A391845 Numbers k such that the least prime not dividing the arithmetic derivative of k is equal to the least prime dividing k.

1, 2, 6, 10, 14, 15, 18, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 50, 51, 54, 57, 58, 62, 65, 66, 69, 70, 74, 78, 82, 86, 87, 90, 93, 94, 95, 98, 102, 106, 110, 111, 114, 118, 122, 123, 126, 129, 130, 134, 138, 141, 142, 146, 150, 154, 155, 158, 159, 161, 162, 166, 170, 174, 177, 178, 182, 183, 185, 186, 190

Offset

1,2

Comments

After the two initial terms, this is a subsequence of A080364. The first terms that appear in A080364 but not here are: 55, 75, 77, 85, 91, 105, ...

Note that the intersection of (A048103\{1} and A373488) is also certainly a subsequence of A080364. See comments in A373487. - Antti Karttunen, Jan 28 2026

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

{k such that A053669(A003415(k)) == A020639(k)}.

Example

For 1, 1' = A003415(1) = 0, and there are no prime that would not divide 0, as there are no prime that would divide 1, therefore we include 1 in this sequence as a special case.
6 = 2*3 is included because the smallest prime that does not divide 6' = 2+3 = 5 is 2, which is also the smallest prime dividing 6.
15 = 3*5 is included because the smallest prime that does not divide 15' = 3+5 = 8 is 3, which is also the smallest prime dividing 15.
5005 = 5*7*11*13 is included because the smallest prime that does not divide 5005' = 2556 = 2^2 * 3^2 * 71 is 5, which is also the smallest prime dividing 5005.

Prog

(PARI) is_A391845 = A391844;

Crossrefs

Cf. A003415, A020639, A053669, A391844 (characteristic function).

Subsequence of A080364 (apart from 1 and 2).

Subsequence of A042968.

Subsequences: A100484\{4}, {intersection of A048103 and A369650}, A392592, A392868.

Cf. also A048103, A370125, A373487, A373488.

Sequence in context: A095270 A364261 A263827 * A281974 A080324 A332951

Adjacent sequences: A391842 A391843 A391844 * A391846 A391847 A391848

Keyword

nonn

Author

Antti Karttunen, Jan 16 2026

Status

approved