A141274 Composite numbers k such that k+1 is prime and the index in the composites is also prime.

12, 28, 36, 60, 100, 130, 138, 150, 172, 180, 198, 222, 270, 292, 348, 396, 420, 432, 456, 520, 540, 568, 612, 618, 640, 682, 732, 768, 786, 820, 838, 880, 928, 996, 1032, 1060, 1068, 1090, 1096, 1212, 1222, 1276, 1300, 1306, 1320, 1372, 1428, 1486, 1548

Offset: 1

Author

Enoch Haga, Jun 21 2008

Keywords

• easy
• nonn

Comments

Suggested by Carlos Rivera's Puzzle 449.

Examples

			a(1) = 12 because 12+1 = 13, prime and the index of 12 is 5, so both the composite 12+1 and the index are prime.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Carlos Rivera, Puzzle 449. Count N composites after p, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A141275.

Programs

Mathematica

Select[Range[4,1548],!PrimeQ[#]&&PrimeQ[#+1]&&PrimeQ[#-PrimePi[#]-2]&] (* James C. McMahon, Jul 19 2025 *)

Python

from sympy import compositepi, isprime
def ok(n):
    return not isprime(n) and isprime(n+1) and isprime(compositepi(n)-1)
print([k for k in range(1549) if ok(k)]) # Michael S. Branicky, Dec 28 2021

Formula

Compute the composite numbers beginning with 4. At 4 the composite index is zero, at 6 it is 1, at 8 it is 2, ... and at 12 it is 5, etc.

Extensions

Edited by N. J. A. Sloane, Jun 21 2008