cp's OEIS Frontend

A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

%I A353399 #5 May 20 2022 08:51:18
%S A353399 1,2,12,20,36,44,56,68,100,124,164,184,208,236,240,268,332,436,464,
%T A353399 484,508,528,608,628,688,716,720,752,764,776,816,844,880,964,1108,
%U A353399 1132,1156,1168,1200,1264,1296,1324,1344,1360,1412,1468,1488,1584,1604,1616,1724
%N A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.
%C A353399 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.
%C A353399 We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
%F A353399 A005361(a(n)) = A353394(a(n)).
%e A353399 The terms together with their prime indices begin:
%e A353399      1: {}
%e A353399      2: {1}
%e A353399     12: {1,1,2}
%e A353399     20: {1,1,3}
%e A353399     36: {1,1,2,2}
%e A353399     44: {1,1,5}
%e A353399     56: {1,1,1,4}
%e A353399     68: {1,1,7}
%e A353399    100: {1,1,3,3}
%e A353399    124: {1,1,11}
%e A353399    164: {1,1,13}
%e A353399    184: {1,1,1,9}
%e A353399    208: {1,1,1,1,6}
%e A353399    236: {1,1,17}
%e A353399    240: {1,1,1,1,2,3}
%t A353399 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353399 red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t A353399 Select[Range[100],Times@@red/@primeMS[#]==Times@@Last/@FactorInteger[#]&]
%Y A353399 Product of prime indices is A003963, counted by A339095.
%Y A353399 The LHS (product of exponents) is A005361, counted by A266477.
%Y A353399 The RHS (product of shadows) is A353394, first appearances A353397.
%Y A353399 A related comparison is A353395, counted by A353396.
%Y A353399 The partitions are counted by A353398.
%Y A353399 Taking indices instead of exponents on the LHS gives A353503.
%Y A353399 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353399 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353399 A124010 gives prime signature, sorted A118914.
%Y A353399 A130091 lists numbers with distinct prime exponents, counted by A098859.
%Y A353399 A181819 gives prime shadow, with an inverse A181821.
%Y A353399 A325131 lists numbers relatively prime to their prime shadow.
%Y A353399 Numbers divisible by their prime shadow:
%Y A353399 - counted by A325702
%Y A353399 - listed by A325755
%Y A353399 - co-recursive version A325756
%Y A353399 - nonprime recursive version A353389
%Y A353399 - recursive version A353393
%Y A353399 - recursive version counted by A353426
%Y A353399 Cf. A000720, A003586, A005117, A143773, A182850, A316428, A320325, A324850.
%K A353399 nonn
%O A353399 1,2
%A A353399 _Gus Wiseman_, May 17 2022