A182861 -id:A182861 - cp's OEIS frontend

A182860 Number of distinct prime signatures represented among the unitary divisors of n.

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 2, 4, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 4, 2, 3, 4, 4, 3, 4, 2, 4, 2, 3, 2, 6, 3, 3, 3, 4, 2, 6, 3, 4, 3, 3, 3, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

a(n) = number of members m of A025487 such that d(m^k) divides d(n^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

a(n) depends only on prime signature of n (cf. A025487).

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  Since a total of 6 distinct prime signatures appear among the unitary divisors of 60, a(60) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A034444, A064553, A085082, A146289, A146290, A182861, A182862, A212180, A328830.

Table[Length@ Union@ Map[Sort[FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 1, 0, e]] &, Select[Divisors@ n, CoprimeQ[#, n/#] &]], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)

A181819(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); }; \\ From A181819
A182860(n) = numdiv(A181819(n)); \\ Antti Karttunen, Jul 19 2017

a(n) = A000005(A181819(n)).
If the canonical factorization of n into prime powers is Product p^e(p), then the formula for d(n^k) is Product_p (ek + 1). (See also A146289, A146290.)
a(n) = A064553(A328830(n)). - Antti Karttunen, Apr 29 2022

A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 1260, 2520, 27720, 138600, 360360, 831600, 10810800, 75675600, 183783600, 1286485200, 24443218800, 38594556000, 424540116000, 733296564000, 8066262204000, 185524030692000, 1693915062840000, 5380196890068000, 38960046445320000, 166786103592108000

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

In other words, the sequence includes k iff A182860(k) > A182860(m) for all m < k.

The records for the number of distinct prime signatures are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, ... (see the link for more values). - Amiram Eldar, Jul 07 2019

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  This makes a total of 6 distinct prime signatures that appear among the unitary divisors of 60.  Since no positive integer smaller than 60 has more than 4 distinct prime signatures appearing among its unitary divisors, 60 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..60
Amiram Eldar, Table of n, a(n), A182860(a(n)) for n = 1..60
Eric Weisstein's World of Mathematics, Unitary Divisor

Subsequence of A025487, A129912, A181826, A182863. See also A034444, A085082, A182860, A182861.

f[1] = 1; f[n_] := Times @@ (Values[Counts[FactorInteger[n][[;; , 2]]]] + 1); fm = 0; s={}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Jan 19 2019 *)

a(14)-a(26) from Amiram Eldar, Jan 19 2019

cp's OEIS Frontend

A182860 Number of distinct prime signatures represented among the unitary divisors of n.

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