A332038 -id:A332038 - cp's OEIS frontend

A332039 Indices of records in A332038.

1, 12, 24, 60, 120, 240, 360, 720, 1440, 2880, 4320, 5760, 7200, 8640, 11520, 14400, 17280, 21600, 25920, 28800, 34560, 43200, 60480, 86400, 120960, 129600, 172800, 241920, 259200, 302400, 345600, 483840, 518400, 604800, 907200, 1036800, 1209600, 1814400, 2419200

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that isigma(x) = k has more solutions x than any smaller k, where isigma(x) is the sum of infinitary divisors of x (A049417).
The infinitary version of A145899.
The corresponding number of solutions for each term is 1, 2, 3, 5, 7, 12, 13, 20, ... (see the link for more values).

			There are 3 solutions to isigma(x) = 24: isigma(14) = isigma(15) = isigma(23) = 24. For all m < 24 there are 2 or fewer solutions to isigma(x) = m, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..53
Amiram Eldar, Table of n, a(n), A332038(a(n)) for n = 1..53

Cf. A145899, A049417, A289132, A332037, A332038, A332041.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); m = 10000; v = Table[0, {m}]; Do[i = isigma[k]; If[i <= m, v[[i]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

A332036 Number of integers whose bi-unitary divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 1, 1, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(12) = 2 since there are 2 solutions to bsigma(x) = 12 (bsigma is A188999): 6 and 11.

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

Cf. A054973, A063974, A188999, A332038, A332040.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); m = 100; v = Table[0, {m}]; Do[b = bsigma[k]; If[b <= m, v[[b]]++], {k, 1, m}]; v

A332040 Number of integers whose exponential divisors sum to n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 3, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(6) = 2 since there are 2 solutions to esigma(x) = 6 (esigma is A051377): 4 and 6.

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

Cf. A054973, A051377, A063974, A332036, A332038.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; m = 100; v = Table[0, {m}]; Do[e = esigma[k]; If[e <= m, v[[e]]++], {k, 1, m}]; v

A332042 Number of integers whose Dedekind psi function (A001615) values are n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(6) = 2 since there are 2 solutions to psi(x) = 6: 4 and 5.

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

Cf. A001615, A054973, A063974, A065463, A332036, A332038, A332040.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); m = 100; v = Table[0, {m}]; Do[i = psi[k]; If[i <= m, v[[i]]++], {k, 1, m}]; v

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044422... (A065463). - Amiram Eldar, Dec 25 2024

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A332039 Indices of records in A332038.

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A332036 Number of integers whose bi-unitary divisors sum to n.

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A332040 Number of integers whose exponential divisors sum to n.

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A332042 Number of integers whose Dedekind psi function (A001615) values are n.

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