A163831 a(n) is the n-th composite minus the sum of the indices of the primes in its prime factorization.

2, 3, 5, 5, 6, 8, 9, 10, 12, 13, 15, 15, 16, 19, 19, 19, 21, 22, 24, 27, 26, 26, 28, 30, 29, 31, 34, 35, 37, 38, 36, 42, 41, 43, 42, 44, 47, 47, 49, 47, 47, 53, 50, 55, 58, 56, 58, 59, 58, 62, 65, 61, 67, 66, 68, 69, 73, 73, 68, 76, 75, 71, 75, 80, 82, 81, 81, 80, 78, 84, 89, 89, 90, 92

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 05 2009

Keywords

• nonn

Comments

For the n-th composite c(n) = A002808(n), determine its canonical factorization c(n) = Product_{j} prime(j)^e_j; then A163515(n) = Sum_{j} e_j*j; a(n) = c(n) - A163515(n).

Examples

			A002808(1) = 4 = prime(1)*prime(1), so a(1) = 4 - (1+1) = 2.
A002808(2) = 6 = prime(1)*prime(2), so a(2) = 6 - (1+2) = 3.
A002808(3) = 8 = prime(1)*prime(1)*prime(1), so a(3) = 8 - (1+1+1) = 5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002808, A163515, A163829.

Programs

Maple

A002808 := proc(n) local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; end if; end do; end if; end proc:
A163515 := proc(n) local c; c := A002808(n) ; pfs := ifactors(c)[2] ; add( op(2,p)*numtheory[pi](op(1,p)), p=pfs) ; end:
A163831 := proc(n) A002808(n)-A163515(n) ; end: seq(A163831(n),n=1..100) ; # R. J. Mathar, Aug 05 2009

Mathematica

cmsi[n_]:=n-Total[(PrimePi/@(Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]))]; cmsi/@Select[Range[100],CompositeQ] (* Harvey P. Dale, Dec 15 2021 *)

Formula

a(n) = A002808(n) - A163515(n).

Extensions

Edited and corrected by R. J. Mathar, Aug 05 2009