A374904 -id:A374904 - cp's OEIS frontend

A374903 Denominator of the mean number of divisors of the divisors of n.

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 1, 2, 4, 4, 1, 2, 1, 2, 1, 4, 4, 2, 4, 1, 4, 2, 1, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 1, 1, 4, 2, 2, 1, 1, 4, 1, 2, 4, 4, 4, 4, 4, 2, 2, 2, 4, 1, 1, 4, 8, 2, 1, 4, 8, 2, 1, 2, 4, 1, 1, 4, 8, 2, 2, 1, 4, 2, 2, 4, 4, 4

Offset: 1

Amiram Eldar, Jul 23 2024

			For n = 2, n has 2 divisors, 1 and 2. Their numbers of divisors are tau(1) = 1 and tau(2) = 2, and their mean number of divisors is (1 + 2)/2 = 3/2. Therefore a(2) = denominator(3/2) = 2.

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

Cf. A374902 (numerators), A374904, A374905.

f[p_, e_] := (e + 2)/2; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = denominator(vecprod(apply(x -> x/2 +1, factor(n)[, 2])));

a(A374904(n)) = 1.

a(A374905(n)) = 1.

A374905 Least integers of their prime signature (A025487) whose divisors have an integer mean number of divisors.

Original entry on oeis.org

1, 4, 12, 16, 36, 64, 72, 144, 180, 192, 256, 288, 576, 720, 900, 960, 1024, 1152, 1260, 1296, 1728, 1800, 2304, 2880, 3072, 3600, 4096, 4608, 5184, 6300, 7200, 8640, 9216, 10800, 11520, 12600, 14400, 15552, 16384, 18432, 20160, 20736, 25200, 25920, 27648, 28800

Offset: 1

Amiram Eldar, Jul 23 2024

nonn

If k is a term then every number with the same prime signature as k is a term of A374904.

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

Intersection of A025487 and A374904.

Cf. A374902, A374903.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/2 + 1; d[1] = 1; d[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Select[lps, d[#] == 1 &]

is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); n == 1 || (prime(#p) == p[#p] && e == vecsort(e, , 4) && denominator(vecprod(apply(x -> x/2 +1, e))) == 1);}

A374906 a(n) is the least number whose divisors have a mean number of divisors that equals n.

Original entry on oeis.org

1, 4, 12, 36, 72, 144, 288, 576, 720, 1728, 4608, 2880, 18432, 7200, 8640, 14400, 294912, 20160, 1179648, 43200, 50400, 115200, 18874368, 100800, 172800, 460800, 181440, 352800, 1207959552, 302400, 4831838208, 705600, 806400, 7372800, 1058400, 907200, 309237645312

Offset: 1

Amiram Eldar, Jul 23 2024

nonn

a(n) is the least number k such that A374902(k)/A374903(k) = A007425(k)/A000005(k) = n.

All the terms are in A025487.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A005179, A007425, A025487, A374902, A374903, A374904, A374905, A374907.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/2 + 1; f[1] = 1; f[n_] := Times @@ f @@@ FactorInteger[n]; s = f /@ lps; lps[[TakeWhile[Flatten[FirstPosition[s, #] & /@ Range[Max[s]]], ! MissingQ[#] &]]]

a(p) = 9 * 2^(p-2) for prime p >= 5.

a(p^e) = 9 * 2^(p^e-2) for prime p >= 3 and e >= 2.

A374907 Number whose divisors have a mean number of divisors that attains a record value.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2880, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 30240, 40320, 50400, 60480, 90720, 100800, 120960, 151200, 181440, 241920, 302400, 362880, 453600

Offset: 1

Amiram Eldar, Jul 23 2024

nonn

First differs from A301414 at n = 454: a(454) = 526399743264198303532032000 is not a term of A301414. Is A301414 a subsequence of this sequence? The first 1073 terms of A301414 are in this sequence.
Indices of records of A374902(k)/A374903(k) = A007425(k)/A000005(k).
All the terms are least integers of their prime signature (A025487) since A374902(k)/A374903(k) depends only on the prime signature of k.
The corresponding record values are 1, 3/2, 2, 9/4, 5/2, 3, 15/4, 4, 9/2, 5, ... .

Amiram Eldar, Table of n, a(n) for n = 1..1085

Subsequence of A025487.

Cf. A000005, A002182, A007425, A301414, A308912, A374902, A374903, A374904, A374905, A374906.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/2 + 1; f[1] = 1; f[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; fmax = -1; Do[f1 = f[lps[[k]]]; If[f1 > fmax, fmax = f1; AppendTo[s, lps[[k]]]], {k, 1, Length[lps]}]; s

cp's OEIS Frontend

A374903 Denominator of the mean number of divisors of the divisors of n.

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A374905 Least integers of their prime signature (A025487) whose divisors have an integer mean number of divisors.

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A374906 a(n) is the least number whose divisors have a mean number of divisors that equals n.

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A374907 Number whose divisors have a mean number of divisors that attains a record value.

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