A369663 -id:A369663 - 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));

A369661 Numbers k whose arithmetic derivative k' is of the form 4m+2, and k' has an even number of prime factors.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 69, 85, 93, 121, 129, 133, 145, 169, 177, 205, 213, 217, 237, 249, 253, 265, 289, 309, 329, 361, 375, 393, 417, 445, 459, 469, 473, 489, 493, 505, 517, 529, 533, 553, 565, 573, 581, 597, 629, 633, 669, 685, 697, 713, 753, 781, 783, 793, 813, 817, 819, 841, 865, 869, 875, 889, 913, 933, 949, 961

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 3 modulo 4.
Numbers k such that A003415(k) is in A369966.
For all n >= 1, A003415((1/2)*A003415(a(n))) is odd.

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

Setwise difference A327862 \ A369662.
Cf. A003415, A369660 (characteristic function), A369966.
Subsequences: A108181, A369663 (terms of the form 4m+3).

PARI
```
\\ See A369660.
```

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
```
\\ See A369665.
```

A369668 Numbers that have an even number of prime factors with multiplicity and whose arithmetic derivative is of the form 4k+3.

Original entry on oeis.org

10, 26, 34, 58, 74, 82, 90, 106, 122, 146, 178, 194, 202, 210, 218, 226, 234, 250, 274, 298, 306, 314, 330, 346, 362, 386, 394, 458, 466, 482, 490, 514, 522, 538, 546, 554, 562, 570, 586, 626, 634, 650, 666, 674, 690, 698, 706, 714, 738, 746, 770, 778, 794, 802, 810, 818, 842, 850, 858, 866, 898, 914, 922, 930, 954

Offset: 1

Antti Karttunen, Feb 08 2024

nonn

All terms are of the form 4m+2 because if n has an even number of odd prime factors, then its arithmetic derivative A003415(n) is even. See A235991 and A235992.

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

Intersection of A028260 and A358774.
Intersection of A358774 and A369966.
Cf. A001222, A003415, A235991, A235992, A369667 (characteristic function).
Cf. also A369663.

PARI
```
\\ See A369667.
```

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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A369661 Numbers k whose arithmetic derivative k' is of the form 4m+2, and k' has an even 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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PARI

A369668 Numbers that have an even number of prime factors with multiplicity and whose arithmetic derivative is of the form 4k+3.

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PARI