A339354 -id:A339354 - cp's OEIS frontend

A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

0, 1, 1, 1, 1, 5, 1, 5, 1, 5, 1, 14, 1, 5, 10, 5, 1, 14, 1, 21, 10, 5, 1, 30, 1, 5, 10, 21, 1, 39, 1, 21, 10, 5, 26, 30, 1, 5, 10, 46, 1, 50, 1, 21, 35, 5, 1, 66, 1, 30, 10, 21, 1, 50, 26, 70, 10, 5, 1, 91, 1, 5, 59, 21, 26, 50, 1, 21, 10, 79, 1, 130, 1, 5, 35, 21, 50, 50, 1, 110

Offset: 1

Ilya Gutkovskiy, Dec 01 2020

nonn

Sum of squares of divisors of n that are smaller than sqrt(n).

Index entries for sequences related to sums of divisors

Cf. A001157, A056924, A067558, A070039, A095118, A339354.

nmax = 80; CoefficientList[Series[Sum[k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^2 &, # < Sqrt[n] &], {n, 80}]

a(n) = sumdiv(n, d, if (d^2 < n, d^2)); \\ Michel Marcus, Dec 02 2020

A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 17, 1, 17, 1, 17, 1, 98, 1, 17, 82, 17, 1, 98, 1, 273, 82, 17, 1, 354, 1, 17, 82, 273, 1, 723, 1, 273, 82, 17, 626, 354, 1, 17, 82, 898, 1, 1394, 1, 273, 707, 17, 1, 1650, 1, 642, 82, 273, 1, 1394, 626, 2674, 82, 17, 1, 2275, 1, 17, 2483, 273, 626

Offset: 1

Ilya Gutkovskiy, Aug 19 2021

nonn

Cf. A001159, A056924, A070039, A279363, A339353, A339354, A347143.

Table[DivisorSum[n, #^4 &, # < Sqrt[n] &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[k^4 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A347142(n) = { my(s=0); fordiv(n,d,if((d^2)>=n,return(s)); s += (d^4)); }; \\ Antti Karttunen, Aug 19 2021

G.f.: Sum_{k>=1} k^4 * x^(k*(k + 1)) / (1 - x^k).

A347162 Sum of cubes of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 153, 1, 1, 28, 1, 126, 28, 1, 1, 28, 126, 1, 28, 1, 1, 153, 1, 1, 28, 1, 126, 28, 1, 1, 28, 126, 344, 28, 1, 1, 153, 1, 1, 371, 1, 126, 28, 1, 1, 28, 469, 1, 28, 1, 1, 153

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001158, A051000, A333805, A333807, A339354, A347161.

Table[DivisorSum[n, #^3 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(2 k - 1)^3 x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
scod[n_]:=Total[Select[Divisors[n],#Harvey P. Dale, Jan 07 2022 *)

a(n) = my(r=sqrt(n)); sumdiv(n, d, if ((d%2) && (dMichel Marcus, Aug 21 2021

G.f.: Sum_{k>=1} (2*k - 1)^3 * x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

A347157 Sum of cubes of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 0, 8, 0, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 160, 0, 8, 27, 8, 125, 35, 0, 8, 27, 133, 0, 35, 0, 8, 152, 8, 0, 35, 0, 133, 27, 8, 0, 35, 125, 351, 27, 8, 0, 160, 0, 8, 370, 8, 125, 35, 0, 8, 27, 476, 0, 35, 0, 8, 152

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001158, A005064, A005067, A333806, A333808, A339354, A347156, A347158.

Table[DivisorSum[n, #^3 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[Prime[k]^3 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^3 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

cp's OEIS Frontend

A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

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A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

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A347162 Sum of cubes of odd divisors of n that are < sqrt(n).

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A347157 Sum of cubes of distinct prime divisors of n that are < sqrt(n).

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