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

%I A132336 #26 Jun 06 2025 12:23:37
%S A132336 0,2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,
%T A132336 252,275,299,324,350,377,405,434,464,495,495,528,562,597,633,670,708,
%U A132336 747,787,828,870,913,957,1002,1048,1095,1143,1192,1242,1293,1345,1398,1452
%N A132336 Sum of the integers from 1 to n, excluding perfect fifth powers.
%H A132336 T. D. Noe, <a href="/A132336/b132336.txt">Table of n, a(n) for n = 1..10000</a>
%F A132336 a(n) = A000217(n) - A000539(r) where r = floor(n^(1/5)).
%F A132336 a(n) = n(n+1)/2 - (2r^6 + 6r^5 + 5r^4 - r^2)/12.
%F A132336 a(n) = A000217(n) - A000539(r) where r= A178487(n). - _R. J. Mathar_, Oct 12 2010
%e A132336 a(1)=0+1, excluding 0 and 1, so a(1)=0.
%e A132336 a(2)=0+1+2, excluding 0 and 1, so a(2)=2.
%e A132336 a(3)=0+1+2+3, excluding 0 and 1, so a(3)=2+3=5.
%p A132336 A000217 := proc(n) n*(n+1)/2 ; end proc:
%p A132336 A000539 := proc(n) (2*n^6+6*n^5+5*n^4-n^2)/12 ; end proc:
%p A132336 A132336 := proc(n) r := floor(n^(1/5)) ; A000217(n)-A000539(r); end proc: seq(A132336(n),n=1..40) ;
%o A132336 (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, ", "))
%o A132336 (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
%o A132336 (Python)
%o A132336 from sympy import integer_nthroot
%o A132336 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
%Y A132336 Cf. A000217, A000539, A178487.
%Y A132336 Different from A000096.
%Y A132336 Cf. A132337.
%K A132336 nonn,easy
%O A132336 1,2
%A A132336 _Cino Hilliard_, Nov 07 2007
%E A132336 Edited by the Assoc. Editors of the OEIS, Oct 12 2010. Thanks to _Daniel Mondot_ for pointing out that the sequence needed editing.