A132336 Sum of the integers from 1 to n, excluding perfect fifth powers.

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 495, 528, 562, 597, 633, 670, 708, 747, 787, 828, 870, 913, 957, 1002, 1048, 1095, 1143, 1192, 1242, 1293, 1345, 1398, 1452

Offset: 1

Author

Cino Hilliard, Nov 07 2007

Keywords

• nonn
• easy

Examples

			a(1)=0+1, excluding 0 and 1, so a(1)=0.
a(2)=0+1+2, excluding 0 and 1, so a(2)=2.
a(3)=0+1+2+3, excluding 0 and 1, so a(3)=2+3=5.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A000539, A178487.

Different from A000096.

Cf. A132337.

Programs

Maple

A000217 := proc(n) n*(n+1)/2 ; end proc:
A000539 := proc(n) (2*n^6+6*n^5+5*n^4-n^2)/12 ; end proc:
A132336 := proc(n) r := floor(n^(1/5)) ; A000217(n)-A000539(r); end proc: seq(A132336(n),n=1..40) ;

PARI

g5(n)=for(x=1, n, r=floor(x^(1/5)); sum5=(2*r^6+6*r^5+5*r^4-r^2)/12; sn=x* (x+1)/2; print1(sn-sum5, ", "))

PARI

a(n) = my(r=sqrtnint(n,5)); n*(n+1)/2 - (2*r^6+6*r^5+5*r^4-r^2)/12; \\ Ruud H.G. van Tol, Nov 02 2023

Python

from sympy import integer_nthroot
def A132336(n): return n*(n+1)-(m:=integer_nthroot(n,5)[0])**2*(m**2*(m*(m+3<<1)+5)-1)//6>>1 # Chai Wah Wu, Jun 06 2025

Formula

a(n) = A000217(n) - A000539(r) where r = floor(n^(1/5)).

a(n) = n(n+1)/2 - (2r^6 + 6r^5 + 5r^4 - r^2)/12.

a(n) = A000217(n) - A000539(r) where r= A178487(n). - R. J. Mathar, Oct 12 2010

Extensions

Edited by the Assoc. Editors of the OEIS, Oct 12 2010. Thanks to Daniel Mondot for pointing out that the sequence needed editing.