cp's OEIS Frontend

A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

%I A353503 #10 May 20 2022 10:18:20
%S A353503 1,2,12,36,40,112,352,832,960,1296,2176,2880,4864,5376,11776,12544,
%T A353503 16128,29696,33792,34560,38400,63488,64000,101376,115200,143360,
%U A353503 151552,159744,335872,479232,704512,835584,1540096,1658880,1802240
%N A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).
%C A353503 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 A353503 A003963(a(n)) = A005361(a(n)).
%e A353503 The terms together with their prime indices begin:
%e A353503      1: {}
%e A353503      2: {1}
%e A353503     12: {1,1,2}
%e A353503     36: {1,1,2,2}
%e A353503     40: {1,1,1,3}
%e A353503    112: {1,1,1,1,4}
%e A353503    352: {1,1,1,1,1,5}
%e A353503    832: {1,1,1,1,1,1,6}
%e A353503    960: {1,1,1,1,1,1,2,3}
%e A353503   1296: {1,1,1,1,2,2,2,2}
%e A353503   2176: {1,1,1,1,1,1,1,7}
%e A353503   2880: {1,1,1,1,1,1,2,2,3}
%e A353503   4864: {1,1,1,1,1,1,1,1,8}
%e A353503   5376: {1,1,1,1,1,1,1,1,2,4}
%t A353503 Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]
%o A353503 (Python)
%o A353503 from itertools import count, islice
%o A353503 from math import prod
%o A353503 from sympy import primepi, factorint
%o A353503 def A353503_gen(startvalue=1): # generator of terms >= startvalue
%o A353503     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 A353503_list = list(islice(A353503_gen(),20)) # _Chai Wah Wu_, May 20 2022
%Y A353503 For shadows instead of exponents we get A003586, counted by A008619.
%Y A353503 The LHS (product of prime indices) is A003963, counted by A339095.
%Y A353503 The RHS (product of prime exponents) is A005361, counted by A266477.
%Y A353503 The version for shadows instead of indices is A353399, counted by A353398.
%Y A353503 These partitions are counted by A353506.
%Y A353503 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353503 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353503 A130091 lists numbers with distinct prime exponents, counted by A098859.
%Y A353503 A124010 gives prime signature, sorted A118914.
%Y A353503 A181819 gives prime shadow, with an inverse A181821.
%Y A353503 A353394 gives product of shadows of prime indices, firsts A353397.
%Y A353503 Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.
%K A353503 nonn
%O A353503 1,2
%A A353503 _Gus Wiseman_, May 17 2022