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

1, 3, 5, 11, 15, 17, 26, 31, 33, 41, 51, 55, 58, 59, 67, 78, 83, 85, 86, 93, 94, 109, 123, 126, 127, 130, 146, 148, 155, 157, 158, 165, 174, 177, 179, 187, 191, 196, 201, 202, 205, 211, 241, 244, 249, 255, 258, 274, 277, 278, 282, 283, 284, 286, 290, 295, 298

Offset: 1

Author

Gus Wiseman, May 17 2022

Keywords

• nonn

Comments

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.

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.

Examples

			The terms together with their prime indices begin:
      1: {}         78: {1,2,6}      158: {1,22}
      3: {2}        83: {23}         165: {2,3,5}
      5: {3}        85: {3,7}        174: {1,2,10}
     11: {5}        86: {1,14}       177: {2,17}
     15: {2,3}      93: {2,11}       179: {41}
     17: {7}        94: {1,15}       187: {5,7}
     26: {1,6}     109: {29}         191: {43}
     31: {11}      123: {2,13}       196: {1,1,4,4}
     33: {2,5}     126: {1,2,2,4}    201: {2,19}
     41: {13}      127: {31}         202: {1,26}
     51: {2,7}     130: {1,3,6}      205: {3,13}
     55: {3,5}     146: {1,21}       211: {47}
     58: {1,10}    148: {1,1,12}     241: {53}
     59: {17}      155: {3,11}       244: {1,1,18}
     67: {19}      157: {37}         249: {2,23}
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.
		

Crossrefs

The prime terms are A006450.

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

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

This is a ranking of the partitions counted by A353396.

Another related comparison is A353399, counted by A353398.

A001222 counts prime factors with multiplicity, distinct A001221.

A003963 gives product of prime indices.

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

A124010 gives prime signature, sorted A118914, product A005361.

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

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

Numbers divisible by their prime shadow:

- counted by A325702

- listed by A325755

- co-recursive version A325756

- nonprime recursive version A353389

- recursive version A353393, counted by A353426

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Times@@red/@primeMS[#]==red[#]&]

Formula

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