A370128 -id:A370128 - cp's OEIS frontend

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

Offset: 1

Antti Karttunen, Feb 05 2022

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.

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

Cf. A003415, A024451, A048103, A060735, A235991, A276085, A276086, A327862, A351088, A351089, A351232, A369656, A369663, A369664, A371104.
Union of A370127 and A370128.
Subsequence of A328118.
Subsequences: A351229, A369959, A369960, A369970 (after its two initial terms).
Cf. also A369650.

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));

A370127 Numbers k such that (A276086(k)/s)^s < k^(s-1), where A276086 is the primorial base exp-function, and s = bigomega(k).

Original entry on oeis.org

30, 32, 36, 60, 210, 212, 216, 240, 420, 2310, 2312, 2313, 2314, 2316, 2318, 2320, 2322, 2324, 2328, 2340, 2344, 2346, 2352, 2370, 2376, 2400, 2520, 2522, 2528, 2550, 2730, 4620, 4624, 4626, 4632, 4650, 4656, 4680, 4830, 4832, 4860, 6930, 30030, 30031, 30032, 30033, 30034, 30035, 30036, 30037, 30038, 30039, 30040

Offset: 1

Antti Karttunen, Feb 22 2024

nonn

Numbers k such that A276086(k) < s * k^((s-1)/s), with s = A001222(k).

For these numbers it must hold that A276086(k) < A003415(k) because (A003415(k)/s)^s >= k^(s-1) [with s = A001222(k)] holds for all k >= 2. See Ufnarovski and Åhlander, Theorem 9, point (4). In other words, this is a subsequence of A351228 \ {6}.

Antti Karttunen, Table of n, a(n) for n = 1..5204
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Setwise difference A351228 \ A370128.
Cf. A001222, A003415, A276086.
Cf. A066576 (subsequence).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA370127(n) = { my(x=A276086(n), s=bigomega(n)); ((x/s)^s < n^(s-1)); };

cp's OEIS Frontend

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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