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A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

%I A048767 #36 Oct 08 2022 00:01:35
%S A048767 1,2,4,3,8,8,16,5,9,16,32,12,64,32,32,7,128,18,256,24,64,64,512,20,27,
%T A048767 128,25,48,1024,64,2048,11,128,256,128,27,4096,512,256,40,8192,128,
%U A048767 16384,96,72,1024,32768,28,81,54,512,192,65536,50,256,80,1024,2048
%N A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.
%C A048767 If the prime power factors p^e of n are replaced by prime(e)^pi(p), then the prime terms q in the sequence pertain to 2^m with m > 1, since pi(2) = 1. - _Michael De Vlieger_, Apr 25 2017
%C A048767 Also the Heinz number of the integer partition obtained by applying the map described in A217605 (which interchanges the parts with their multiplicities) to the integer partition with Heinz number n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The image of this map (which is the union of this sequence) is A130091. - _Gus Wiseman_, May 04 2019
%H A048767 Alois P. Heinz, <a href="/A048767/b048767.txt">Table of n, a(n) for n = 1..10000</a>
%H A048767 Stephan Wagner, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p13">The Number of Fixed Points of Wilf's Partition Involution</a>, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.
%e A048767 For n=6, 6 = (2^1)*(3^1), a(6) = ([first prime]^pi(2))*([first prime]^pi(3)) = (2^1)*(2^2) = 8.
%e A048767 From _Gus Wiseman_, May 04 2019: (Start)
%e A048767 For n = 1..20, the prime indices of n together with the prime indices of a(n) are the following:
%e A048767    1: {} {}
%e A048767    2: {1} {1}
%e A048767    3: {2} {1,1}
%e A048767    4: {1,1} {2}
%e A048767    5: {3} {1,1,1}
%e A048767    6: {1,2} {1,1,1}
%e A048767    7: {4} {1,1,1,1}
%e A048767    8: {1,1,1} {3}
%e A048767    9: {2,2} {2,2}
%e A048767   10: {1,3} {1,1,1,1}
%e A048767   11: {5} {1,1,1,1,1}
%e A048767   12: {1,1,2} {1,1,2}
%e A048767   13: {6} {1,1,1,1,1,1}
%e A048767   14: {1,4} {1,1,1,1,1}
%e A048767   15: {2,3} {1,1,1,1,1}
%e A048767   16: {1,1,1,1} {4}
%e A048767   17: {7} {1,1,1,1,1,1,1}
%e A048767   18: {1,2,2} {1,2,2}
%e A048767   19: {8} {1,1,1,1,1,1,1,1}
%e A048767   20: {1,1,3} {1,1,1,2}
%e A048767 (End)
%p A048767 A048767 := proc(n)
%p A048767     local a,p,e,f;
%p A048767     a := 1 ;
%p A048767     for f in ifactors(n)[2] do
%p A048767         p := op(1,f) ;
%p A048767         e := op(2,f) ;
%p A048767         a := a*ithprime(e)^numtheory[pi](p) ;
%p A048767     end do:
%p A048767     a ;
%p A048767 end proc: # _R. J. Mathar_, Nov 08 2012
%t A048767 Table[{p, k} = Transpose@ FactorInteger[n]; Times @@ (Prime[k]^PrimePi[p]), {n, 58}] (* _Ivan Neretin_, Jun 02 2016 *)
%t A048767 Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; e >= 0 :> Prime[e]^PrimePi[p]] &, 65] (* _Michael De Vlieger_, Apr 25 2017 *)
%Y A048767 Cf. A008477, A048768, A048769, A064988.
%Y A048767 Cf. A056239, A098859, A109298, A112798, A118914, A130091, A217605, A320348, A325326, A325337.
%K A048767 easy,nonn,mult
%O A048767 1,2
%A A048767 _Naohiro Nomoto_
%E A048767 a(1)=1 prepended by _Alois P. Heinz_, Jul 26 2015