A351229 -id:A351229 - 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));

A351089 Number of iterations of map x -> A003415(x) needed to reach a number >= A276086(n), when starting from x = n, or -1 if such number is never reached.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, 0, -1, 2, -1, -1, -1, 2, -1, -1, 6, 4, -1, -1, -1, 6, -1, -1, -1, 6, -1, 11, -1, 8, -1, 0, 0, 0, -1, -1, 5, 0, -1, -1, 5, 7, -1, -1, -1, 4, 8, -1, -1, 4, -1, 10, 10, 8, -1, 7, 10, 8, -1, -1, -1, 0, -1, -1, 8, 2, -1, -1, -1, 6, 11, -1, -1, 6, -1, 10, 10, 8, -1, -1, -1, 7, 9, -1, -1, 7, -1, 14, 11, 9

Offset: 0

Antti Karttunen, Feb 05 2022

			a(0) = -1 because A003415^(k)(0) = 0 for all values of k >= 0 (i.e., regardless of how many times we apply the arithmetic derivative), and 0 < A276086(0) = 1.
a(1) = -1 because A003415^(k)(1) = 0 for all values of k >= 1, and both 1 and 0 are less than A276086(1) = 2.
a(4) = -1 because A003415^(k)(4) = 4 for all values of k >= 0 (i.e., regardless of how many times we apply the arithmetic derivative), and 4 < A276086(4) = 9.
a(6) = 0 because 6 is already >= A276086(6) = 5 before any iterations.
a(8) = 2 because it takes two iterations with A003415 as 8 -> 12 -> 16 to obtain a number >= A276086(8) = 15.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A349908 (positions of records), A351226 (positions of zeros), A351229 (positions of ones).

Cf. also A351088.

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); };
A351089(n) = { my(u=A276086(n),i=0,prev_n=-1); while(n>0, if(n>=u, return(i)); prev_n = n; n = A003415(n); if(n==prev_n, return(-1)); i++); (-1); };

A351227 Numbers k for which A276086(k) > k, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Antti Karttunen, Feb 05 2022

Index entries for sequences related to primorial base

Cf. A276086, A351226 (complement), A351229 (subsequence).

Indices of positive terms in A351225.

Select[Range[0, 87], Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m > #] &] (* Michael De Vlieger, Feb 05 2022 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351227(n) = (A276086(n)>n);

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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A351089 Number of iterations of map x -> A003415(x) needed to reach a number >= A276086(n), when starting from x = n, or -1 if such number is never reached.

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A351227 Numbers k for which A276086(k) > k, where A276086 is the primorial base exp-function.

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