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

1, 2, 12, 20, 36, 44, 56, 68, 100, 124, 164, 184, 208, 236, 240, 268, 332, 436, 464, 484, 508, 528, 608, 628, 688, 716, 720, 752, 764, 776, 816, 844, 880, 964, 1108, 1132, 1156, 1168, 1200, 1264, 1296, 1324, 1344, 1360, 1412, 1468, 1488, 1584, 1604, 1616, 1724

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: {}
     2: {1}
    12: {1,1,2}
    20: {1,1,3}
    36: {1,1,2,2}
    44: {1,1,5}
    56: {1,1,1,4}
    68: {1,1,7}
   100: {1,1,3,3}
   124: {1,1,11}
   164: {1,1,13}
   184: {1,1,1,9}
   208: {1,1,1,1,6}
   236: {1,1,17}
   240: {1,1,1,1,2,3}
		

Crossrefs

Product of prime indices is A003963, counted by A339095.

The LHS (product of exponents) is A005361, counted by A266477.

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

A related comparison is A353395, counted by A353396.

The partitions are counted by A353398.

Taking indices instead of exponents on the LHS gives A353503.

A001222 counts prime factors with multiplicity, distinct A001221.

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

A124010 gives prime signature, sorted A118914.

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

A181819 gives prime shadow, with an inverse A181821.

A325131 lists numbers relatively prime to their prime shadow.

Numbers divisible by their prime shadow:

- counted by A325702

- listed by A325755

- co-recursive version A325756

- nonprime recursive version A353389

- recursive version A353393

- recursive version counted by A353426

Cf. A000720, A003586, A005117, A143773, A182850, A316428, A320325, A324850.

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[#]==Times@@Last/@FactorInteger[#]&]

Formula

A005361(a(n)) = A353394(a(n)).