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Search: a007510 -id:a007510
Displaying 1-10 of 97 results found. page 1 2 3 4 5 6 7 8 9 10
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A117244 Single (or isolated or non-twin) primes (A007510) that are not Chen primes (A109611). +20
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MAPLE
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
MATHEMATICA
Select[Range[1500], PrimeQ[#] && !PrimeQ[#-2] && PrimeOmega[#+2] > 2 &] (* Amiram Eldar, Oct 19 2021 *)
PROG
(PARI) isok(p) = isprime(p) && !isprime(p-2) && !isprime(p+2) && (bigomega(p+2) > 2); \\ Michel Marcus, Oct 19 2021
CROSSREFS
Cf. A007510, A109611.
KEYWORD
nonn
AUTHOR
Jani Melik, Apr 22 2006
EXTENSIONS
Terms beyond 397 from R. J. Mathar, Dec 09 2009
Offset corrected by Amiram Eldar, Oct 19 2021
STATUS
approved
A153478 Sum of first n isolated (or single) primes A007510. +20
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
MATHEMATICA
Accumulate[Select[Prime[Range[100]], !PrimeQ[#-2]&&!PrimeQ[#+2]&]] (* Harvey P. Dale, Feb 08 2011 *)
CROSSREFS
Cf. A000040, A007504, A007510, A048598, A071148, A153474.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Dec 27 2008
STATUS
approved
A162308 Number of twin primes A001097 smaller than the non-twin prime A007510(n). +20
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..69.
EXAMPLE
a(2)=7 counts the numbers 3, 5, 7, 11, 13, 17, 19 below 23=A007510(2).
MAPLE
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:
seq(A162308(n), n=1..120) ; # R. J. Mathar, Jul 02 2009
CROSSREFS
Cf. A000040, A073425.
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Jul 01 2009
EXTENSIONS
Edited by R. J. Mathar, Jul 02 2009
STATUS
approved
A064110 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. +20
0
1, 2, 23, 263, 2917, 38639, 603311, 11093633, 236524303, 5782539281 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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 ..."
REFERENCES
"Isolated Primes", by Richard L. Francis, J. Rec. Math., 11 (1978), 17-22.
LINKS
Table of n, a(n) for n=0..9.
CROSSREFS
Cf. A007097, A063502.
KEYWORD
hard,nonn
AUTHOR
Lubomir Alexandrov, Sep 07 2001
EXTENSIONS
a(9) from Sean A. Irvine, Jun 12 2023
STATUS
approved
A133076 Successive digits of isolated primes A007510(n). +20
0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..82.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
MATHEMATICA
With[{prs=Prime[Range[100]]}, Flatten[IntegerDigits/@Complement[prs, Flatten[Select[Partition[prs, 2, 1], Last[#]-First[#]==2&]]]]] (* Harvey P. Dale, Sep 20 2011 *)
CROSSREFS
Cf. A033308, A007510.
KEYWORD
easy,nonn,base,less
AUTHOR
Omar E. Pol, Nov 10 2007
STATUS
approved
A167514 Index of prime(n) in A007510 or in A001097. +20
0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A007510 U A001097 = A000040.
LINKS
Table of n, a(n) for n=1..75.
FORMULA
A007510(a(n))=n or A001097(a(n))=n.
CROSSREFS
Cf. A000040, A001097, A007510.
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Nov 05 2009
EXTENSIONS
Definition corrected by R. J. Mathar, May 30 2010
STATUS
approved
A374591 Even numbers that can be written as the sum of two isolated primes (A007510). +20
0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..54.
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
Cf. A007510.
KEYWORD
nonn,changed
AUTHOR
Marc Groz, Jul 12 2024
STATUS
approved
A067774 Primes p such that p+2 is not a prime. +10
41
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
Except for a(1)=2, a(n+1)=A049591(n).
a(n) ~ n log n. - Charles R Greathouse IV, Jul 01 2013
MATHEMATICA
f[n_]:=PrimeQ[n+2]; lst={}; Do[p=Prime[n]; If[ !f[p], AppendTo[lst, p]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 17 2009 *)
PROG
(Magma) [p: p in PrimesUpTo(400) | NextPrime(p)-p ne 2]; // Bruno Berselli, Apr 09 2013
(PARI) is(n)=isprime(n)&&!isprime(n+2) \\ Charles R Greathouse IV, Jul 01 2013
CROSSREFS
Cf. A049591.
Complement of A001359 in A000040, A025584, A007510.
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Feb 06 2002
EXTENSIONS
Better description from Vladeta Jovovic, Dec 14 2002
STATUS
approved
A167706 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. +10
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Equals A007510 U A014574.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
a(n) ~ n log n. - Charles R Greathouse IV, Apr 29 2015
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A007510, A014574.
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Nov 10 2009, Nov 14 2009
EXTENSIONS
Corrected (97 inserted) by R. J. Mathar, Nov 16 2009
STATUS
approved
A132231 Primes congruent to 7 (mod 30). +10
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Primes ending in 7 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 07 2009
Intersection of A030432 and A002476. - Ray Chandler, Apr 07 2009
Only from 4927 on, there are more composite numbers than primes in {7+30k}, see A227869. - M. F. Hasler, Nov 02 2013
Terms are non-twin primes A007510, except for 7. - Jonathan Sondow, Oct 27 2017
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages
Omar E. Pol, Prime numbers in a sieve with 30 columns
Omar E. Pol, Sobre el patrón de los números primos
FORMULA
a(n) = A158573(n)*30 + 7. - Ray Chandler, Apr 07 2009
a(n) = A211890(4,n-1) for n <= 5. - Reinhard Zumkeller, Jul 13 2012
MATHEMATICA
Select[30*Range[0, 100]+7, PrimeQ] (* Harvey P. Dale, Feb 01 2012 *)
Select[Prime[Range[1000]], MemberQ[{7}, Mod[#, 30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
PROG
(Haskell)
a132231 n = a132231_list !! (n-1)
a132231_list = [x | k <- [0..], let x = 30 * k + 7, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012
(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
CROSSREFS
Cf. A007510, A007528, A010051, A068229, A117047, A129807, A039949, A132230-A132236, A214360.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 15 2007
EXTENSIONS
Extended by Ray Chandler, Apr 07 2009
STATUS
approved
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Last modified August 24 19:29 EDT 2024. Contains 375417 sequences. (Running on oeis4.)