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A353395 Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.

%I A353395 #4 May 20 2022 08:51:05
%S A353395 1,3,5,11,15,17,26,31,33,41,51,55,58,59,67,78,83,85,86,93,94,109,123,
%T A353395 126,127,130,146,148,155,157,158,165,174,177,179,187,191,196,201,202,
%U A353395 205,211,241,244,249,255,258,274,277,278,282,283,284,286,290,295,298
%N A353395 Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.
%C A353395 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 A353395 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 A353395 A181819(a(n)) = A353394(a(n)) = Product_i A181819(A112798(a(n),i)).
%e A353395 The terms together with their prime indices begin:
%e A353395       1: {}         78: {1,2,6}      158: {1,22}
%e A353395       3: {2}        83: {23}         165: {2,3,5}
%e A353395       5: {3}        85: {3,7}        174: {1,2,10}
%e A353395      11: {5}        86: {1,14}       177: {2,17}
%e A353395      15: {2,3}      93: {2,11}       179: {41}
%e A353395      17: {7}        94: {1,15}       187: {5,7}
%e A353395      26: {1,6}     109: {29}         191: {43}
%e A353395      31: {11}      123: {2,13}       196: {1,1,4,4}
%e A353395      33: {2,5}     126: {1,2,2,4}    201: {2,19}
%e A353395      41: {13}      127: {31}         202: {1,26}
%e A353395      51: {2,7}     130: {1,3,6}      205: {3,13}
%e A353395      55: {3,5}     146: {1,21}       211: {47}
%e A353395      58: {1,10}    148: {1,1,12}     241: {53}
%e A353395      59: {17}      155: {3,11}       244: {1,1,18}
%e A353395      67: {19}      157: {37}         249: {2,23}
%e A353395 For example, 126 is in the sequence because its prime indices {1,2,2,4} have shadows {1,2,2,3}, with product 12, which is also the prime shadow of 126.
%t A353395 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353395 red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t A353395 Select[Range[100],Times@@red/@primeMS[#]==red[#]&]
%Y A353395 The prime terms are A006450.
%Y A353395 The LHS (prime shadow) is A181819, with an inverse A181821.
%Y A353395 The RHS (product of shadows) is A353394, first appearances A353397.
%Y A353395 This is a ranking of the partitions counted by A353396.
%Y A353395 Another related comparison is A353399, counted by A353398.
%Y A353395 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353395 A003963 gives product of prime indices.
%Y A353395 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353395 A124010 gives prime signature, sorted A118914, product A005361.
%Y A353395 A130091 lists numbers with distinct prime exponents, counted by A098859.
%Y A353395 A324850 lists numbers divisible by the product of their prime indices.
%Y A353395 Numbers divisible by their prime shadow:
%Y A353395 - counted by A325702
%Y A353395 - listed by A325755
%Y A353395 - co-recursive version A325756
%Y A353395 - nonprime recursive version A353389
%Y A353395 - recursive version A353393, counted by A353426
%Y A353395 Cf. A000005, A000961, A003586, A005117, A143773, A182850, A316428, A316438, A320325, A325131, A339095.
%K A353395 nonn
%O A353395 1,2
%A A353395 _Gus Wiseman_, May 17 2022