A369664 -id:A369664 - 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));

A369662 Numbers k whose arithmetic derivative k' is of the form 4m+2, and k' has an odd number of prime factors.

Original entry on oeis.org

65, 77, 135, 141, 161, 185, 201, 209, 221, 301, 305, 315, 321, 341, 351, 365, 377, 381, 413, 437, 453, 481, 485, 495, 497, 501, 537, 545, 589, 649, 681, 689, 717, 721, 729, 735, 737, 745, 749, 785, 789, 849, 855, 893, 901, 905, 917, 921, 975, 989, 999, 1035, 1037, 1073, 1081, 1101, 1121, 1133, 1141, 1157, 1165

Offset: 1

Antti Karttunen, Feb 06 2024

nonn

Equally, numbers k whose arithmetic derivative k' is congruent to 2 modulo 4 and A276085(k') is congruent to 1 modulo 4.

A003415((1/2)*A003415(a(n))) is always even.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Setwise difference A327862 \ A369661.
Cf. A003415, A276085.
Subsequences: A369664 (terms of the form 4m+1).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA369662(n) = { my(d=A003415(n)); (2==(d%4) && (bigomega(d)%2)); };

A369666 Numbers k > 1 for which A276085(A003415(k)) == k (mod 4), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

Original entry on oeis.org

6, 8, 10, 12, 15, 22, 24, 30, 34, 40, 42, 50, 56, 58, 60, 64, 65, 66, 70, 77, 78, 82, 84, 86, 104, 112, 114, 118, 120, 122, 126, 128, 130, 132, 136, 140, 141, 142, 146, 152, 154, 161, 168, 174, 180, 182, 184, 185, 188, 189, 194, 196, 201, 202, 204, 206, 209, 214, 220, 221, 222, 228, 230, 232, 236, 238, 242, 246, 250

Offset: 1

Antti Karttunen, Feb 06 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A276085, A369665 (characteristic function).
Cf. A369663 and A369664 (subsequences).
Cf. also A351228.

PARI
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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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A369662 Numbers k whose arithmetic derivative k' is of the form 4m+2, and k' has an odd number of prime factors.

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A369666 Numbers k > 1 for which A276085(A003415(k)) == k (mod 4), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

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