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.

A249759 Primes p such that sigma(p-1) is a prime q.

Original entry on oeis.org

3, 5, 17, 65537
Offset: 1

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Author

Jaroslav Krizek, Nov 13 2014

Keywords

Comments

Subsequence of {A023194(n)+1}.
Conjectures: 1) sequence is finite; 2) sequence is a subsequence of A019434 (Fermat primes); 3) sequence consists of Fermat primes p such that sigma(p-1) is a Mersenne prime; 4) a(n) = (A249761(n)+3)/2.
3 is the only prime p such that sigma(p+1) is prime, i.e., 3 is the only prime p such that sigma(p-1) and sigma(p+1) are both primes.
Conjecture: 3 is the only number n such that n and sigma(n+1) are both prime.
Primes p such that A051027(p-1) = sigma(sigma(p-1)) = 2*(p-1). Subsequence of A256438. - Jaroslav Krizek, Mar 29 2015
From Jaroslav Krizek, Mar 17 2016: (Start)
Primes p such that A000203(A000010(p)) = sigma(phi(p)) is a prime.
Prime terms from A062514 and A270413, A270414, A270415 and A270416. (End)
From Jaroslav Krizek, Nov 27 2016: (Start)
Corresponding values of primes q are in A249761: 3, 7, 31, 131071, ...
Conjecture: subsequence of A256438 and A278741.
Conjecture: also primes p such that tau(p-1) is a prime q; corresponding values of primes q are 2, 3, 5, 17. (End)

Examples

			Prime 17 is in the sequence because sigma(17-1) = sigma(16) = 31 (prime).

Crossrefs

Cf. A000203, A000668, A019434, A023194, A249760, A249761.

Programs

  • Magma
    [p: p in PrimesUpTo(1000000) | IsPrime(SumOfDivisors(p-1))]
  • Maple
    with(numtheory): A249759:=n->`if`(isprime(n) and isprime(sigma(n-1)), n, NULL): seq(A249759(n), n=1..6*10^5); # Wesley Ivan Hurt, Nov 14 2014
  • Mathematica
    Select[Range[10^5], PrimeQ[#]&& PrimeQ[DivisorSigma[1, # - 1]] &] (* Vincenzo Librandi, Nov 14 2014 *)
Select[Prime[Range[7000]],PrimeQ[DivisorSigma[1,#-1]]&] (* Harvey P. Dale, Jun 14 2020 *)
  • PARI
    lista(nn) = {forprime(p=1, nn, if (isprime(sigma(p-1)), print1(p, ", ")););} \\ Michel Marcus, Nov 14 2014

Formula

a(n) = A249760(n) + 1.
Sigma(a(n)-1) = A249761(n).

A359134 a(n) = Sum_{d|n} (2*d)^(n/d - 1).

Original entry on oeis.org

1, 3, 5, 13, 17, 55, 65, 201, 293, 779, 1025, 3365, 4097, 12303, 17781, 49681, 65537, 204547, 262145, 791549, 1095429, 3145751, 4194305, 12897625, 16787217, 50331675, 68788805, 201591509, 268435457, 815505231, 1073741825, 3223326753, 4355433957, 12884901923
Offset: 1

Views

Author

Seiichi Manyama, Jan 13 2023

Keywords

Links

Crossrefs

Cf. A087909, A278741, A349970, A359733, A359796.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (2*#)^(n/# - 1) &]; Array[a, 30] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (2*d)^(n/d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*k*x^k)))

Formula

G.f.: Sum_{k>0} x^k / (1 - 2 * k * x^k).
If p is prime, a(p) = 1 + 2^(p-1).

A279882 a(n) = 2^(prime(n) + 1) - 1.

Original entry on oeis.org

7, 15, 63, 255, 4095, 16383, 262143, 1048575, 16777215, 1073741823, 4294967295, 274877906943, 4398046511103, 17592186044415, 281474976710655, 18014398509481983, 1152921504606846975, 4611686018427387903, 295147905179352825855, 4722366482869645213695
Offset: 1

Views

Author

Jaroslav Krizek, Dec 21 2016

Keywords

Comments

Numbers whose binary representation is 1 repeated (prime(n)+1) times.
The only prime term is 7.

Examples

			For n=3; a(3) = 2^(prime(3) + 1) - 1 = 2^(5 + 1) - 1 = 2^6 - 1 = 63.

Crossrefs

Cf. A101304 (2^(prime(n)+1)+1), A098102 (2^(prime(n)-1)-1), A278741 (2^(prime(n)-1)+1).

Programs

Formula

a(n) = A101304(n) - 2.
a(n) = A000225(A008864(n)). - Felix Fröhlich, Dec 21 2016
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