 194, 196, 198, 200, 202, 204, 206, 210, 214, 216, 220
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OFFSET
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1,1
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LINKS
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EXAMPLE
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4 = 2 + 2 is a term, as 2 is the smallest isolated prime.
60 = 23 + 37 is the smallest term that is the sum of two distinct isolated primes.
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MATHEMATICA
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Lim=220; ip=Select[Prime[Range[Lim]], NoneTrue[#+{2, -2}, PrimeQ]&] ; ipp[a_]:={a, a}; Select[Union[Total/@Join[ipp/@ip, Subsets[ip, {2}]]], EvenQ[#]&&#<=Lim&] (* James C. McMahon, Aug 10 2024 *)
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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A067774
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Primes p such that p+2 is not a prime.
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+10
41
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2, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389
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OFFSET
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1,1
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COMMENTS
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Primes n such that n!*B(n+1) is an integer where B(k) are the Bernoulli numbers.
All primes except for the lower members of twin primes - i.e. remove 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, ... - Gerard Schildberger, Feb 13 2005
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LINKS
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FORMULA
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Except for a(1)=2, a(n+1)=A049591(n).
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MATHEMATICA
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PROG
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(Magma) [p: p in PrimesUpTo(400) | NextPrime(p)-p ne 2]; // Bruno Berselli, Apr 09 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A167706
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The single or isolated numbers. The union of single (or isolated or non-twin) primes and single (or isolated or average of twin prime pairs) nonprimes.
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+10
21
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2, 4, 6, 12, 18, 23, 30, 37, 42, 47, 53, 60, 67, 72, 79, 83, 89, 97, 102, 108, 113, 127, 131, 138, 150, 157, 163, 167, 173, 180, 192, 198, 211, 223, 228, 233, 240, 251, 257, 263, 270, 277, 282, 293, 307, 312, 317, 331, 337, 348, 353, 359, 367, 373, 379, 383, 389
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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MATHEMATICA
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With[{nn = 78}, {2}~Join~Union[Transpose[Select[Partition[Prime@ Range@ nn, 3, 1], And[#[[2]] - #[[1]] != 2, #[[3]] - #[[2]] != 2] &]][[2]], Map[Mean, Select[Partition[Prime@ Range@ nn, 2, 1], Differences@ # == {2} &]]]] (* Michael De Vlieger, Feb 22 2017, after Harvey P. Dale at A007510 and A014574 *)
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PROG
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(PARI) is(n)=if(n%6, (isprime(n) && !isprime(n-2) && !isprime(n+2)) || n==4, isprime(n-1) && isprime(n+1)) \\ Charles R Greathouse IV, Apr 29 2015
(PARI) lista(pmax) = {my(p1 = 2, p2 = 3); print1(2, ", "); forprime(p3 = 5, pmax, if(p2 == p1 + 2, print1(p1 + 1, ", ")); if(p2 != p1 + 2 && p2 != p3 - 2, print1(p2, ", ")); p1 = p2; p2 = p3); } \\ Amiram Eldar, May 17 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A132231
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Primes congruent to 7 (mod 30).
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+10
15
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7, 37, 67, 97, 127, 157, 277, 307, 337, 367, 397, 457, 487, 547, 577, 607, 727, 757, 787, 877, 907, 937, 967, 997, 1087, 1117, 1237, 1297, 1327, 1447, 1567, 1597, 1627, 1657, 1747, 1777, 1867, 1987, 2017, 2137, 2287, 2347, 2377, 2437, 2467, 2557, 2617, 2647
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OFFSET
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1,1
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COMMENTS
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Primes ending in 7 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 07 2009
Only from 4927 on, there are more composite numbers than primes in {7+30k}, see A227869. - M. F. Hasler, Nov 02 2013
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LINKS
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FORMULA
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MATHEMATICA
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Select[Prime[Range[1000]], MemberQ[{7}, Mod[#, 30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
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PROG
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(Haskell)
a132231 n = a132231_list !! (n-1)
a132231_list = [x | k <- [0..], let x = 30 * k + 7, a010051' x == 1]
(Magma) [p: p in PrimesUpTo(3000) | p mod 30 eq 7 ]; // Vincenzo Librandi, Aug 14 2012
(PARI) forstep(p=7, 1999, 30, isprime(p)&&print1(p", ")) \\ M. F. Hasler, Nov 02 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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