A355031 -id:A355031 - cp's OEIS frontend

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

			a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.
a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.
a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

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

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

A355032 a(n) is the maximum number of prime signatures of numbers with n divisors that have the same number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

			a(2) = 1 since the numbers with 2 divisors are all primes and thus have only 1 prime signature.
a(36) = 2 since numbers with 36 divisors have 2 prime signatures, p1^5 * p2^5 and p1 * p2 * p3^8, that correspond to numbers with 10 prime divisors (counted with multiplicity).
a(72) = 3 since numbers with 72 divisors have 3 prime signatures, p1 * p2^5 * p3^5, p1^2 * p2^2 * p3^7 and p1 * p2 * p3 * p4^8, that correspond to numbers with 11 prime divisors (counted with multiplicity).

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

Cf. A000005, A001222, A118914, A162247, A355028, A355030, A355031, A355033.

Table[Max[Tally[Total[#-1]& /@ f[n]][[;;,2]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(A355033(n)) = n.

A355033 a(n) is the least number k such that A355032(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 36, 72, 288, 960, 720, 1440, 3456, 2880, 6912, 5760, 10080, 11520, 8640, 24192, 21600, 47520, 17280, 28800, 20160, 62208, 46080, 82944, 34560, 50400, 40320, 57600, 51840, 110880, 126720, 141120, 69120, 60480, 248832, 86400, 80640, 233280, 237600, 103680, 100800

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A000005, A001222, A162247, A355030, A355031, A355032.

m[n_] := Max[Tally[Total[# - 1] & /@ f[n]][[;; , 2]]]; seq[len_, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, k}, While[c < len && n < max, k = m[n]; If[k <= len && s[[k]] == 0, c++; s[[k]] = n]; n++]; s]; seq[30, 10^6] (* using the function f by T. D. Noe at A162247 *)

cp's OEIS Frontend

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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A355032 a(n) is the maximum number of prime signatures of numbers with n divisors that have the same number of prime divisors (counted with multiplicity).

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A355033 a(n) is the least number k such that A355032(k) = n, or -1 if no such k exists.

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