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.

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

Author

Chai Wah Wu, Jul 20 2016

Keywords

• sign
• hard

Comments

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

Examples

			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.
		

Crossrefs

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

Formula

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

Extensions

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

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