A347143 -id:A347143 - cp's OEIS frontend

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

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).

A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 1, 626, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 2402, 626, 82, 1, 1, 82, 626, 2402, 82, 1, 1, 707, 1, 1, 2483, 1, 626, 82, 1, 1, 82, 3027, 1, 82, 1, 1, 707

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

			a(18) = 82 as the odd divisors of 18 are the divisors of 9 which are 1, 3 and 9. Of those, 1 and 3 are <= sqrt(18) so we find the sum of fourth powers of 1 and 3 then add them i.e., a(18) = 1^4 + 3^4 = 82. - _David A. Corneth_, Feb 24 2024

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001159, A051001, A069288, A069289, A347143, A347172, A347173, A347174.

Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(2 k - 1)^4 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = {
	my(s = sqrtint(n), res);
	n>>=valuation(n, 2);
	d = divisors(n);
	for(i = 1, #d,
		if(d[i] <= s,
			res += d[i]^4
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

G.f.: Sum_{k>=1} (2*k - 1)^4 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).

A347160 Sum of 4th powers of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 16, 0, 16, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 625, 16, 81, 16, 0, 722, 0, 16, 81, 16, 625, 97, 0, 16, 81, 641, 0, 97, 0, 16, 706, 16, 0, 97, 2401, 641, 81, 16, 0, 97, 625, 2417, 81, 16, 0, 722, 0, 16, 2482, 16, 625, 97, 0, 16, 81, 3042

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001159, A005065, A005068, A063962, A097974, A098002, A347143, A347158, A347159.

Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && PrimeQ[#] &], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k]^4 x^(Prime[k]^2)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^4 * x^(prime(k)^2) / (1 - x^prime(k)).

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

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A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).

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A347160 Sum of 4th powers of distinct prime divisors of n that are <= sqrt(n).

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