A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

9, 36, 125, 225, 441, 1089, 1260, 1521, 1980, 2340, 2401, 2601, 2772, 3060, 3249, 3276, 3420, 4140, 4284, 4761, 4788, 5148, 5220, 5580, 5796, 6660, 6732, 7308, 7380, 7524, 7569, 7740, 7812, 7956, 8460, 8649, 8892, 9108, 9324, 9540, 10332, 10620, 10764, 10836

Offset

1,1

Comments

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.

Said differently, these are nonprime numbers > 1 whose prime shadow is a divisor that is either a prime number or a number already in the sequence.

Links

Table of n, a(n) for n=1..44.

Example

The initial terms and their prime indices:
     9: {2,2}
    36: {1,1,2,2}
   125: {3,3,3}
   225: {2,2,3,3}
   441: {2,2,4,4}
  1089: {2,2,5,5}
  1260: {1,1,2,2,3,4}
  1521: {2,2,6,6}
  1980: {1,1,2,2,3,5}

Mathematica

red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n, red[n]]&&suQ[red[n]];
Select[Range[2, 2000], suQ[#]&&!PrimeQ[#]&]

Crossrefs

The first term that is not a perfect power A001597 is 1260.

Without the recursion we have A325755 (a superset), counted by A325702.

Before removing the primes we had A353393.

These partitions are counted by A353426 minus one.

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.

A181819 gives prime shadow, with an inverse A181821.

A182850 and A323014 give frequency depth, counted by A225485 and A325280.

A325131 lists numbers relatively prime to their prime shadow.

Cf. A000005, A000040, A005117, A316413, A316428, A324850, A353394, A353395, A353397, A353399.

Sequence in context: A168569 A213283 A188162 * A023872 A034557 A285241

Adjacent sequences: A353386 A353387 A353388 * A353390 A353391 A353392

Keyword

nonn

Author

Gus Wiseman, May 15 2022

Status

approved