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

1,1

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).

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A002808, A163515, A163829.

Sequence in context: A363156 A214881 A364085 * A326399 A192419 A376757

Adjacent sequences: A163828 A163829 A163830 * A163832 A163833 A163834

Keyword

nonn

Author

Juri-Stepan Gerasimov, Aug 05 2009

Extensions

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

Status

approved