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A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

%I A357983 #11 Oct 04 2024 08:51:29
%S A357983 1,2,5,4,11,10,23,8,25,22,31,20,47,46,55,16,59,50,103,44,115,62,97,40,
%T A357983 121,94,125,92,137,110,127,32,155,118,253,100,197,206,235,88,179,230,
%U A357983 233,124,275,194,257,80,529,242,295,188,419,250,341,184,515,274
%N A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.
%C A357983 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 shifting a number's prime indices along a function; see the Mathematica program.
%e A357983 First, we have
%e A357983 - 4 = prime(1) * prime(1),
%e A357983 - A000040(1) = 2,
%e A357983 - A064988(4) = prime(2) * prime(2) = 9.
%e A357983 Similarly, A064988(3) = 5. Next,
%e A357983 - 35 = prime(3) * prime(4),
%e A357983 - A064988(3) = 5,
%e A357983 - A064988(4) = 9,
%e A357983 - a(35) = prime(5) * prime(9) = 253.
%t A357983 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357983 mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
%t A357983 Array[mtf[mtf[Prime]],100]
%Y A357983 Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.
%Y A357983 Applying the transformation only once gives A064988.
%Y A357983 The union is A076610 (numbers whose prime indices are themselves prime).
%Y A357983 For partition numbers instead of primes we have A357979.
%Y A357983 A000040 lists the primes.
%Y A357983 A056239 adds up prime indices, row-sums of A112798.
%Y A357983 Cf. A000720, A003964, A063834, A215366, A296150, A299201, A299202, A357975.
%K A357983 nonn,mult
%O A357983 1,2
%A A357983 _Gus Wiseman_, Oct 24 2022