%I #44 Jul 08 2022 21:53:45

%S 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,

%T 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,

%U 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

%N 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).

%C 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

%H Antti Karttunen, <a href="/A329382/b329382.txt">Table of n, a(n) for n = 1..10201</a>

%H Antti Karttunen, <a href="/A329382/a329382.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

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

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

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

%F From _Antti Karttunen_, Jan 14 2020: (Start)

%F 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.

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

%F (End)

%F a(n) = A003963(A122111(n)). - _Gus Wiseman_, Mar 27 2022

%t 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 *)

%o (PARI)

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

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

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

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

%o (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

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

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

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

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

%Y These are the row products of A321649 and of A321650.

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

%Y A008480 counts permutations of prime indices, conjugate A321648.

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

%Y A124010 gives prime signature, sorted A118914, sum A001222.

%Y A238744 gives the conjugate of prime signature, rank A238745.

%Y Cf. A000701, A000720, A001221, A046682, A175508, A290822, A303975, A316524, A324850, A352486-A352491, A353570.

%K nonn

%O 1,4

%A _Antti Karttunen_, Nov 17 2019