A329382 - OEIS

A329382

Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 4, 1, 6, 1, 3, 4, 2, 1, 4, 8, 2, 9, 3, 1, 6, 1, 5, 4, 2, 8, 8, 1, 2, 4, 4, 1, 6, 1, 3, 9, 2, 1, 5, 16, 12, 4, 3, 1, 12, 8, 4, 4, 2, 1, 8, 1, 2, 9, 6, 8, 6, 1, 3, 4, 12, 1, 10, 1, 2, 18, 3, 16, 6, 1, 5, 16, 2, 1, 8, 8, 2, 4, 4, 1, 12, 16, 3, 4, 2, 8, 6, 1, 24, 9, 16, 1, 6, 1, 4, 18

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OFFSET

1,4

COMMENTS

Also the product of parts of the conjugate of the integer partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (3,2) with Heinz number 15 has conjugate (2,2,1) with product a(15) = 4. - Gus Wiseman, Mar 27 2022

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10201

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to primorial numbers

FORMULA

a(n) = A005361(A108951(n)).

A329605(n) >= a(n) >= A329617(n) >= A329378(n).

a(A019565(n)) = A284001(n).

From Antti Karttunen, Jan 14 2020: (Start)

If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > k3 > ... > kx, then a(n) = e(k1)^(k1-k2) * (e(k1)+e(k2))^(k2-k3) * (e(k1)+e(k2)+e(k3))^(k3-k4) * ... * (e(k1)+e(k2)+...+e(kx))^kx.

a(n) = A000005(A331188(n)) = A329605(A052126(n)).

(End)

a(n) = A003963(A122111(n)). - Gus Wiseman, Mar 27 2022

MATHEMATICA

Table[Times @@ FactorInteger[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]][[All, -1]], {n, 105}] (* Michael De Vlieger, Jan 21 2020 *)

PROG

(PARI)

A005361(n) = factorback(factor(n)[, 2]); \\ from A005361

A034386(n) = prod(i=1, primepi(n), prime(i));

A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951

A329382(n) = A005361(A108951(n));

(PARI) A329382(n) = if(1==n, 1, my(f=factor(n), e=0, m=1); forstep(i=#f~, 1, -1, e += f[i, 2]; m *= e^(primepi(f[i, 1])-if(1==i, 0, primepi(f[i-1, 1])))); (m)); \\ Antti Karttunen, Jan 14 2020

CROSSREFS

Cf. A000005, A005361, A019565, A034386, A108951, A284001, A331188, A329378, A329605, A329617.

This is the conjugate version of A003963 (product of prime indices).

The solutions to a(n) = A003963(n) are A325040, counted by A325039.

The Heinz number of the conjugate partition is given by A122111.

These are the row products of A321649 and of A321650.

A000700 counts self-conj partitions, ranked by A088902, complement A330644.

A008480 counts permutations of prime indices, conjugate A321648.

A056239 adds up prime indices, row sums of A112798 and of A296150.

A124010 gives prime signature, sorted A118914, sum A001222.

A238744 gives the conjugate of prime signature, rank A238745.