A328321 Numbers n for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is strictly positive.

4, 6, 10, 12, 14, 15, 16, 20, 21, 22, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 48, 50, 51, 52, 54, 55, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 91, 92, 93, 94, 95, 99, 100, 102, 105, 106, 108, 110, 111, 112, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 132, 133

Offset: 1

Author

Antti Karttunen, Oct 13 2019

Keywords

• nonn

Comments

Numbers n for which A051903(A003415(n)) >= A051903(n), i.e., numbers such that taking their arithmetic derivative does not decrease their "degree", A051903, the maximal exponent in prime factorization.

Examples

			10 = 2*5 has maximal exponent (A051903) 1, and its arithmetic derivative A003415(10) = 2+5 = 7 also has maximal exponent 1, thus 10 is included in this sequence.
15 = 3*5 has maximal exponent 1, and its arithmetic derivative A003415(15) = 3+5 = 8 = 2^3 has maximal exponent 3, thus 15 is included in this sequence.
For 8 = 2^3, its arithmetic derivative A003415(8) = 12 = 2^2 * 3, and as 2 < 3 (highest exponent of 12 is less than that of 8), 8 is NOT included here, and from this we also see that A100716 is not a subsequence of this sequence.
		

Links

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

Crossrefs

Cf. A003415, A051903, A100716, A328302, A328310, A328311.

Cf. A328320 (complement), A051674, A157037, A328304, A328305 (subsequences).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
isA328321(n) = (A328311(n)>0);