A353394 Product of prime shadows of prime indices of n (with multiplicity).

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset

1,3

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.

Links

Table of n, a(n) for n=1..86.

Formula

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

Positions where a(n) = A003963(n) are A003586.

Positions where a(n) = A005361(n) are A353399, counted by A353398.

Positions where a(n) = A181819(n) are A353395, counted by A353396.

Example

We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.

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]];
Table[Times@@red/@primeMS[n], {n, 100}]

Crossrefs

Positions of first appearances are A353397.

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.

A181819 gives prime shadow, with an inverse A181821.

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

A325131 lists numbers relatively prime to their prime shadow.

A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.

Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.

Sequence in context: A103284 A071287 A355731 * A349494 A072084 A359587

Adjacent sequences: A353391 A353392 A353393 * A353395 A353396 A353397

Keyword

nonn

Author

Gus Wiseman, May 17 2022

Status

approved