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.

A164319 Primes p such that the sum of divisors of p+1 is larger than 2*p.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 167, 173, 179, 191, 197, 199, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 347, 349, 353, 359, 367, 379, 383, 389, 401, 419
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

Keywords

Comments

For a subset of these, namely p=179, 239, 359, 419, etc, sigma(p+1) is even larger than 3*p.

Examples

			For p=3, the sum of divisors of p+1 is A000203(4)=7 > 2*3, so p=3 is in the sequence.

Links

Crossrefs

Cf. A008333, A085498, A164318.

Programs

  • Mathematica
    f[n_]:=Plus@@Divisors[n]; lst={};Do[p=Prime[n];If[f[p+1]>2*p,AppendTo[lst, p]],{n,6!}];lst
Select[Prime[Range[100]], DivisorSigma[1, # + 1] > 2 # &] (* G. C. Greubel, Sep 13 2017 *)
  • PARI
    lista(nn) = forprime(p=2, nn, if (sigma(p+1) > 2*p, print1(p, ", "))); \\ Michel Marcus, Sep 13 2017

Extensions

Edited by R. J. Mathar, Aug 21 2009

A349762 Numbers k such that phi(k) = A000010(k) is an abundant number (A005101) and d(k) = A000005(k) is a deficient number (A005100).

Original entry on oeis.org

13, 19, 21, 25, 26, 27, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 49, 54, 55, 56, 57, 61, 62, 65, 66, 67, 70, 71, 73, 74, 77, 78, 79, 81, 82, 86, 87, 88, 89, 91, 93, 95, 97, 100, 101, 103, 104, 105, 109, 110, 111, 112, 113, 114, 115, 119, 122, 123, 125, 127, 129
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 3^(p^2-1) where p is a prime.

Examples

			13 is a term since phi(13) = 12 is an abundant number, sigma(12) = 28 > 2*12 = 24, and d(13) = 2 is a deficient number, sigma(2) = 3 < 2*2 = 4.

Links

Crossrefs

Cf. A000005, A000010, A005100, A005101, A349758, A349759, A349761.
A164318 is a subsequence.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := abQ[EulerPhi[n]] && defQ[DivisorSigma[0, n]]; Select[Range[150], q]

A164320 Primes p such that sums of divisors of the two adjacent integers are each > 2*p.

Original entry on oeis.org

19, 31, 41, 71, 79, 89, 101, 103, 113, 127, 139, 197, 199, 223, 271, 281, 307, 349, 353, 367, 379, 401, 439, 449, 461, 463, 491, 499, 521, 571, 607, 617, 619, 641, 643, 701, 727, 739, 761, 769, 811, 821, 859, 881, 911, 919, 929, 941, 953, 967, 991, 1039, 1061
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

Keywords

Examples

			For p=19, the sum of the divisors of 18 is A000203(18)=39 > 2*19, and the sum of the divisors
of 20 is A000203(20)= 42 > 2*19, so p=19 is in the sequence.

Links

Crossrefs

Cf. A164318, A164319.

Programs

  • Mathematica
    f[n_]:=Plus@@Divisors[n]; lst={};Do[p=Prime[n];If[f[p-1]>2*p&&f[p+1]> 2*p,AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[200]],DivisorSigma[1,#-1]>2#&&DivisorSigma[ 1,#+1]>2#&] (* Harvey P. Dale, Nov 10 2011 *)

Formula

Intersection of A164318 and A164319.

Extensions

References to unrelated sequences removed by R. J. Mathar, Aug 21 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.