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.

A247838 Numbers k such that sigma(sigma(k)) is prime.

Original entry on oeis.org

3, 2667, 3937, 57337, 172011, 253921, 677207307, 1073602561, 732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969
Offset: 1

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Author

Jaroslav Krizek, Sep 28 2014

Keywords

Comments

Numbers k such that A051027(k) is a prime p.
Prime 3 is the only prime p such that sigma(sigma(p)) is a prime q.
Conjecture: Subsequence of A046528 (numbers that are a product of distinct Mersenne primes).
Corresponding values of primes p: 7, 8191, 8191, 131071, 524287, 524287, ... (A247822). Conjecture: values of primes p is equal to Mersenne primes (A000668).
732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969 and 196751176038481899983340171 are terms. - Jaroslav Krizek, Mar 25 2015
a(9) > 10^10. - Michel Marcus, Feb 13 2020
a(13) > 10^19. - Giovanni Resta, Feb 14 2020

Examples

			2667 is a term because sigma(sigma(2667)) = sigma(4096) = 8191 (i.e., prime).

Crossrefs

Cf. A000203, A023194, A063103, A000668, A046528, A051027, A247821, A247822, A247954.

Programs

  • Magma
    [n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(n)))];
  • Maple
    with(numtheory): A247838:=n->`if`(isprime(sigma(sigma(n))),n,NULL): seq(A247838(n), n=1..10^5); # Wesley Ivan Hurt, Oct 02 2014
  • Mathematica
    Select[Range[260000],PrimeQ[DivisorSigma[1,DivisorSigma[1,#]]]&] (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Jan 18 2024 *)
  • PARI
    isok(n) = isprime(sigma(sigma(n))); \\ Michel Marcus, Oct 01 2014

Formula

a(n) = 2*A247821(n)-1.

Extensions

a(7)-a(8) from Michel Marcus, Oct 02 2014
a(9)-a(12) from Giovanni Resta, Feb 14 2020

A063836 Numbers k such that usigma(sigma(k)) is prime.

Original entry on oeis.org

3, 217, 57337
Offset: 1

Views

Author

Robert G. Wilson v, Aug 21 2001

Keywords

Comments

Also, numbers k such that sigma(k) + 1 is a Fermat prime (A019434). Equivalently, numbers k that are a product of distinct Mersenne primes (A000668), k = (2^p_1 - 1) * (2^p_2 - 1) * ... * (2^p_m - 1), p_i are in A000043 and m >= 1, such that p_1 + p_2 + ... + p_m = 2^s and 2^(2^s) + 1 is prime. - Amiram Eldar, Jan 25 2025

Crossrefs

Cf. A000043, A000203, A000668, A019434, A034448, A061765, A063103.
Subsequence of A046528.

Programs

  • Mathematica
    us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Do[m = n; If[ PrimeQ[ us[ DivisorSigma[1, n]]], Print[n]], {n, 1, 10^7/4} ]
  • PARI
    usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
isok(k) = isprime(usigma(sigma(k))); \\ Amiram Eldar, Jan 25 2025

A175906 Numbers n of the form 2^(A000043-1)*A046528 such that sigma(n) is a perfect number.

Original entry on oeis.org

12, 10924032, 16125952, 3757637632, 45091651584, 66563866624, 727145809044307968, 1152771972099211264, 845044701535107443245558061611352064
Offset: 1

Views

Author

Manuel Valdivia, Oct 12 2010

Keywords

Comments

sigma(sigma(A046528(2,12,13,18,21,22,56,57,175,176,177,..))) is Mersenne prime.

Examples

			45091651584=2^18*3*7*8191, sigma(45091651584)=137438691328 is perfect number.

Crossrefs

Cf. A000396, A000203, A000668, A000043, A046528, A063103.

Formula

a(n)= 2^( A000043(2,5,5,6,7,7,8,8,9,9,9,..)-1)*A046528(2,12,13,18,21,22,56,57,175,176,177,..).
Showing 1-3 of 3 results.

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