A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227, 229, 233, 239, 241, 251

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

Equals A353389 U A000040.

Example

The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   36: {1,1,2,2}

Maple

pshadow:= proc(n) local F, i;
  F:= ifactors(n)[2];
  mul(ithprime(i), i=F[.., 2])
end proc:
filter:= proc(n) local s;
  if isprime(n) then return true fi;
  s:= pshadow(n);
  n mod s = 0 and member(s, R)
end proc:
R:= {}:
for i from 2 to 2000 do if filter(i) then R:= R union {i} fi od:
sort(convert(R, list)); # Robert Israel, Mar 10 2025

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, 200], suQ[#]&]

Crossrefs

The first term that is not a prime power A000961 is 36.

The first term that is not a prime or a perfect power A001597 is 1260. - Corrected by Robert Israel, Mar 10 2025

The non-recursive version is A325755, counted by A325702.

Removing all primes gives A353389.

These partitions are counted by A353426.

The version for compositions is A353431.

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.

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

A181819 gives prime shadow, with an inverse A181821.

A325131 lists numbers relatively prime to their prime shadow.

Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

Sequence in context: A095415 A326149 A301987 * A316857 A318589 A132630

Adjacent sequences: A353390 A353391 A353392 * A353394 A353395 A353396

Keyword

nonn

Author

Gus Wiseman, May 15 2022

Status

approved