%I #23 Dec 16 2024 01:57:11

%S 1,1,1,1,2,1,2,1,3,1,2,1,3,1,3,2,3,1,1,5,1,2,3,3,2,1,2,2,1,2,2,2,1,2,

%T 1,2,1,1,6,1,4,2,5,1,1,1,1,3,3,1,3,7,1,6,1,2,3,2,1,1,1,3,2,4,1,1,1,1,

%U 1,1,1,9,1,1,1,6,2,1,1,1,4,1,8,4,2,2,3,1,1,1,3,9,1,2,1,10,1,2,1,1

%N Number of solutions x to prime(n) = sigma(x) - 1, where prime(n) is the n-th prime.

%C prime(n) itself is always the largest solution, but often composite solutions also occur.

%C If a(n) = 1, then the single solution is prime(n).

%H Amiram Eldar, <a href="/A066075/b066075.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%F a(n) = A054973(prime(n)+1). - _Amiram Eldar_, Dec 16 2024

%e For n = 96, prime(96) = 503, 503 = sigma(x)-1 has 10 solutions together with 503: {204, 220, 224, 246, 284, 286, 334, 415, 451, 503}, so a(96) = 10.

%o (PARI) { for (n=1, 1000, a=1; for (x=1, prime(n) - 1, if (prime(n) == (sigma(x) - 1), a++)); write("b066075.txt", n, " ", a) ) } \\ _Harry J. Smith_, Nov 10 2009

%o (PARI) a(n) = invsigmaNum(prime(n)+1); \\ _Amiram Eldar_, Dec 16 2024, using _Max Alekseyev_'s invphi.gp

%Y Number of solutions to A000040(n) = A000203(x) - 1.

%Y Cf. A000040, A000203, A054973, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

%K nonn

%O 1,5

%A _Labos Elemer_, Dec 03 2001