A066075 Number of solutions x to prime(n) = sigma(x) - 1, where prime(n) is the n-th prime.

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, 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, 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

Offset: 1

Author

Labos Elemer, Dec 03 2001

Keywords

• nonn

Comments

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

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

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

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

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

Programs

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

PARI

a(n) = invsigmaNum(prime(n)+1); \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

Formula

a(n) = A054973(prime(n)+1). - Amiram Eldar, Dec 16 2024