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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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%I #70 Mar 25 2024 21:29:08

%S 6,30,32,36,60,210,212,213,214,216,240,420,2310,2312,2313,2314,2315,

%T 2316,2317,2318,2319,2320,2322,2324,2328,2340,2342,2343,2344,2346,

%U 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

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

%C 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.

%H Antti Karttunen, <a href="/A351228/b351228.txt">Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))</a>

%H Victor Ufnarovski and Bo Ã…hlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%o (PARI)

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

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

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

%Y Cf. A003415, A024451, A048103, A060735, A235991, A276085, A276086, A327862, A351088, A351089, A351232, A369656, A369663, A369664, A371104.

%Y Union of A370127 and A370128.

%Y Subsequence of A328118.

%Y Subsequences: A351229, A369959, A369960, A369970 (after its two initial terms).

%Y Cf. also A369650.

%K nonn,base

%O 1,1

%A _Antti Karttunen_, Feb 05 2022