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A218277
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Convolution of level 3 of the divisor function.
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4
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0, 0, 0, 1, 3, 4, 10, 15, 24, 33, 45, 65, 77, 102, 143, 155, 180, 268, 255, 315, 434, 435, 462, 695, 593, 735, 960, 918, 945, 1437, 1160, 1395, 1825, 1692, 1668, 2549, 1995, 2385, 3073, 2775, 2730, 4190, 3157, 3747, 4739, 4290, 4140, 6355, 4686, 5523, 7044
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OFFSET
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1,5
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COMMENTS
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Named W3(n) by S. Alaca and K. S. Williams.
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LINKS
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FORMULA
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a(n) = Sum_{m<3n} sigma(m)*sigma(n-3*m).
a(n) = sigma3(n)/24 - n*sigma(n)/12 + sigma(n)/24 + 3*sigma3(n/3)/8 - n*sigma(n/3)/4 + sigma(n/3)/24.
a(n) = (1/72)*(31*sigma_3(n) - sigma_3(3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)). - Ridouane Oudra, Mar 21 2021
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MAPLE
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f:= n -> add(numtheory:-sigma(m)*numtheory:-sigma(n-3*m), m=1..floor((n-1)/3)):
with(numtheory): seq((1/72)*(31*sigma[3](n) - sigma[3](3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)), n=1..50); # Ridouane Oudra, Mar 21 2021
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MATHEMATICA
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a[n_] := Sum[DivisorSigma[1, m] DivisorSigma[1, n-3m], {m, 1, (n-1)/3}];
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PROG
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(PARI) lista(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/3), sigma(m)*sigma(i-3*m)); print1(s , ", "); ); }
(PARI) lista(n) = {for (i=1, n, v = sigma(i, 3)/24 - i*sigma(i)/12 + sigma(i)/24; if (i%3 == 0, v += 3*sigma(i/3, 3)/8 - i*sigma(i/3)/4 + sigma(i/3)/24); print1(v , ", "); ); }
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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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