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%I #7 Oct 25 2022 07:34:47
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,31,18,19,20,21,22,23,24,25,26,
%T 27,28,29,30,59,32,33,62,35,36,37,38,39,40,127,42,79,44,45,46,47,48,
%U 49,50,93,52,53,54,55,56,57,58,211,60,61,118,63,64,65,66
%N Second MTF-transform of A000041. Replace prime(k) with prime(A357977(k)) in the prime factorization of n.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define the MTF-transform as applying a function horizontally along a number's prime indices; see the Mathematica program.
%e We have:
%e - 51 = prime(2) * prime(7),
%e - A357977(2) = 2,
%e - A357977(7) = 11,
%e - a(51) = prime(2) * prime(11) = 93.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
%t Array[mtf[mtf[PartitionsP]],100]
%Y Other multiplicative sequences: A003961, A045966, A064988, A064989, A357980.
%Y Applying the transformation only once gives A357977, strict A357978.
%Y For primes instead of partition numbers we have A357983.
%Y A000040 lists the primes.
%Y A056239 adds up prime indices, row-sums of A112798.
%Y Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975.
%K nonn,mult
%O 1,2
%A _Gus Wiseman_, Oct 24 2022
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