A384994 Positive numbers whose prime indices of their prime factors, together with their exponents are distinct.

1, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 189, 191, 193, 197, 199

Offset

1,2

Comments

This sequence is infinite because it contains all the odd primes.

Links

Table of n, a(n) for n=1..60.

Example

1 is a term because it has no prime factors and thus no duplicates.
189 is a term because 189 = 3^3 * 7^1 = prime(2)^3 * prime(4)^1 and 1, 2, 3 and 4 are distinct.
9 is not a term because 9 = 3^2 = prime(2)^2 where the number 2 occurs twice.

Mathematica

q[k_] := Module[{f = FactorInteger[k]}, UnsameQ @@ Join[PrimePi[f[[;; , 1]]], f[[;; , 2]]]]; Select[Range[200], q] (* Amiram Eldar, Sep 08 2025 *)

Prog

(Python)
from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return len(L := [primepi(p) for p in f] + list(f.values())) == len(set(L))
print([k for k in range(1, 200) if ok(k)])
(PARI) isok(k) = my(f=factor(k), v=concat(apply(primepi, f[, 1]), f[, 2])); #v == #Set(v); \\ Michel Marcus, Sep 08 2025

Crossrefs

Subsequence of A130091.

Cf. A000040, A049084.

Sequence in context: A108372 A325131 A270342 * A066542 A003310 A038525

Adjacent sequences: A384991 A384992 A384993 * A384995 A384996 A384997

Keyword

easy,nonn

Author

Kalle Siukola, Sep 07 2025

Status

approved