A063103 Numbers k such that sigma(usigma(k)) is prime.

3, 8, 2667, 3937, 57337, 172011, 253921, 677207307, 1073602561, 732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969, 196751176038481899983340171, 223076247804911695439842851, 262302377656070899470360793, 262336402488441531425882329

Offset

1,1

Comments

a(2) = 8 is the only even term. Each of the odd terms is a product of distinct Mersenne primes (A046528), that have Mersenne exponents (A000043) p_1, p_2, ..., p_m (m >= 1) such that p_1 + p_2 + ... + p_m + 1 is also a Mersenne exponent. - Amiram Eldar, Jan 25 2025

Links

Amiram Eldar, Table of n, a(n) for n = 1..97 (terms below 10^1000)

Example

k = 8: usigma(8) = 9 and sigma(9) = 13, a prime.
k = 2667: usigma(2667) = 4096 and sigma(4096) = 8191, a prime.

Mathematica

us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Do[m = n; If[ PrimeQ[ DivisorSigma[1, us[n]]], Print[n]], {n, 1, 10^7} ]

Prog

(PARI) u(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)); for(n=1, 10^7, if(isprime(sigma(u(n))), print(n)))
(Magma) us:=func<n|&+[d:d in Divisors(n)| Gcd(d, n div d) eq 1]>; [k:k in [1..10^6]| IsPrime(DivisorSigma(1, us(k)))]; // Marius A. Burtea, Feb 13 2020

Crossrefs

Cf. A000203, A034448, A063845.

Cf. A000043, A000668, A046528.

Sequence in context: A354119 A380504 A278974 * A058847 A088110 A122759

Adjacent sequences: A063100 A063101 A063102 * A063104 A063105 A063106

Keyword

nonn

Author

Jason Earls, Aug 07 2001

Extensions

a(8)-a(9) from Donovan Johnson, Jul 16 2012

a(10)-a(13) from Manuel Valdivia, Sep 28 2012

a(14)-a(17) from Amiram Eldar, Jan 25 2025

Status

approved