A158275 -id:A158275 - cp's OEIS frontend

A158274 Numerators of antiharmonic means of divisors of n.

1, 5, 5, 3, 13, 25, 25, 17, 7, 65, 61, 15, 85, 125, 65, 11, 145, 35, 181, 13, 125, 305, 265, 85, 21, 425, 41, 75, 421, 325, 481, 65, 305, 725, 325, 21, 685, 181, 425, 221, 841, 625, 925, 61, 91, 1325, 1105, 55, 43, 35

Offset: 1

Jaroslav Krizek, Mar 15 2009

Numbers k such that sigma_2(k)/sigma_1(k) = A001157(k)/A000203(k) are integers are in A020487.

			Antiharmonic means of divisors of n>=1: 1, 5/3, 5/2, 3, 13/2, 25/6, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A020487, A065473, A152649, A158275 (denominators)

Table[Numerator[DivisorSigma[2, n]/DivisorSigma[1, n]], {n, 50}] (* Ivan Neretin, May 22 2015 *)

a(n) = numerator(sigma(n,2)/sigma(n)); \\ Amiram Eldar, Nov 21 2022

Antiharmonic mean of divisors of number n = Product (p_i^e_i) is sigma_2(n)/sigma_1(n) = A001157(n)/A000203(n) = Product ((p_i^(e_i+1)+1)/(p_i+1)).

Sum_{k=1..n} a(k)/A158275(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (3*p-2)/(p^3)) = A152649 * A065473 = 0.387941... . - Amiram Eldar, Nov 21 2022

A379812 a(n) = sigma_1(n) * sigma_2(n).

Original entry on oeis.org

1, 15, 40, 147, 156, 600, 400, 1275, 1183, 2340, 1464, 5880, 2380, 6000, 6240, 10571, 5220, 17745, 7240, 22932, 16000, 21960, 12720, 51000, 20181, 35700, 32800, 58800, 25260, 93600, 30784, 85995, 58560, 78300, 62400, 173901, 52060, 108600, 95200, 198900, 70644

Offset: 1

Amiram Eldar, Jan 03 2025

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. Balasubramanian and K. Ramachandra, The place of an identity of Ramanujan in prime number theory, Proceedings of the Indian Academy of Sciences, Section A, Vol. 83. No. 4 (1976), pp. 156-165.
Yoichi Motohashi, On an identity of Ramanujan, Hardy-Ramanujan Journal, Vol. 41 (2019), pp. 48-49.
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
Bertram Martin Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, Vol. s2-21, No. 1 (1923), pp. 235-255.

Cf. A000203, A001157, A072861, A086666, A092346, A158274, A158275, A356533, A356534, A379813, A379814.

a[n_] := Times @@ DivisorSigma[{1, 2}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 2);}

a(n) = A000203(n) * A001157(n).
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(2*e+2)-1) / ((p-1) * (p^2-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) / zeta(2*s-3).
In general, Dirichlet g.f. of  sigma_i(n) * sigma_j(n): zeta(s) * zeta(s-i) * zeta(s-j) * zeta(s-i-j) / zeta(2*s-i-j) (Ramanujan, 1916).
Sum_{k=1..n}  a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(3) * zeta(4) / zeta(5) = zeta(3) * Pi^6 / (540*zeta(5)) = 2.06386841111121962734... .
In general, Sum_{k=1..n}  sigma_i(k) * sigma_j(k) ~ c(i,j) * n^(i+j+1) / (i+j+1), for i, j >= 1, where c(i,j) = zeta(i+1) * zeta(j+1) * zeta(i+j+1) / zeta(i+j+2).
G.f.: Sum_{k>=1} Sum_{l>=1} k*l^2*x^lcm(k, l)/(1 - x^lcm(k, l)). - Miles Wilson, Jul 10 2025

A176803 a(n) = the smallest natural numbers m such that product of antiharmonic mean of the divisors of n and antiharmonic mean of the divisors of m are integers, a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 4, 0, 1, 4, 0, 0, 4, 1, 100, 0, 0, 9, 0, 0, 1, 100, 4, 0, 1, 0, 0, 0, 0, 1, 25, 0, 0, 325, 0

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

Antiharmonic mean of the divisors of number n is rational number b(n) = A001157(n) / A000203(n) = A158274(n) / A158275(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A020487: a(A020487(n)) = 1. a(n) = 1 iff A158275(n) = 1. a(n) = 0 for infinitely many n. a(n) = 0 for even A158275(n).

			For n = 10; b(10) = 65/9, a(n) = 100 because b(100) = 63; 65/9 * 63 = 455 (integer).

cp's OEIS Frontend

A158274 Numerators of antiharmonic means of divisors of n.

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A379812 a(n) = sigma_1(n) * sigma_2(n).

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A176803 a(n) = the smallest natural numbers m such that product of antiharmonic mean of the divisors of n and antiharmonic mean of the divisors of m are integers, a(n) = 0 if no such number exists.

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