A010553 -id:A010553 - cp's OEIS frontend

A347494 a(n) is the number of instances where tau(k)|tau(n); n>=1, kA000005.

0, 1, 2, 1, 3, 4, 4, 6, 2, 7, 5, 8, 6, 10, 11, 1, 7, 11, 8, 13, 14, 15, 9, 17, 3, 17, 18, 16, 10, 21, 11, 19, 21, 22, 23, 4, 12, 25, 26, 29, 13, 31, 14, 23, 24, 29, 15, 17, 4, 27, 31, 28, 16, 37, 33, 39, 34, 35, 17, 50, 18, 38, 32, 1, 39, 46, 19, 34, 41, 49, 20, 59

Offset: 1

David James Sycamore, Aug 31 2021

nonn

a(n) >= 1, for n >= 2; equality only when n = 2^(p-1) for any prime p. More generally, if p is prime(m), q any prime, and n=p^(q-1) then tau(n) = q, and the only numbers k < m such that tau(k)|tau(n) are 1 and q. Every prime < p contributes 1 to the count of a(n), and so does 1 itself, therefore a(n) = m-1+1 = m; see formula. Since for a given m, this holds for all primes q, it follows that every m > 0 appears in the sequence infinitely many times.

			a(1) = 0 because there is no k < 1 such that tau(k)|tau(1).
a(2) = 1, since there is only one instance of tau(k)|tau(2), namely k=1.
a(3) = 2, since there are two instances of tau(k)|tau(3), namely k=1 and k=2.
a(4) = 3, since there is only one instance of tau(k)|tau(4), namely k=1, etc.

David A. Corneth, Table of n, a(n) for n = 1..10001

Cf: A000005, A010553, A047983, A010553.

With[{s = DivisorSigma[0, Range[72]]}, Array[Count[Mod[#2, s[[Range[#1 - 1]]]], 0] & @@ {#, s[[#]]} &, Length[s] - 1, 2]] (* Michael De Vlieger, Sep 09 2021 *)

first(n) = {my(l = List(), res = vector(n)); for(i = 1, n, nd = numdiv(i); if(nd > #l, for(i = #l + 1, nd, listput(l, 0) ) ); d = divisors(nd); for(j = 1, #d, res[i] += l[d[j]] ); l[nd]++; ); res } \\ David A. Corneth, Sep 03 2021

a(prime(m)^(q-1)) = m for m >= 1 and any prime q.

A307715 Decimal expansion of Sum_{t>0} log((t + 1)/t)^2.

Original entry on oeis.org

9, 7, 7, 1, 8, 9, 1, 8, 3, 2, 6, 8, 9, 3, 6, 5, 5, 4, 4, 5, 7, 8, 8, 5, 7, 4, 9, 4, 7, 6, 4, 3, 4, 7, 4, 8, 0, 7, 7, 3, 9, 2, 5, 0, 6, 4, 7, 4, 7, 2, 3, 9, 0, 1, 7, 7, 0, 2, 0, 9, 8, 9, 7, 5, 5, 3, 1, 8, 4, 4, 5, 2, 9, 3, 9, 2, 3, 9, 3, 3, 5, 6, 2, 9, 0, 1, 2, 3, 2, 1, 0, 7, 9, 7, 4, 3, 2, 0, 3, 3, 5, 9, 2, 3, 2

Offset: 0

Stefano Spezia, Apr 24 2019

This constant appears at several places in the literature:
1) In the asymptotic formula of the number of minimal covering systems with exactly n elements (see Theorem 1.1 in Balister, Bollobás, Morris, Sahasrabudhe and Tiba) and
2) in the maximal size of the iterated divisor function
(see Theorem 1 in Buttkewitz, Elsholtz, Ford and Schlage-Puchta) and
3) in the maximal order of the iterated r_2 function, which counts the number of representations as sums of 2 squares (see Theorems 2.1. and 2.3 in Elsholtz, M. Technau and N. Technau). - Modified by C. Elsholtz, Apr 15 2025

			0.9771891832689365544578857494764347480773925064747239017702...

P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The structure and number of Erdős covering systems, arXiv:1904.04806 [math.CO], 2019. J. Eur. Math. Soc. 26, 75-109 (2024).
Y. Buttkewitz, C. Elsholtz, K. Ford, and J.-C. Schlage-Puchta, The maximal order of iterated multiplicative functions, arxiv:1108.1815 [math.NT], 2011. IMRN issue 17, 4051-4061 (2012).
C. Elsholtz, M. Technau, and N. Technau, The maximal order of iterated multiplicative functions, arxiv:1709.04799 [math.NT], 2017. Mathematika, 65 (4) 2019, 990-1009.
P. Erdős, Egy kongruenciarendszerekrol szóló problémáról, (On a problem concerning congruence-systems, in Hungarian), Mat. Lapok, 4 (1952), 122-128.

Cf. A001008, A002805, A010553, A080340, A094076, A275489, A294593, A296195, A335831.

First[RealDigits[NSum[(Log[(t + 1)/t])^2, {t, 1, Infinity}, NSumTerms -> 100, Method -> {"NIntegrate", "MaxRecursion" -> 10}, WorkingPrecision -> 100]]]

sumpos(t=1, log((t + 1)/t)^2) \\ Michel Marcus, Apr 26 2019

From Amiram Eldar, Jun 17 2023: (Start)
Equals 2 * Sum_{k>=1} H(k) * (zeta(k+1)-1) / (k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals -Sum_{k>=1} zeta'(2*k) / k. (End)

cp's OEIS Frontend

A347494 a(n) is the number of instances where tau(k)|tau(n); n>=1, kA000005.

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