A163636 The sum of all odd numbers from 2n-1 up to the n-th odd nonprime.

1, 24, 60, 112, 153, 171, 253, 275, 336, 448, 525, 555, 640, 672, 828, 864, 969, 1155, 1197, 1320, 1449, 1495, 1632, 1680, 1728, 1875, 2133, 2407, 2580, 2640, 2700, 2760, 2820, 2880, 3069, 3264, 3328, 3672, 3740, 3808, 3876, 4248, 4320, 4551, 4625, 4864

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 02 2009

Keywords

• nonn
• easy

Examples

			a(1)=1. a(2)=3+5+7+9=24. a(3)=5+7+9+11+13+15=60.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005408, A014076.

Programs

Maple

A014076 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then RETURN(a) ; fi; od: fi; end:
A163636 := proc(n) local onpr; onpr := A014076(n) ; (onpr+2*n-1)*(onpr-2*n+3)/4; end: seq(A163636(n),n=1..80) ; # R. J. Mathar, Aug 08 2009

Mathematica

A014076 := Select[Range[1, 10299, 2], PrimeOmega[#] != 1 &]; Table[(A014076[[n]] + 2*n - 1)*(A014076[[n]] - 2*n + 3)/4, {n, 1, 50}] (* G. C. Greubel, Jul 31 2017 *)
Module[{nn=201,onp},onp=Select[Range[1,nn,2],!PrimeQ[#]&];Table[Total[ Range[ 2n-1,onp[[n]],2]],{n,Length[onp]}]] (* Harvey P. Dale, Jul 03 2020 *)

Python

from sympy import primepi
def A163636(n):
    if n == 1: return 1
    m, k, n2 = n-1, primepi(n) + n - 1 + (n>>1), (n<<1)-1
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return (lambda x: (x+n2)*(x-n2+2)>>2)(m) # Chai Wah Wu, Jul 31 2024

Formula

a(n) = A005408(n-1)+A005408(n)+...+A014076(n);

a(n) = ( A014076(n)+2*n-1 ) *( A014076(n)-2*n+3 )/4.

Extensions

Edited and a(21) corrected by R. J. Mathar, Aug 08 2009