A332040 -id:A332040 - cp's OEIS frontend

A332041 Indices of records in A332040.

1, 6, 30, 330, 390, 2730, 5460, 12090, 60060, 92820, 223860, 1021020, 1922700, 3805620, 13458900, 41861820, 110362980, 113573460, 227146920, 251170920, 502341840, 563603040, 888287400, 1270629360, 1776574800, 3310889400, 23107724640, 27939071160, 33754921200, 36419783400

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that esigma(x) = k has more solutions x than any smaller k, where esigma(x) is the sum of exponential divisors of x (A051377).
The exponential version of A145899.
The corresponding number of solutions for each term is 1, 2, 5, 6, 8, 9, 10, 12, 15, 16, 19, 22, 27, 29, 35, 37, 38, 44, 45, 47, 50, 51, 52, 53, 66, 80, 83, 89, 95, 102.

			There are 2 solutions to esigma(x) = 6: esigma(4) = esigma(6) = 6. For all m < 6 there are no more than one solution to esigma(x) = m, thus 6 is in the sequence.

Cf. A051377, A145899, A289132, A332037, A332039, A332040.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; m = 10000; v = Table[0, {m}]; Do[sig = esigma[k]; If[sig <= m, v[[sig]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

a(26)-a(30) from Giovanni Resta, Feb 06 2020

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

A332038 Number of integers whose infinitary 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, 1, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(12) = 2 since there are 2 solutions to isigma(x) = 12 (isigma is A049417): 6 and 11.

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

Cf. A054973, A049417, A063974, A332036, A332040.

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 = 100; v = Table[0, {m}]; Do[i = isigma[k]; If[i <= m, v[[i]]++], {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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A332041 Indices of records in A332040.

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

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A332038 Number of integers whose infinitary divisors sum to n.

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

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