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Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.
7

%I #4 May 20 2022 08:51:05

%S 1,3,5,11,15,17,26,31,33,41,51,55,58,59,67,78,83,85,86,93,94,109,123,

%T 126,127,130,146,148,155,157,158,165,174,177,179,187,191,196,201,202,

%U 205,211,241,244,249,255,258,274,277,278,282,283,284,286,290,295,298

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

%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.

%C 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 A181819(a(n)) = A353394(a(n)) = Product_i A181819(A112798(a(n),i)).

%e The terms together with their prime indices begin:

%e 1: {} 78: {1,2,6} 158: {1,22}

%e 3: {2} 83: {23} 165: {2,3,5}

%e 5: {3} 85: {3,7} 174: {1,2,10}

%e 11: {5} 86: {1,14} 177: {2,17}

%e 15: {2,3} 93: {2,11} 179: {41}

%e 17: {7} 94: {1,15} 187: {5,7}

%e 26: {1,6} 109: {29} 191: {43}

%e 31: {11} 123: {2,13} 196: {1,1,4,4}

%e 33: {2,5} 126: {1,2,2,4} 201: {2,19}

%e 41: {13} 127: {31} 202: {1,26}

%e 51: {2,7} 130: {1,3,6} 205: {3,13}

%e 55: {3,5} 146: {1,21} 211: {47}

%e 58: {1,10} 148: {1,1,12} 241: {53}

%e 59: {17} 155: {3,11} 244: {1,1,18}

%e 67: {19} 157: {37} 249: {2,23}

%e 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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];

%t Select[Range[100],Times@@red/@primeMS[#]==red[#]&]

%Y The prime terms are A006450.

%Y The LHS (prime shadow) is A181819, with an inverse A181821.

%Y The RHS (product of shadows) is A353394, first appearances A353397.

%Y This is a ranking of the partitions counted by A353396.

%Y Another related comparison is A353399, counted by A353398.

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

%Y A003963 gives product of prime indices.

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

%Y A124010 gives prime signature, sorted A118914, product A005361.

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

%Y A324850 lists numbers divisible by the product of their prime indices.

%Y Numbers divisible by their prime shadow:

%Y - counted by A325702

%Y - listed by A325755

%Y - co-recursive version A325756

%Y - nonprime recursive version A353389

%Y - recursive version A353393, counted by A353426

%Y Cf. A000005, A000961, A003586, A005117, A143773, A182850, A316428, A316438, A320325, A325131, A339095.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 17 2022