A274962 -id:A274962 - cp's OEIS frontend

A274963 Numbers n such that both sigma(n) and sigma(n) - 2 are primes.

4, 9, 16, 25, 729, 1681, 3481, 7921, 10201, 17161, 552049, 579121, 1423249, 5812921, 7091569, 7447441, 9066121, 9765625, 10374841, 10569001, 11895601, 22572001, 38105929, 43546801, 46689889, 52258441, 75151561, 82065481, 86918329, 90649441, 94458961, 94926049

Offset: 1

Jaroslav Krizek, Jul 12 2016

nonn

Intersection of A249485 and A023194.
The next term, if it exists, must be greater than 10^8.
Each term is a square.
Most of the terms seem to be of the form p^2 for some prime p. Out of the first 10539 terms, 6 of them are not of the form p^2. - Chai Wah Wu, Jul 13 2016

			729 is in the sequence because sigma(729) = 1093 and 1091 are both primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10539

Cf. A000203, A023194, A249485, A274962.

[n: n in[1..10^7] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n)-2)]

from sympy import isprime, divisor_sigma
A274963_list = [n for n, s in ((d**2, divisor_sigma(d**2)) for d in range(1,10**3)) if isprime(s) and isprime(s-2)] # Chai Wah Wu, Jul 13 2016

A275237 Smallest number k > 0 such that sigma(x) and sigma(x)+2 are both prime, where x = (6k+1)^(6n+4), or -1 if no such k exists.

Original entry on oeis.org

1, 348, 436, 6018, 5880, -1, 4612, 26921, 16166, 81111, -1, 426260, -1, 181876, 227180, -1, 12836, 287388, 2317, -1, -1, 1128403, 668927, -1, 5295, -1, -1, 490118, 2217967, 1607226, -1, 1212183, 100728, -1, -1, -1, -1, 1191713, 43475567, 165965, -1, 2915491, 361885, 4159496, 3398061, -1, 88930, -1, 10451327, -1, -1

Offset: 0

Chai Wah Wu, Jul 20 2016

If x is a number such that sigma(x) and sigma(x)+2 are both prime (A274962), then x = 2 or x is of the form (6k+1)^(6r+4) where 6k+1 is prime.
For p = 6*k+1, sigma(p^34) = (46656*k^6 + 54432*k^5 + 27216*k^4 + 7560*k^3 + 1260*k^2 + 126*k + 7) * (1296*k^4 + 1080*k^3 + 360*k^2 + 60*k + 5) * c(k), thus a(5) = -1. - Altug Alkan , Jul 21 2016
Similarly a(12) = a(19) = a(23) = a(26) = a(33) = a(34) = -1. Furthermore, for all r > 0, a(5*r) = -1 since sigma((6k+1)^(30r+4)) = ((6*k+1)^(6*r) + ((6*k+1)^(6*r) -1)/(6*k))*(1296*k^4*(6*k + 1)^(24*r) + 864*k^3*(6*k + 1)^(24*r) + 216*k^3*(6*k + 1)^(18*r) + 216*k^2*(6*k + 1)^(24*r) + 108*k^2*(6*k + 1)^(18*r) + 36*k^2*(6*k + 1)^(12*r) + 24*k*(6*k + 1)^(24*r) + 18*k*(6*k + 1)^(18*r) + 12*k*(6*k + 1)^(12*r) + 6*k*(6*k + 1)^(6*r) + (6*k + 1)^(24*r) + (6*k + 1)^(18*r) + (6*k + 1)^(12*r) + (6*k + 1)^(6*r) + 1). - Chai Wah Wu, Jul 21 2016

			For n = 0, x = 7^4 is the smallest fourth power such that sigma(x) and sigma(x)+2 are both prime, thus a(0) = 1.

Cf. A000203, A023194, A249763, A274962, A274963.

a(A059324(n)) = -1. - Altug Alkan, Aug 13 2016

a(31)-a(37) from Chai Wah Wu, Aug 01 2016

a(38)-a(50) from Chai Wah Wu, Aug 18 2016

cp's OEIS Frontend

A274963 Numbers n such that both sigma(n) and sigma(n) - 2 are primes.

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