A063783 -id:A063783 - cp's OEIS frontend

A066100 Primes p such that p^6 + p^3 + 1 is prime.

2, 3, 11, 191, 269, 383, 509, 809, 827, 887, 1409, 1427, 1787, 1907, 1949, 2141, 2243, 2339, 2357, 2477, 2591, 2699, 2789, 4073, 4517, 4643, 4787, 5171, 5237, 5501, 5531, 5693, 6311, 6329, 6359, 6911, 6947, 7019, 7253, 7349, 7499, 7577, 7691, 7907, 8819

Offset: 1

Labos Elemer, Dec 04 2001

nonn

Original name: "Primes p such that the sum of the cubes of the divisors of p^2 is prime."
Primes p such that sigma_3(p^2) is prime.
It appears that squares of these primes give A063783, those numbers whose sum of cubes of divisors is prime.

			p=11: p^2=121, cubes of divisors of p^2 = {p^6, p^3, 1}, sigma_3(p^2) = p^6 + p^3 + 1 = 1771561 + 1331 + 1 = 1772893 = q, a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Paolo Santonastaso and Ferdinando Zullo, Linearized trinomials with maximum kernel, Journal of Pure and Applied Algebra, Vol. 226, No. 3 (2022), 106842; arXiv preprint, arXiv:2012.14861 [math.NT], 2020-2021.

Cf. A000040, A001158, A063783.

Select[Prime@ Range@ 1200, PrimeQ@ DivisorSigma[3, #^2] &] (* Michael De Vlieger, Jul 16 2017 *)

isok(p) = { isprime(p) && isprime(sigma(p^2, 3)) } \\ Harry J. Smith, Nov 13 2009

a(n) = sqrt(A063783(n)). - Amiram Eldar, Aug 16 2024

Name replaced with simpler description offered in an Oct 10 2010 comment by James R. Buddenhagen by Jon E. Schoenfield, Jul 17 2017

A063784 Primes that are the sum of cubes of divisors of some integer.

Original entry on oeis.org

73, 757, 1772893, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 7824668707707203971, 8443914727229480773, 32564717507686012813, 48095468363380957093, 54811417636756749151

Offset: 1

Labos Elemer, Aug 17 2001

nonn

Primes of the form p^6 + p^3 + 1 where p is a prime. - Amiram Eldar, Aug 16 2024

			sigma_3(9) = 1 + 27 + 729 = 757, a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001158, A023194, A060883, A063783, A066100.

Select[Table[p^6 + p^3 + 1, {p, Prime[Range[500]]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(q=p^6 + p^3 + 1), write("b063784.txt", n++, " ", q); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009

Primes of form p = sigma_3(k).
From Amiram Eldar, Aug 16 2024: (Start)
a(n) = A001158(A063783(n)).
a(n) = A060883(A066100(n)). (End)

A252040 Least number k such that the sum of the n-th powers of divisors of k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 2, 4, 2, 3418801, 0, 64, 2, 4, 0, 21611482313284249, 0, 38580382095460899155325996786287338133521, 0, 0, 2, 2908327322588596409721563830760333292497745890881, 0, 366742604046618684582110328334808145525757868881, 0, 0, 0, 264638735999480827156185738343112840094453729311824181089

Offset: 1

Derek Orr, Dec 12 2014

nonn

a(n) = 2 if and only if 2^n + 1 is in A019434.
From Jinyuan Wang, Jan 30 2021: (Start)
a(n) = 0 if n > 1 is not a prime power. Proof: note that sigma_n(k) = Product_{i=1..m} (1 + p_i^n + ... + p_i^(n*e_i)), where k = Product_{i=1..m} p_i^e_i. We only need to prove when n > 1 is not a prime power and e > 1, s = Sum_{i=0..e-1} p^(n*i) = (p^(n*e) - 1)/(p^n - 1) is composite. If e is prime, then s is divisible by (p^(e^(t+1)) - 1)/(p^(e^t) - 1), where t is the e-adic valuation of n. If e is composite, then s is divisible by (p^(n*q) - 1)/(p^n - 1), where q is a prime factor of e.
Corollary: k must be of the form p^(e - 1) when n = e^t, where p and e are primes. Therefore, a(2^t) = 0 if 2^2^t + 1 is composite. (End)

			2 has two divisors, 2 and 1. 2^3 + 1^3 = 9 is not prime.
3 has two divisors, 3 and 1. 3^3 + 1^3 = 28 is not prime.
4 has three divisors, 4, 2, and 1. 4^3 + 2^3 + 1^3 = 73 is prime. So, a(3) = 4.

Jinyuan Wang, Table of n, a(n) for n = 1..50

Cf. A000203, A001157, A001158, A001159, A001160, A019434.

Cf. A023194 (sigma(n) is prime), A063783 (sigma_3(n) is prime).

a252040[n_Integer] := If[PrimePowerQ[n] && (p=First@ First@ FactorInteger[n])>2, q=2; While[!PrimeQ[DivisorSigma[n, q^(p-1)]], q=NextPrime[q]]; q^(p-1), 2*Boole[PrimeQ[2^n+1]]]; a252040 /@ Range[10] (* Michael De Vlieger, Dec 13 2014 *) (* modified by Jinyuan Wang, Jan 30 2021 *)

a(n) = if(isprimepower(n, &p) && p>2, my(q=2); while(!ispseudoprime(sigma(q^(p-1), n)), q=nextprime(q+1)); q^(p-1), 2*isprime(2^n+1)); \\ Modified by Jinyuan Wang, Jan 25 2021

Name edited by and more terms from Jinyuan Wang, Jan 30 2021

A073807 Number of divisors of sum of cube of divisors.

Original entry on oeis.org

1, 3, 6, 2, 12, 18, 8, 12, 2, 20, 18, 12, 8, 24, 36, 4, 32, 6, 24, 24, 24, 30, 36, 72, 4, 24, 32, 16, 24, 60, 36, 16, 60, 48, 60, 4, 16, 72, 24, 80, 24, 72, 24, 36, 24, 60, 60, 24, 8, 12, 96, 16, 20, 96, 80, 96, 50, 40, 72, 72, 16, 108, 16, 8, 54, 100, 12, 64, 108, 100, 32, 24

Offset: 1

Labos Elemer, Aug 13 2002

nonn

a(n) = 2 for n in A063783. - Robert Israel, Jul 12 2023

			For n=10: D={1,2,5,10}, 1+8+125+1000=1134, divisors(1134)={1,2,3,6,7,9,14,18,21,27,42,54,63,81,126,162,189,378,567,1134} so a(10)=20.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A001158, A062068, A063783, A073802-A073813.

f:= n -> numtheory:-tau(numtheory:-sigma[3](n)):
map(f, [$1..100]); # Robert Israel, Jul 12 2023

Table[DivisorSigma[0,DivisorSigma[3,n]],{n,80}] (* Harvey P. Dale, Dec 13 2011 *)

a(n) = numdiv(sigma(n, 3)); \\ Michel Marcus, Jul 13 2023

a(n) = A000005(A001158(n)).

cp's OEIS Frontend

A066100 Primes p such that p^6 + p^3 + 1 is prime.

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A063784 Primes that are the sum of cubes of divisors of some integer.

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A252040 Least number k such that the sum of the n-th powers of divisors of k is prime, or 0 if there is no such k.

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A073807 Number of divisors of sum of cube of divisors.

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