A351228 - OEIS

A351228

Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

6, 30, 32, 36, 60, 210, 212, 213, 214, 216, 240, 420, 2310, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2322, 2324, 2328, 2340, 2342, 2343, 2344, 2346, 2348, 2349, 2352, 2370, 2372, 2376, 2400, 2520, 2522, 2523, 2524, 2526, 2528, 2550, 2552, 2730, 4620, 4622, 4623, 4624, 4626, 4628, 4632, 4650, 4652, 4656

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OFFSET

1,1

COMMENTS

Conjecture: Apart from the initial 6, the rest of terms are the numbers k for which A003415(k) > A276086(k), thus giving the positions of zeros in A351232. In other words, it seems that only k=6 satisfies A003415(k) = A276086(k). See also comments in A351088.

From Antti Karttunen, Jan 15 2026: (Start)

Sequence A276086(A391938(.)) sorted into ascending order, thus, if the above conjecture is correct, then apart from the initial 6 this is the complement of (A000040 U A391939).

Note how the terms of this sequence seem to be rare, while those of A391938 seem to be common.

Compare also to the pair A391939 - A373848.

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))

Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Index entries for sequences related to primorial base

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

isA351228(n) = (A003415(n)>=A276086(n));