cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A188130 Primes p such that 6p+1 divides the Mersenne number M(p)=A000225(p).

Original entry on oeis.org

5, 37, 73, 233, 397, 461, 557, 577, 601, 761, 1013, 1321, 1361, 1381, 1453, 1693, 1777, 1993, 2417, 2593, 2621, 2897, 3037, 3181, 3457, 3581, 3593, 4001, 4273, 4441, 4517, 4597, 4801, 4813, 4861, 4933, 5197, 5393, 5557, 5717, 5801, 6173, 6277, 6353, 6373, 6841, 6977, 7573, 7853, 7901, 8353, 8377, 9613, 10321, 10357
Offset: 1

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Author

M. F. Hasler, Mar 21 2011

Keywords

Comments

These primes are such that p=1 (mod 4) and 6p+1 is prime, but there are other primes with these properties (13, 17, ...) not in this sequence.
There are no primes p such that 4p+1 divides M(p), but those for which 2p+1 divides M(p) are the Lucasian primes A002515, and those for which 10p+1 divides M(p) are listed in A188133.

Links

Crossrefs

Cf. A002515, A188133.
Primes in A038844.

Programs

  • Mathematica
    Select[Range[10^4], PrimeQ[#] && PowerMod[2, #, 6# + 1] == 1 &] (* Amiram Eldar, Nov 13 2019 *)
  • PARI
    forprime(p=1,1e5,Mod(2,p*6+1)^p-1||print1(p", "))

A350702 Primes p such that 14*p + 1 divides 2^p - 1.

Original entry on oeis.org

929, 1433, 2393, 2609, 2657, 4373, 4793, 6029, 7529, 10133, 10433, 10949, 10973, 13049, 13109, 16829, 18869, 20873, 22349, 23417, 24137, 26717, 27737, 27893, 28433, 28517, 30977, 33809, 33857, 37217, 38189, 38237, 39209, 39749, 41453, 41813, 42569, 43313, 43613
Offset: 1

Views

Author

Karl-Heinz Hofmann, Jan 27 2022

Keywords

Comments

Known divisors of Mersenne(p) (2^p-1 or Mp for short) are of the form 2*k*p+1. See crossrefs for other k's. If k == 2 (mod 4), there are no such divisors in general. Here k is 14/2 = 7.

Examples

			See LINKS for example of a(13).

Links

Crossrefs

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A188133 (k = 5).
Cf. A000040, A001348.

Programs

  • Mathematica
    Select[Range[45000], PrimeQ[#] && PowerMod[2, #, 14*# + 1] == 1 &] (* Amiram Eldar, Jan 27 2022 *)
  • PARI
    forprime(p=1, 1e6, if (Mod(2, p*14+1)^p==1, print1(p,", ")))
  • Python
    from sympy import sieve
print([p for p in sieve[1:1000000] if pow(2, p, 14*p+1) == 1])

Formula

{p = A000040(i): 14*p+1 | A001348(i)}.

A350703 a(n) is the least integer k such that (2*n*k+1) | (2^k-1).

Original entry on oeis.org

3, 18, 5, 9, 15, 50, 40, 16, 7, 156, 60, 25, 180, 102, 113, 81, 10, 50, 29, 159, 51, 56, 24, 36, 47, 90, 337, 72, 55, 106, 33, 102, 780, 28, 117, 25, 155, 540, 60, 104, 223, 1012, 168, 180, 91, 540, 3132, 47, 510, 412, 154, 45, 80, 432, 201, 36, 90, 144, 97, 53, 279, 880
Offset: 1

Views

Author

Karl-Heinz Hofmann, Feb 03 2022

Keywords

Comments

The formula 2nk+1 is used to find trivial factors of Mersenne(p). Here it is used for all exponents (prime exponents and not prime exponents).
Mersenne primes of A000043 can be found in this sequence too (except for 2). E.g.: a(1, 3, 9, 315, 3855, 13797) = A000043(2..7).
If n mod 4 = 2 then a(n) must be composite.

Examples

			a(5) = 15: 2^15 - 1 = 32767; 2*5*15 + 1 = 151; 32767 mod 151 = 0, and there are no numbers < 15 which satisfy the requirement for n = 5.

Links

Crossrefs

Cf. A000043, A002515, A188130, A122095, A188133, A350702.

Programs

  • Mathematica
    a[n_] := Module[{k = 1}, While[PowerMod[2, k, 2*n*k + 1] != 1, k++]; k];  Array[a, 62] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = my(k=1); while (Mod(2, 2*n*k+1)^k != 1, k++); k; \\ Michel Marcus, Feb 03 2022
  • Python
    def A350703(k,expo):
    while pow(2, expo, 2*k*expo+1) != 1: expo += 1
    return expo
print([A350703(k,1) for k in range(1, 63)])
Showing 1-3 of 3 results.

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