A008332 -id:A008332 - cp's OEIS frontend

A164318 Primes p such that the sum of divisors of p-1 is larger than 2*p.

13, 19, 31, 37, 41, 43, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 139, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 271, 277, 281, 283, 307, 313, 331, 337, 349, 353, 367, 373, 379, 397, 401, 409, 421, 433, 439, 449, 457, 461, 463, 487, 491

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

			p=13 is in the sequence because A000203(12) = A008332(6) = 28 > 2*p.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A008332, A140557.

Select[Prime[Range[100]],DivisorSigma[1,#-1]>2#&]  (* Harvey P. Dale, Mar 31 2011 *)

Edited by R. J. Mathar, Aug 21 2009

A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

Original entry on oeis.org

-1, 1, 5, 8, 14, 22, 25, 31, 28, 48, 56, 73, 78, 76, 56, 80, 74, 138, 112, 120, 159, 136, 102, 156, 210, 185, 168, 126, 240, 212, 248, 212, 226, 240, 226, 300, 314, 283, 204, 252, 222, 474, 296, 412, 339, 388, 472, 360, 270, 472, 378, 368, 634, 396, 427, 316, 404, 592, 534, 628, 436, 434, 582, 480, 684, 456, 700, 836

Offset: 1

Labos Elemer, Mar 03 2004

The sequence differs from A065394 since it is not monotonic.

			a(1) = sigma(phi(2))- phi(sigma(2)) = sigma(1)-phi(3) = 1-2 = -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000203, A033632, A065393, A065394, A065395, A008331, A008332, A092584, A092585, A092586, A092587, A092588, A092589.

[DivisorSigma(1,EulerPhi(p))-EulerPhi(DivisorSigma(1,p)): p in PrimesUpTo(400)]; // Bruno Berselli, Oct 20 2015

Table[DivisorSigma[1, p-1] - EulerPhi[p+1], {p, Prime[Range[100]]}] (* Amiram Eldar, Jun 09 2024 *)

a(n) = sigma(prime(n)-1) - phi(prime(n)+1) = A008332(n) - A008331(n). - Amiram Eldar, Jun 09 2024

A174843 Irregular triangle in which row n lists the divisors of prime(n)-1.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 6, 1, 2, 5, 10, 1, 2, 3, 4, 6, 12, 1, 2, 4, 8, 16, 1, 2, 3, 6, 9, 18, 1, 2, 11, 22, 1, 2, 4, 7, 14, 28, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 4, 6, 9, 12, 18, 36, 1, 2, 4, 5, 8, 10, 20, 40, 1, 2, 3, 6, 7, 14, 21, 42, 1, 2, 23, 46, 1, 2, 4, 13, 26, 52, 1, 2, 29, 58, 1, 2, 3, 4

Offset: 1

T. D. Noe, Mar 30 2010

Row n begins with 1, ends with prime(n)-1, and has A008328(n) terms.

			The first 10 rows:
  1
  1, 2
  1, 2, 4
  1, 2, 3, 6
  1, 2, 5, 10
  1, 2, 3, 4, 6, 12
  1, 2, 4, 8, 16
  1, 2, 3, 6, 9, 18
  1, 2, 11, 22
  1, 2, 4, 7, 14, 28

T. D. Noe, Rows n=1..500 of triangle, flattened

Cf. A008328 (row lengths), A008332 (row sums).

Flatten[Table[Divisors[p-1], {p, Prime[Range[100]]}]]

row(n) = divisors(prime(n)-1); \\ Amiram Eldar, May 02 2025

A067758 Numbers k such that sigma(prime(k) - 1) == 0 (mod k).

Original entry on oeis.org

1, 4, 9, 27, 42, 60, 64, 70, 78, 144, 168, 180, 216, 238, 260, 318, 558, 600, 672, 960, 1008, 1053, 1260, 1620, 1806, 2112, 3318, 3780, 4608, 5544, 12152, 40084, 40095, 41664, 47040, 48825, 49176, 51870, 59832, 60528, 71040, 99008, 100356, 113904, 132000, 159000

Offset: 1

Benoit Cloitre, Feb 05 2002

nonn

There are 91 terms up to and including 10 million. - Harvey P. Dale, Oct 04 2016

Amiram Eldar, Table of n, a(n) for n = 1..163 (terms 1..91 from Harvey P. Dale)

Cf. A000203, A006093, A008332.

Select[Range[120000],Divisible[DivisorSigma[1,Prime[#]-1],#]&] (* Harvey P. Dale, Oct 04 2016 *)

for(k=1,120000, if(sigma(prime(k)-1)%k==0,print1(k,",")))

More terms from Klaus Brockhaus, Feb 07 2002

A326391 Lesser of twin primes p >= 3 for which sigma(p+1)/sigma(p-1) reaches record value, where sigma(n) is the divisor sum function (A000203).

Original entry on oeis.org

3, 7559, 42839, 55439, 110879, 415799, 1713599, 1940399, 2489759, 6652799, 6846839, 15855839, 31600799, 85765679, 232792559, 845404559, 1470268799, 6299092799, 10708457759, 17459441999, 32125373279, 135019684799, 439977938399, 449755225919, 1799020903679, 2126560035599, 2835413380799, 6278415343199

Offset: 1

Amiram Eldar, Sep 11 2019

nonn

Garcia et al. proved that assuming Dickson's conjecture, {sigma(p+1)/sigma(p-1) : p and p+2 are prime} is dense in [0, oo), and thus this sequence is infinite.

			The values of sigma(p+1)/sigma(p-1) for the first terms are 2.333... < 2.539... < 2.621... < 2.734... < 2.836...

Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe Udell, Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function, arXiv:1906.05927 [math.NT], 2019.
Wikipedia, Dickson's conjecture.

Cf. A000203, A001359, A006512, A008332, A008333, A326356.

s = {}; rm = 0; p = 2; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; r = DivisorSigma[1, p + 1]/DivisorSigma[1, p - 1]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^6}]; s

a(22)-a(28) from Giovanni Resta, Nov 01 2019

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A164318 Primes p such that the sum of divisors of p-1 is larger than 2*p.

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A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

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A174843 Irregular triangle in which row n lists the divisors of prime(n)-1.

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A067758 Numbers k such that sigma(prime(k) - 1) == 0 (mod k).

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A326391 Lesser of twin primes p >= 3 for which sigma(p+1)/sigma(p-1) reaches record value, where sigma(n) is the divisor sum function (A000203).

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