Search: a356043 -id:a356043
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A356045
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Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} sigma_k(j) * floor(n/j).
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+10
5
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1, 1, 4, 1, 5, 7, 1, 7, 10, 13, 1, 11, 18, 21, 16, 1, 19, 40, 45, 28, 25, 1, 35, 102, 123, 72, 48, 28, 1, 67, 280, 393, 250, 138, 57, 38, 1, 131, 798, 1371, 1020, 540, 189, 83, 44, 1, 259, 2320, 5025, 4498, 2514, 885, 301, 101, 53, 1, 515, 6822, 18963, 20652, 12828, 4917, 1553, 403, 129, 56
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. of column k: (1/(1-x)) * Sum_{j>=1} sigma_k(j) * x^j/(1 - x^j).
T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k * tau(j/d).
T(n,k) = Sum_{j=1..n} Sum_{d|j} sigma_k(d).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
4, 5, 7, 11, 19, 35, ...
7, 10, 18, 40, 102, 280, ...
13, 21, 45, 123, 393, 1371, ...
16, 28, 72, 250, 1020, 4498, ...
25, 48, 138, 540, 2514, 12828, ...
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PROG
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(PARI) T(n, k) = sum(j=1, n, sigma(j, k)*(n\j));
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, d^k*numdiv(j/d)));
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, sigma(d, k)));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A356126
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a(n) = Sum_{k=1..n} k * sigma_3(k).
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+10
2
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1, 19, 103, 395, 1025, 2537, 4945, 9625, 16438, 27778, 42430, 66958, 95532, 138876, 191796, 266692, 350230, 472864, 603204, 787164, 989436, 1253172, 1533036, 1926156, 2319931, 2834263, 3386143, 4089279, 4796589, 5749149, 6672701, 7871069, 9101837, 10605521
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} k^4 * binomial(floor(n/k)+1,2).
G.f.: (1/(1-x)) * Sum_{k>=1} k^4 * x^k/(1 - x^k)^2.
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MATHEMATICA
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a[n_] := Sum[k * DivisorSigma[3, k], {k, 1, n}]; Array[a, 34] (* Amiram Eldar, Jul 28 2022 *)
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PROG
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(PARI) a(n) = sum(k=1, n, k*sigma(k, 3));
(PARI) a(n) = sum(k=1, n, k^4*binomial(n\k+1, 2));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k/(1-x^k)^2)/(1-x))
(Python)
from math import isqrt
def A356126(n): return ((-(s:=isqrt(n))**2*(s+1)**2*((s<<1)+1)*(s*(3*(s+1))-1)>>1)+sum(k*(q:=n//k)*(q+1)*(15*k**3+((q<<1)+1)*(q*(3*(q+1))-1)) for k in range(1, s+1)))//30 # Chai Wah Wu, Oct 24 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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