Displaying 1-10 of 19 results found.
Number of prime factors (without multiplicity) in Mersenne composites A065341.
+20
13
2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 5, 4, 5, 2, 4, 3, 4, 5, 3, 2, 2, 3, 6, 2, 4, 4, 6, 2, 5, 3, 4, 2, 2, 3, 2, 3, 2, 5, 3, 4, 4, 3, 5, 2, 3, 3, 6, 5, 2, 2, 5, 3, 9, 4, 3, 5, 2, 8, 4, 4, 3, 5, 2, 4, 6, 3, 4, 2, 7, 3, 4, 4, 2, 5, 4, 5, 3, 5, 4, 3, 6, 4, 3, 4, 3, 4, 4
COMMENTS
Currently the smallest prime exponent p for which 2^p-1 is incompletely factored is p = 1213. - Gord Palameta, Aug 06 2018
LINKS
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project: cofactor of M1213 is C297.
MATHEMATICA
k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; AppendTo[k, d]], {n, 1, 40}]; k
PROG
(PARI) forprime(p=1, 1e3, if(!ispseudoprime(2^p-1), print1(omega(2^p-1), ", "))) \\ Felix Fröhlich, Aug 12 2014
Mersenne composites ( A065341) with exactly 3 prime factors.
+20
9
536870911, 8796093022207, 140737488355327, 9007199254740991, 2361183241434822606847, 9444732965739290427391, 604462909807314587353087
MATHEMATICA
k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k
Mersenne composites ( A065341) with exactly 2 prime factors.
+20
7
2047, 8388607, 137438953471, 2199023255551, 576460752303423487, 147573952589676412927, 9671406556917033397649407, 158456325028528675187087900671, 2535301200456458802993406410751
MAPLE
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) = 2) then
RETURN (i)
MATHEMATICA
k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k
PROG
(PARI) forprime(p=1, 1e2, if(bigomega(2^p-1)==2, print1(2^p-1, ", "))) \\ Felix Fröhlich, Aug 12 2014
a(n) is the largest proper divisor of the Mersenne composite A065341(n).
+20
3
89, 178481, 2304167, 616318177, 164511353, 20408568497, 59862819377, 1416003655831, 3203431780337, 761838257287, 10334355636337793, 21514198099633918969, 224958284260258499201, 57912614113275649087721
COMMENTS
Note that not all the largest divisors are primes.
MATHEMATICA
a = {}; Do[m = 2^Prime[n] - 1; If[PrimeQ[m], null, AppendTo[a, Divisors[m][[ -2]]]], {n, 1, 40}]; a
Number of prime factors (with multiplicity) of Mersenne composites ( A065341).
+20
2
2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 5, 4, 5, 2, 4, 3, 4, 5, 3, 2, 2, 3, 6, 2, 4, 4, 6, 2, 5, 3, 4, 2, 2, 3, 2, 3, 2, 5, 3, 4, 4, 3, 5, 2, 3, 3, 6, 5, 2, 2, 5, 3, 9, 4, 3, 5, 2, 8, 4, 4, 3, 5, 2, 4, 6, 3, 4, 2, 7, 3, 4, 4, 2, 5, 4, 5, 3, 5, 4
COMMENTS
If the conjecture that all Mersenne composites are squarefree is true, then this sequence is identical to A135975. - Felix Fröhlich, Aug 24 2014
MATHEMATICA
a = {}; Do[If[PrimeQ[n] && !PrimeQ[2^n - 1], w = 2^n - 1; c = FactorInteger[w]; d = Length[c]; b = 0; Do[b = b + c[[k]][[2]], {k, 1, d}]; AppendTo[a, b]], {n, 2, 150}]; a
PrimeOmega/@Select[2^Prime[Range[100]]-1, !PrimeQ[#]&] (* Harvey P. Dale, Nov 01 2016 *)
PROG
(PARI) forprime(p=2, 1e3, if(!ispseudoprime(2^p-1), print1(bigomega(2^p-1), ", "))) \\ Felix Fröhlich, Aug 24 2014
Mersenne composites A065341 with 4 or more prime factors.
+20
1
10384593717069655257060992658440191, 2854495385411919762116571938898990272765493247, 182687704666362864775460604089535377456991567871
MAPLE
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) > 3) then
RETURN (i)
MATHEMATICA
k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d >3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k
Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.
+10
19
341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489, 88357, 90751
COMMENTS
Every semiprime in A001567 is in this sequence (see Sierpiński). a(61) = 294409 is the first term having more than two prime factors. See A178997 for super-Poulet numbers having more than two prime factors. - T. D. Noe, Jan 11 2011 Composite numbers n such that 2^d == 2 (mod n) for every d|n. - Thomas Ordowski, Sep 04 2016 Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - Thomas Ordowski, Sep 06 2016 Composite numbers n = p(1)^e(1)*p(2)^e(2)*...*p(k)^e(k) such that 2^gcd(p(1)-1,p(2)-1,...,p(k)-1) == 1 (mod n). - Thomas Ordowski, Sep 12 2016 Nonsquarefree terms are divisible by the square of a Wieferich prime (see A001220). These include 1194649, 12327121, 5654273717, 26092328809, 129816911251. - Robert Israel, Sep 13 2016
REFERENCES
W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.
MAPLE
filter:= = proc(n)
not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
end proc:
select(filter, [seq(i, i=3..10^5, 2)]); # Robert Israel, Sep 13 2016
MATHEMATICA
Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]
PROG
(PARI) is(n)=if(isprime(n), return(0)); fordiv(n, d, if(Mod(2, d)^d!=2, return(0))); n>1 \\ Charles R Greathouse IV, Aug 27 2016
Primes p such that 2^p-1 has exactly 2 prime factors.
+10
7
11, 23, 37, 41, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997, 1061, 1063
MATHEMATICA
k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, Prime[n]]]], {n, 1, 40}]; k
Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.
+10
4
5, 9, 12, 13, 17, 19, 23, 25, 26, 27, 29, 32, 33, 34, 35, 39, 45, 46, 49, 53, 57, 58, 60, 62, 69, 74, 75, 82, 88, 93, 99, 129, 140, 152, 164, 166, 168, 178, 179
MATHEMATICA
k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, n]]], {n, 1, 40}]; k
Select[Range[40], PrimeNu[2^Prime[#]-1]==2&] (* Harvey P. Dale, Jul 07 2013 *)
Second prime factor, if it exists, of Mersenne numbers.
+10
3
89, 178481, 1103, 616318177, 164511353, 9719, 4513, 69431, 3203431780337, 761838257287, 48544121, 2298041, 202029703, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393
EXAMPLE
The 5th Mersenne number 2^11 - 1 = 23*89 and 89 is the second prime divisor.
The 9th Mersenne number 2^23 - 1 = 47*178481 and 178481 is the second prime divisor.
Notice 23, 89 congruent to 1 mod 11 and 47, 178481 congruent to 1 mod 23.
PROG
(PARI) mersenne(b, n, d) = { c=0; forprime(x=2, n, c++; y = b^x-1; f=factor(y); v=component(f, 1); ln = length(v); if(ln>=d, print1(v[d]", ")); ) }
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