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.

A259023 Numbers n such that Product_{d|n} d = k^2 for some k > n and simultaneously number k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

24, 54, 56, 88, 154, 174, 238, 248, 266, 296, 328, 374, 378, 430, 442, 472, 488, 494, 498, 510, 568, 582, 584, 680, 710, 730, 742, 786, 856, 874, 894, 918, 962, 986, 1038, 1246, 1270, 1406, 1434, 1442, 1446, 1542, 1558, 1586, 1598
Offset: 1

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Author

Jaroslav Krizek, Sep 01 2015

Keywords

Comments

Product_{d|n} d is the product of divisors of n (A007955).
If 1+ Product_{d|k} d for k > 2 is a prime p, then p-1 is a square.
With number 2 complement of A259021 with respect to A118369.
See A258897 - divisorial primes of the form 1 + Product_{d|a(n)} d.

Examples

			The number 24 is in sequence because A007955(24) = 331776 = 576^2 and simultaneously 331777 is prime.

Links

Crossrefs

Subsequence of A048943 (product of divisors of n is a square) and A118369 (numbers n such that Prod_{d|n} d + 1 is prime).
Cf. A007955, A258455, A259020, A259021, A258897.

Programs

  • Magma
    [n: n in [1..2000] | &*(Divisors(n)) ne n^2 and IsSquare(&*(Divisors(n))) and IsPrime(&*(Divisors(n))+1)];
  • PARI
    A007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2))
is(n)=my(t=A007955(n)); t>n^2 && issquare(t) && isprime(t+1) \\ Charles R Greathouse IV, Sep 01 2015

A259020 Numbers k such that k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

1, 6, 10, 14, 26, 74, 94, 134, 146, 206, 314, 326, 386, 466, 576, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2916, 2974, 2986, 3046, 3106, 3134, 3136, 3214, 3254, 3274, 3314, 3326
Offset: 1

Views

Author

Jaroslav Krizek, Sep 01 2015

Keywords

Comments

The divisorial primes are primes of the form p = 1 + Product_{d|k} d = 1 + A007955(k) for some k.
Supersequence of A259021. Subsequence of A005574. First deviation from A259021 is at a(15).

Examples

			The number 6 is in sequence because prime 37 = 6^2 + 1 is prime of the form p = 1 + Product_{d|k} d = 1 + A007955(k) for k = 6.

Links

Crossrefs

Cf. A005574, A007955, A048943, A118369, A258455, A258897, A259021, A259023.

Programs

  • Magma
    Set(Sort([1] cat [Floor(Sqrt(&*(Divisors(n)))): n in [3..10000] | IsPrime(&*(Divisors(n))+1)]));

A363059 Numbers k such that the number of divisors of k^2 equals the number of divisors of phi(k), where phi is the Euler totient function.

Original entry on oeis.org

1, 5, 57, 74, 202, 292, 394, 514, 652, 1354, 2114, 2125, 3145, 3208, 3395, 3723, 3783, 4053, 4401, 5018, 5225, 5298, 5425, 5770, 6039, 6363, 6795, 6918, 7564, 7667, 7676, 7852, 7964, 8585, 9050, 9154, 10178, 10535, 10802, 10818, 10954, 11223, 12411, 13074, 13634
Offset: 1

Views

Author

Amiram Eldar, May 16 2023

Keywords

Comments

Numbers k such that A048691(k) = A062821(k).
Amroune et al. (2023) characterize solutions to this equation and prove that Dickson's conjecture implies that this sequence is infinite.
They show that the only squarefree semiprime terms are 57, 514 and some of the numbers of the form 2*(4*p^2+1), where p and 4*p^2+1 are both primes (a subsequence of A259021).

Examples

			5 is a term since both 5^2 = 25 and phi(5) = 4 have 3 divisors.

Links

Crossrefs

Cf. A000005, A000010, A048691, A062821.
Cf. A052291, A259021.

Programs

  • Mathematica
    Select[Range[15000], DivisorSigma[0, #^2] == DivisorSigma[0, EulerPhi[#]] &]
  • PARI
    is(n) = numdiv(n^2) == numdiv(eulerphi(n));
Showing 1-3 of 3 results.

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