A371104 -id:A371104 - 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));

A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

Original entry on oeis.org

2, 9, 26, 122, 1382, 21446, 204566, 9699686, 90387605

Offset: 1

Antti Karttunen, Jun 22 2024

For n > 1, the index of the next term in A373849, after its sixth term 0, that is a multiple of A002110(n), as for n >= 1, the smallest k such that A003415(k) == A276086(k) mod A002110(n) gives the sequence 1, 6, 6, 6, 6, 6, 6, 6, ..., because A003415(6) = A276086(6).
Provided that such k exists for every n (and the escape clause is not needed), then the sequence is by necessity monotonic. If it is strictly monotonic, then it implies that k=6 is the only k such that A003415(k) = A276086(k). See also comments in A351228.
Note that if we instead search for the smallest k such that A276086(k) is a multiple of A002110(n) we obtain A143293, partial sums of the primorial numbers. See also A368703.

Cf. A002110, A003415, A143293, A276086, A373849.

Cf. also A351228, A368703, A371104, A373848.

A002110(n) = prod(i=1,n,prime(i));
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); };
A373603(n) = { my(m=A002110(n), c=2); for(i=1,oo,if(0==((A276086(i)-A003415(i))%m), c--; if(0==c, return(i)))); };

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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