A283052 -id:A283052 - cp's OEIS frontend

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

1, 2, 3, 4, 7, 8, 12, 16, 24, 32, 36, 48, 56, 64, 72, 96, 108, 128, 144, 192, 200, 216, 288, 360, 400, 432, 504, 576, 648, 720, 792, 800, 864, 1008, 1080, 1152, 1296, 1440, 1512, 1584, 1728, 1800, 1944, 2016, 2160, 2304, 2592, 2880, 3024, 3240, 3456, 3600

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Analogous to highly abundant numbers (A002093) with the sum of the coreful divisors function (A057723) instead of the sum of divisors function (A000203).

			The first 10 values of A057723(n) for n=1..10 are: 1, 2, 3, 6, 5, 6, 7, 14, 12, 10. The record values, 1, 2, 3, 6, 7 and 14 occur at 1, 2, 3, 4, 7 and 8, the first 6 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..533 (terms below 10^10)

Cf. A007947, A057723, A283052, A308053.

Similar sequences: A002093, A285614, A292983, A327634, A328134, A329883.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 3600}]; seq

A377138 Powerful numbers that have more divisors than any smaller powerful number.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 576, 864, 900, 1728, 1800, 3600, 5400, 7200, 10800, 14400, 21600, 32400, 43200, 64800, 86400, 88200, 176400, 264600, 352800, 529200, 705600, 1058400, 1587600, 2116800, 3175200, 4233600, 6350400, 8467200, 10584000, 12700800

Offset: 1

Amiram Eldar, Oct 17 2024

nonn

First differs from A283052 at n = 12.
Indices of records in A357669.
The corresponding record values are 1, 3, 4, 5, 6, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 36, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..548
Amiram Eldar, Table of n, a(n), A357669(a(n)) for n = 1..548

Subsequence of A001694 and A025487.

Cf. A000005, A002182, A283052, A357669, A363194.

f[p_, e_] := If[e == 1, 1, e + 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; With[{v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq]

cp's OEIS Frontend

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

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