A366917
a(n) = Sum_{k=1..n} (-1)^k*k^3*floor(n/k).
Original entry on oeis.org
-1, 6, -22, 49, -77, 119, -225, 358, -399, 483, -849, 1139, -1059, 1349, -2179, 2500, -2414, 2885, -3975, 4971, -4661, 4663, -7505, 8819, -6932, 8454, -11986, 12438, -11952, 12744, -17048, 20399, -16897, 17501, -25843, 27904, -22750, 25270, -36274, 37184, -31738
Offset: 1
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f:= proc(n) local k; add((-1)^k * k^3 * floor(n/k), k=1..n) end proc;
map(f, [$1..100]); # Robert Israel, Dec 29 2023
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a[n_]:=Sum[ (-1)^k*k^3*Floor[n/k],{k,n}]; Array[a,41] (* Stefano Spezia, Oct 29 2023 *)
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a(n) = sum(k=1, n, (-1)^k*k^3*(n\k)); \\ Michel Marcus, Oct 29 2023
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from math import isqrt
def A366917(n): return (-(t:=isqrt(m:=n>>1))**3*(t+1)**2+sum((q:=m//k)*((k**3<<2)+q*(q*(q+2)+1)) for k in range(1,t+1))<<2)+((s:=isqrt(n))**3*(s+1)**2 - sum((q:=n//k)*((k**3<<2)+q*(q*(q+2)+1)) for k in range(1,s+1))>>2)
A366919
a(n) = Sum_{k=1..n} (-1)^k*k^n*floor(n/k).
Original entry on oeis.org
-1, 2, -22, 203, -2285, 33855, -609345, 12420372, -284964519, 7347342215, -209807114169, 6554034238459, -222469737401739, 8159109186320903, -321461264348047819, 13538455640979049698, -606976994365011212414, 28864017965496692865925, -1451086990386146504580735
Offset: 1
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a[n_]:=Sum[ (-1)^k*k^n*Floor[n/k],{k,n}]; Array[a,19] (* Stefano Spezia, Oct 29 2023 *)
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a(n) = sum(k=1, n, (-1)^k*k^n*(n\k)); \\ Michel Marcus, Oct 29 2023
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from math import isqrt
from sympy import bernoulli
def A366919(n): return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<
A366936
Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^j*j^k*floor(n/j).
Original entry on oeis.org
-1, -1, -1, -1, 0, -3, -1, 2, -4, -2, -1, 6, -8, 1, -4, -1, 14, -22, 11, -5, -4, -1, 30, -68, 49, -15, -1, -6, -1, 62, -214, 203, -77, 15, -9, -4, -1, 126, -668, 841, -423, 119, -35, 4, -7, -1, 254, -2062, 3491, -2285, 807, -225, 48, -9, -7, -1, 510, -6308, 14449
Offset: 1
Array begins:
-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...
-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, ...
-3, -4, -8, -22, -68, -214, -668, -2062, -6308, -19174, ...
-2, 1, 11, 49, 203, 841, 3491, 14449, 59483, 243481, ...
-4, -5, -15, -77, -423, -2285, -12135, -63677, -331143, -1709645, ...
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from math import isqrt
from itertools import count, islice
from sympy import bernoulli
def A366936_T(n,k):
if k:
return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))<>1))**2<<1)+((sum(m//k for k in range(1, t+1))<<1)-sum(n//k for k in range(1, s+1))<<1)
def A366936_gen(): return (A366936_T(k+1,n-k-1) for n in count(1) for k in range(n))
A366936_list = list(islice(A366936_gen(),30))
Showing 1-3 of 3 results.