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Numbers whose product of prime indices equals their product of prime exponents (prime signature).
12

%I #10 May 20 2022 10:18:20

%S 1,2,12,36,40,112,352,832,960,1296,2176,2880,4864,5376,11776,12544,

%T 16128,29696,33792,34560,38400,63488,64000,101376,115200,143360,

%U 151552,159744,335872,479232,704512,835584,1540096,1658880,1802240

%N Numbers whose product of prime indices equals their product of prime exponents (prime signature).

%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. A number's prime signature (row n A124010) is the sequence of positive exponents in its prime factorization.

%F A003963(a(n)) = A005361(a(n)).

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 12: {1,1,2}

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

%e 112: {1,1,1,1,4}

%e 352: {1,1,1,1,1,5}

%e 832: {1,1,1,1,1,1,6}

%e 960: {1,1,1,1,1,1,2,3}

%e 1296: {1,1,1,1,2,2,2,2}

%e 2176: {1,1,1,1,1,1,1,7}

%e 2880: {1,1,1,1,1,1,2,2,3}

%e 4864: {1,1,1,1,1,1,1,1,8}

%e 5376: {1,1,1,1,1,1,1,1,2,4}

%t Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]

%o (Python)

%o from itertools import count, islice

%o from math import prod

%o from sympy import primepi, factorint

%o def A353503_gen(startvalue=1): # generator of terms >= startvalue

%o return filter(lambda n: n == 1 or prod((f:=factorint(n)).values()) == prod(primepi(p)**e for p,e in f.items()), count(max(startvalue,1)))

%o A353503_list = list(islice(A353503_gen(),20)) # _Chai Wah Wu_, May 20 2022

%Y For shadows instead of exponents we get A003586, counted by A008619.

%Y The LHS (product of prime indices) is A003963, counted by A339095.

%Y The RHS (product of prime exponents) is A005361, counted by A266477.

%Y The version for shadows instead of indices is A353399, counted by A353398.

%Y These partitions are counted by A353506.

%Y A001222 counts prime factors with multiplicity, distinct A001221.

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

%Y A130091 lists numbers with distinct prime exponents, counted by A098859.

%Y A124010 gives prime signature, sorted A118914.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A353394 gives product of shadows of prime indices, firsts A353397.

%Y Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 17 2022