A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

1, 2, 12, 36, 40, 112, 352, 832, 960, 1296, 2176, 2880, 4864, 5376, 11776, 12544, 16128, 29696, 33792, 34560, 38400, 63488, 64000, 101376, 115200, 143360, 151552, 159744, 335872, 479232, 704512, 835584, 1540096, 1658880, 1802240

Offset

1,2

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. A number's prime signature (row n A124010) is the sequence of positive exponents in its prime factorization.

Links

Table of n, a(n) for n=1..35.

Formula

A003963(a(n)) = A005361(a(n)).

Example

The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   352: {1,1,1,1,1,5}
   832: {1,1,1,1,1,1,6}
   960: {1,1,1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  2176: {1,1,1,1,1,1,1,7}
  2880: {1,1,1,1,1,1,2,2,3}
  4864: {1,1,1,1,1,1,1,1,8}
  5376: {1,1,1,1,1,1,1,1,2,4}

Mathematica

Select[Range[1000], Times@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]

Prog

(Python)
from itertools import count, islice
from math import prod
from sympy import primepi, factorint
def A353503_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or prod((f:=factorint(n)).values()) == prod(primepi(p)**e for p, e in f.items()), count(max(startvalue, 1)))
A353503_list = list(islice(A353503_gen(), 20)) # Chai Wah Wu, May 20 2022

Crossrefs

For shadows instead of exponents we get A003586, counted by A008619.

The LHS (product of prime indices) is A003963, counted by A339095.

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

The version for shadows instead of indices is A353399, counted by A353398.

These partitions are counted by A353506.

A001222 counts prime factors with multiplicity, distinct A001221.

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

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

A124010 gives prime signature, sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A353394 gives product of shadows of prime indices, firsts A353397.

Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.

Sequence in context: A200543 A363453 A246426 * A176583 A011379 A338610

Adjacent sequences: A353500 A353501 A353502 * A353504 A353505 A353506

Keyword

nonn

Author

Gus Wiseman, May 17 2022

Status

approved