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Numbers k such that sigma(sigma(2k-1)) is a prime p.
7

%I #24 Sep 08 2022 08:46:09

%S 2,1334,1969,28669,86006,126961,338603654,536801281,

%T 366479720500691270,375344017599431990,500461553802019261,

%U 554079264075351985

%N Numbers k such that sigma(sigma(2k-1)) is a prime p.

%C Numbers n such that A000203(A000203(2n-1)) = A000203(A008438(n-1)) = A051027(2n-1) is a prime p.

%C Corresponding values of primes p are 7, 8191, 8191, 131071, 524287, 524287, ... (= A247822). Conjecture: The primes p are Mersenne primes (A000668).

%C sigma(sigma(2*a(9)-1)) > 10^16.

%C If the above conjecture is true, the next terms are 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985, 98375588019240949991670086, ... . - _Hiroaki Yamanouchi_, Oct 01 2014

%C a(13) > 5*10^18. - _Giovanni Resta_, Feb 14 2020

%F a(n) = (A247838(n) +1) / 2.

%F a(n)-1 = numbers n such that sigma(sigma(2n+1)) is a prime p: 1, 1333, 1968, 28668, 86005, 126960, ...

%e Number 1334 is in sequence because sigma(sigma(2*1334-1)) = sigma(sigma(2667)) = sigma(4096) = 8191, i.e., prime.

%t Select[Range[10^6], PrimeQ[DivisorSigma[1, DivisorSigma[1, 2 # - 1]]] &] (* _Robert Price_, May 17 2019 *)

%o (Magma) [n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1)))]

%o (PARI)

%o for(n=1,10^7,if(ispseudoprime(sigma(sigma(2*n-1))),print1(n,", "))) \\ _Derek Orr_, Sep 29 2014

%Y Cf. A000203, A008438, A247790, A247791, A247820, A247822, A247823, A247954.

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Sep 28 2014

%E a(7)-a(8) from _Hiroaki Yamanouchi_, Oct 01 2014

%E a(9)-a(12) from _Giovanni Resta_, Feb 14 2020