A010553 -id:A010553 - cp's OEIS frontend

A141551 Numbers k with property that if d divides k, then tau(tau(d)) also divides k.

1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 180, 192, 204, 228, 240, 252, 276, 288, 300, 324, 348, 360, 372, 396, 444, 468, 480, 492, 516, 564, 576, 588, 600, 612, 636, 684, 708, 720, 732, 804, 828, 840, 852, 876, 900, 948, 960, 972, 996, 1044, 1068

Offset: 1

N. J. A. Sloane, Sep 12 2008

nonn

tau() = A000005().

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A010553, A141586.

with(numtheory); isA141551 := proc(n) local dvs,d; dvs := divisors(n) ; for d in dvs do if not tau(tau(d)) in dvs then RETURN(false): fi; od: RETURN(true); end:
t1:=[]; for n from 1 to 60000 do if isA141551(n) then t1:=[op(t1),n]; fi; od:

aQ[n_] := AllTrue[Divisors[n], Divisible[n, DivisorSigma[0, DivisorSigma[0, #]]] &]; Select[Range[1000], aQ] (* Amiram Eldar, Jul 08 2019 *)

A193348 Number of odd divisors of tau(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3

Offset: 1

Michel Lagneau, Jul 23 2011

			a(36) = 3 because tau(36) = 9 and the 3 odd divisors are {1, 3, 9}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A010553, A036537, A193349.

a[n_] := Block[{d = Divisors[DivisorSigma[0,n]]}, Count[OddQ[d], True]]; Table[a[n], {n, 80}]

PARI
```
a(n)=sumdiv(sigma(n,0),d,d%2);
```

a(n)=n=numdiv(n);numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Jul 30 2011

a(n) = A001227(A000005(n)). - Reinhard Zumkeller, Jul 25 2011
From Amiram Eldar, Aug 12 2024: (Start)
a(n) = 1 if and only if n is in A036537.
a(n) = A010553(n) if and only if n is a square. (End)

A053021 Number of divisors function applied twice to n!.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 12, 12, 12, 16, 24, 24, 30, 30, 42, 36, 40, 48, 56, 80, 48, 72, 80, 112, 112, 144, 120, 216, 252, 168, 189, 168, 80, 224, 168, 288, 320, 121, 192, 440, 480, 384, 408, 624, 792, 864, 960, 1152, 864, 728, 504, 780, 840, 1080, 960, 840, 972

Offset: 1

Labos Elemer, Feb 24 2000

nonn

			a(7) = d(d(7!)) = d(60) = 12;
a(8) = d(d(8!)) = d(96) = 12.

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

Cf. A000005, A000142, A027423, A010553.

DivisorSigma[0,DivisorSigma[0,Range[60]!]] (* Harvey P. Dale, May 20 2013 *)

a(n) = numdiv(numdiv(n!)); \\ Michel Marcus, May 01 2016

a(n) = A010553(n!) = A000005(A027423(n)) = A000005(A000005(A000142(n))).

A111407 a(n) = f(f(n+1)) - f(f(n)), where f(0) = 0 and f(m) = tau(m) = A000005(m) for m > 0.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, -1, 1, -1, 1, -1, 2, -2, 1, 0, -1, 0, 2, -2, 2, -1, 0, -1, 2, -2, 1, 0, 1, -2, 2, -2, 2, -1, 0, 0, 0, -1, 1, 0, 1, -2, 2, -2, 2, 0, -1, -1, 2, -2, 2, -1, 1, -2, 2, -1, 1, -1, 0, -1, 4, -4, 1, 1, -2, 1, 1, -2, 2, -1, 1, -2, 4, -4, 1, 1, 0, -1, 1, -2, 2, -2, 1, -1, 4, -3, 0, 0, 1, -2, 4, -3, 1, -1, 0, 0, 3, -4, 2, 0, -1, -1

Offset: 0

N. J. A. Sloane, Nov 12 2005

sign

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A000005, A111405.

First differences of A010553.

f = numdiv;
a(n) = f(f(n+1)) - f(f(n));
concat([1], vector(166,n,a(n))) \\ Joerg Arndt, Jul 06 2013

f(n) = if(!n,n,numdiv(n));
A111407(n) = f(f(n+1)) - f(f(n)); \\ Antti Karttunen, Oct 07 2017

Description clarified by Antti Karttunen, Oct 07 2017

A139130 a(n) = Sum_{k=1..n} d(d(k)), where d(k) = number of divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 14, 17, 19, 22, 24, 28, 30, 33, 36, 38, 40, 44, 46, 50, 53, 56, 58, 62, 64, 67, 70, 74, 76, 80, 82, 86, 89, 92, 95, 98, 100, 103, 106, 110, 112, 116, 118, 122, 126, 129, 131, 135, 137, 141, 144, 148, 150, 154, 157, 161, 164, 167, 169, 175, 177, 180

Offset: 1

Leroy Quet, Jun 05 2008

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., Vol. 49, No. 2 (1948), 333-340. See Eq. 1.6. [From _N. J. A. Sloane_, Mar 12 2009]
Paul Erdős, On the sum Sum_{n=1..x} d[d(n)], Math. Student, Vol. 36 (1968), pp. 227-229.
E. Heppner, Über die Iteration von Teilerfunktionen, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.

