A190867 -id:A190867 - cp's OEIS frontend

A335850 Cubefull highly composite numbers: numbers with a record number of cubefull divisors (A190867).

1, 8, 16, 32, 64, 128, 256, 512, 1024, 1728, 2592, 5184, 7776, 10368, 15552, 20736, 31104, 46656, 62208, 93312, 124416, 186624, 248832, 373248, 559872, 746496, 1119744, 1492992, 2239488, 2985984, 3359232, 4478976, 6718464, 8957952, 13436928, 17915904, 26873856

Offset: 1

Amiram Eldar, Jun 26 2020

nonn

The analogous sequence of squarefull highly composite numbers is the sequence of highly powerful numbers (A005934).
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, ... (see the link for more values).
Also, indices of records in A361430, i.e., numbers k with a record number of coreful divisors d such that k/d is also a coreful divisor of k (a coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, see A307958). - Amiram Eldar, Aug 15 2023

Amiram Eldar, Table of n, a(n) for n = 1..832
Amiram Eldar, Table of n, a(n), A190867(a(n)) for n = 1..832

Subsequence of A025487.

Cf. A002182, A005934, A190867, A307958, A361430.

f[p_, e_] := Max[1, e-1] ; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); s = {}; dm = 0; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 10^5}]; s

d(n) = vecprod(apply(x->max(1, x-1), factor(n)[, 2]));
lista(kmax) = {my(dm = 0, d1); for(k = 1, kmax, d1 = d(k); if(d1 > dm, dm = d1; print1(k, ", ")));} \\ Amiram Eldar, Aug 15 2023

A365498 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 06 2023

The number of unitary divisors of n that are cubefree numbers (A004709). - Amiram Eldar, Sep 06 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A365498, Sequence Machine.

Cf. A001620, A004709, A056671, A365488, A365499.

Cf. A034444, A158522, A190867, A286324, A307428, A368172, A368248, A368885, A369310.

f[p_, e_] := If[e <= 2, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 06 2023 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X - X^3))[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,
f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...
and gamma is the Euler-Mascheroni constant A001620.
Multiplicative with a(p^e) = 2 if e <= 2, and 1 otherwise. - Amiram Eldar, Sep 06 2023
From Vaclav Kotesovec, Jan 27 2025: (Start)
Following formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 1000000 terms correctly:
a(n) = A056671(n) * A368885(n).
a(n) = A034444(n) / A368248(n).
a(n) = A158522(n) / A307428(n).
a(n) = A369310(n) / A190867(n).
a(n) = A286324(n) / A368172(n). (End)

A385005 The sum of the cubefull divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 15 2025

The sum of the terms in A036966 that divide n.

The number of these divisors is A190867(n), and the largest of them is A360540(n).

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

Cf. A036966, A190867, A183097, A360540.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), this sequence (cubefull), A385006 (biquadratefree).

f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

Original entry on oeis.org

1, 16, 81, 384, 625, 640, 896, 1296, 1408, 1664, 2176, 2401, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, 6784, 7552, 7808, 8576, 9088, 9216, 9344, 10000, 10112, 10624, 10935, 11392, 12416, 12928, 13184, 13696, 13952, 14464, 14641, 15309, 16256, 16768

Offset: 1

Amiram Eldar, Feb 25 2023

nonn

Numbers k such that A073184(k) = A190867(k).
Numbers whose largest cubefree divisor (A007948) and cubefull part (A360540) have the same number of divisors (A000005).
If k and m are coprime terms, then k*m is also a term.
The characteristic function of this sequence depends only on prime signature.
1 is the only cubefree (A004709) term.
Includes the 4th powers of squarefree numbers (1 and A113849).
The 4th powers of primes (A030514) are the only terms that are prime powers (A246655).
Numbers of the for m*p^(3*2^k+1), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k, are all terms. In particular, this sequence includes numbers of the form p^7*q, where p != q are primes (A179664), and numbers of the form p^13*q*r where p, q, and r are distinct primes.
The corresponding numbers of cubefree (or 3-full) divisors are 1, 3, 3, 6, 3, 6, 6, 9, 6, 6, 6, 3, 6, 6, ... .

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

Cf. A000005, A004709, A007948, A073184, A190867, A246655, A360540, A360902.

Subsequences: A030514, A113849, A179664, A360907.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; Select[Range[10^4], q]

is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }

A360907 Numbers k such that k and k+1 both have the same number of cubefree divisors and 3-full divisors.

Original entry on oeis.org

916352, 3002751, 13080447, 22598271, 26110592, 28909952, 45706112, 49472127, 52890624, 53391231, 56190591, 58471552, 63468927, 65148543, 67947903, 69780608, 84744063, 89376128, 93142143, 94974848, 143530623, 143683712, 145770111, 155847807, 165925503, 177109375

Offset: 1

Amiram Eldar, Feb 25 2023

nonn

Numbers k such that k and k+1 are both terms of A360906.

			48 is a term since A073184(916352) = A190867(916352) = 6 and A073184(916353) = A190867(916353) = 6.

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

Subsequence of A360906.

Cf. A073184, A190867.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; seq[kmax_] := Module[{s = {}, k = 1, q1 = q[1], q2}, Do[q2 = q[k]; If[q1 && q2, AppendTo[s, k-1]]; q1 = q2, {k, 2, kmax}]; s]; seq[2*10^5]

is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }
lista(kmax) = {my(is1 = is(1), i2); for(k=2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2); }

cp's OEIS Frontend

A335850 Cubefull highly composite numbers: numbers with a record number of cubefull divisors (A190867).

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A365498 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).

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A385005 The sum of the cubefull divisors of n.

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A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

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A360907 Numbers k such that k and k+1 both have the same number of cubefree divisors and 3-full divisors.

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