A365889 Numbers k whose least prime divisor divides its exponent in the prime factorization of k.

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 220, 228, 236, 240, 244, 252, 256, 260, 268, 272, 276, 284, 292, 297, 300, 304, 308

Offset

1,1

Comments

Numbers k such that A020639(k) | A067029(k). - The second A-number corrected by Antti Karttunen, Jan 16 2026.

The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = ((prime(n)-1)/(prime(n)*(prime(n)^prime(n)-1))) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4 and 5, d(n) = 1/6, 1/78, 1/11715, 4/14411985 and 8/10984499318485.

The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.17957281768342725732... .

Numbers k such that A053669(A276085(k)) is not A020639(k). - Antti Karttunen, Jan 16 2026

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

4 = 2^2 is a term since its least prime factor, 2, divides its exponent, 2.
16 = 2^4 is a term since its least prime factor, 2, divides its exponent, 4.

Mathematica

q[n_] := Divisible @@ Reverse[FactorInteger[n][[1]]]; Select[Range[2, 400], q]

Prog

(PARI) is(n) = {my(f = factor(n)); n > 1 && !(f[1, 2] % f[1, 1]); }

Crossrefs

Cf. A020639, A053669, A067029, A276085.

Subsequence of A342090, which is a subsequence of A100716.

Subsequences: A108269 (even terms), A365883, A365890, A365891.

Sequence in context: A310564 A053362 A342090 * A310565 A267958 A077770

Adjacent sequences: A365886 A365887 A365888 * A365890 A365891 A365892

Keyword

nonn,easy

Author

Amiram Eldar, Sep 22 2023

Status

approved