Cf. A000005, A010553, A006218.

with(numtheory): a:= n-> add(tau(tau (k)), k=1..n): seq(a(n), n=1..70); # Alois P. Heinz, Aug 28 2008

Table[Sum[DivisorSigma[0,DivisorSigma[0,k]],{k,1,n}],{n,1,62}] (* Geoffrey Critzer, Sep 28 2013 *)
Accumulate[Table[DivisorSigma[0, DivisorSigma[0, k]], {k, 1, 62}]] (* Amiram Eldar, Jan 15 2024 *)

a(n) = sum(k = 1, n, numdiv(numdiv(k))); \\ Michel Marcus, Sep 28 2013

a(n) = b * n * log(log(n)) + Sum_{k=0..floor(sqrt(n))} b_k * n/log(n)^k + O(n * exp(-c*sqrt(log(n)))), where b, b_k and c are constants (Heppner, 1974). - Amiram Eldar, Jan 15 2024

More terms from Alois P. Heinz, Aug 28 2008

A163107 a(n) = tau(tau(sigma(n))), where tau = A000005, the number of divisors, and sigma = A000203, the sum of divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 3, 3, 2, 4, 4, 4, 3, 4, 4, 2, 4, 3, 4, 4, 4, 3, 4, 6, 2, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 3, 3, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 4, 3, 3, 6, 4, 4, 5, 6, 5, 4, 6, 6, 5, 3, 6, 4, 2, 6, 4, 4, 6, 6, 4, 6, 4, 3, 4, 4, 6, 6, 5, 4, 4, 2, 6, 6, 6, 6, 6, 5, 6, 6, 6, 4, 5, 4, 4, 5, 6, 4, 4, 6, 3, 4, 5, 4, 5, 4

Offset: 1

Jaroslav Krizek, Jul 20 2009

nonn

Repeated application of tau (number of divisors) and sigma (sum of divisors).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A000203, A010553, A062068.

with(numtheory) : A163107 := proc(n) tau(tau(sigma(n))) ; end: seq(A163107(n),n=1..120) ; # R. J. Mathar, Jul 27 2009

DivisorSigma[0,DivisorSigma[0,DivisorSigma[1,Range[110]]]] (* Harvey P. Dale, Nov 25 2016 *)

A163107(n) = numdiv(numdiv(sigma(n))); \\ Antti Karttunen, Nov 07 2017

a(n) = A000005(A000005(A000203(n))) = A000005(A062068(n)) = A010553(A000203(n)).

More terms from R. J. Mathar, Jul 27 2009

A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152

Offset: 1

Matthew Vandermast, Dec 04 2010

nonn

In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n.
A141586 is a subsequence. Is A110821 a subsequence?
Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551.
a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022

			9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors).
Cf. A010553 (d(d(n))), A036450 (d^3(n)), A036452 (d^4(n)), A036453 (d^5(n)).
Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).

is_A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022

Edited by M. F. Hasler, Apr 16 2022

A193347 Number of even divisors of tau(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 3, 0, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 0, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 0, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 0, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 0, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 2, 1, 3

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

			a(24) = 3 because tau(24) = 8 and the 3 even divisors are {2, 4, 8}.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A010553, A183063, A193348, A193350.

f[n_] := Block[{d = Divisors[DivisorSigma[0,n]]}, Count[EvenQ[d], True]]; Table[f[n], {n, 80}]

PARI
```
a(n)=sumdiv(sigma(n,0),d,(1-d%2));
```

a(n) = A183063(A000005(n)). - Antti Karttunen, May 28 2017
From Amiram Eldar, Jan 27 2025: (Start)
a(n) = 0 if and only if n is a square.
a(n) = A010553(n) - A193348(n). (End)

A193987 Least number k such that tau(tau(k)) = n.

Original entry on oeis.org

1, 2, 6, 12, 120, 60, 7560, 360, 1260, 2520, 294053760, 5040, 128501493120, 332640, 110880, 55440, 106858629141264000, 277200, 188391763176048432000, 720720, 21621600, 27935107200, 1356699703068812438127792000, 3603600, 857656800, 18632716502400, 227026800, 183783600

Offset: 1

T. D. Noe, Aug 10 2011

nonn

Here tau is the number of divisors function, A000005. Such a k always exists because an upper bound is 2^(2^n-1). For n < 19, and small composite numbers, terms can be found among the highly composite numbers, A002182. The b-file in A005179 is useful when tau^(-1)(n) is small.

Cf. A010553 (tau(tau(n))), A000005, A002182, A005179.

a(27) corrected by Amiram Eldar, Jan 20 2025

A307305 Self-composition of the number of divisors function (A000005).

Original entry on oeis.org

1, 4, 12, 34, 92, 246, 640, 1660, 4264, 10914, 27732, 70247, 177466, 447570, 1126344, 2828465, 7089391, 17746456, 44384884, 110927184, 276993616, 691007612, 1722214602, 4289021667, 10675557184, 26561494820, 66063726382, 164248795485, 408168287028, 1013819012498

Offset: 1

Ilya Gutkovskiy, Apr 01 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..760
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A000005, A010553, A055507, A307306.

g[x_] := g[x] = Sum[x^k/(1 - x^k), {k, 1, 30}]; a[n_] := a[n] = SeriesCoefficient[g[g[x]], {x, 0, n}]; Table[a[n], {n, 30}]

G.f.: g(g(x)), where g(x) = Sum_{k>=1} x^k/(1 - x^k) is the g.f. of A000005.

cp's OEIS Frontend

A141551 Numbers k with property that if d divides k, then tau(tau(d)) also divides k.

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A193348 Number of odd divisors of tau(n).

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A053021 Number of divisors function applied twice to n!.

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A111407 a(n) = f(f(n+1)) - f(f(n)), where f(0) = 0 and f(m) = tau(m) = A000005(m) for m > 0.

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A139130 a(n) = Sum_{k=1..n} d(d(k)), where d(k) = number of divisors of k.

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A163107 a(n) = tau(tau(sigma(n))), where tau = A000005, the number of divisors, and sigma = A000203, the sum of divisors of n.

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A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

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