A335831 -id:A335831 - cp's OEIS frontend

A010553 a(n) = tau(tau(n)).

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 3, 4, 2, 4, 2, 4, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 4, 2

Offset: 1

N. J. A. Sloane

Ramanujan (1915) posed the problem of finding the extreme large values of a(n). Buttkewitz et al. determined the maximal order of log a(n).

Every number eventually appears. Sequence A193987 gives the least term where each number appears. - T. D. Noe, Aug 10 2011

S. Ramanujan, Highly composite numbers. Proc. London Math. Soc., series 2, 14 (1915), 347-409. Republished in Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 78-128.

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 2000 terms from Enrique Pérez Herrero)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.5.
Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta, A problem of Ramanujan, Erdos and Katai on the iterated divisor function, arXiv:1108.1815 [math.NT], Aug 08 2011.

Cf. A000005, A036450, A193987 (least number k such that tau(tau(k)) = n), A335831.

Maple
```
with(numtheory): f := n->tau(tau(n));
```

Table[Nest[DivisorSigma[0, #] &, n, 2], {n, 81}] (* Michael De Vlieger, Dec 24 2015 *)

A010553(n)=numdiv(numdiv(n)); \\ Enrique Pérez Herrero, Jul 13 2010

a(n) = A000005(A000005(n)). a(1) = 1, a(p) = 2 for p = primes (A000040), a(pq) = 3 for pq = product of two distinct primes (A006881), a(pq...z) = k + 1 for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = A000005(k+1) for p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027). - Jaroslav Krizek, Jul 17 2009
a(A007947(n)) = 1 + A001221(n); (n>1). - Enrique Pérez Herrero, May 30 2010
Asymptotically, Max_{i<=n} log(tau(tau(i))) = sqrt(log(n))/log_2(n) * (c + O(log_3(n)/log_2(n)) where c = 8*Sum_{j>=1} log^2 (1 + 1/j)) ~ 2.7959802335... [Buttkewitz et al.].

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

A010553 a(n) = tau(tau(n)).

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