A369650 -id:A369650 - cp's OEIS frontend

A327859 a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

1, 1, 2, 2, 9, 2, 18, 2, 25, 5, 10, 2, 225, 2, 30, 15, 21, 2, 750, 2, 625, 45, 50, 2, 525, 45, 150, 3750, 21, 2, 14, 2, 18375, 75, 250, 25, 49, 2, 750, 225, 735, 2, 630, 2, 875, 210, 1250, 2, 385875, 75, 1050, 375, 13125, 2, 36750, 225, 1029, 1125, 14, 2, 1029, 2, 42, 5250, 2941225, 125, 98, 2, 1225, 1875, 78750

Offset: 0

Antti Karttunen, Sep 30 2019

Sequence contains only terms of A048103.
Are there fixed points other than 1, 2, 10, 15, 5005? (There are none in the range 5006 .. 402653184.) See A369650.
Records occur at n = 0, 2, 4, 6, 8, 12, 18, 27, 32, 48, 64, 80, 144, 224, 256, 336, 448, 480, 512, 1728, ... (see also A131117).
a(n) and n are never multiples of 9 at the same time, thus the fixed points certainly exclude any terms of A008591. For a proof, consider my comment in A047257 and that A003415(9*n) is always a multiple of 3. - Antti Karttunen, Feb 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A008591, A048103, A131117, A276086, A327858, A327860, A341517 [= mu(a(n))], A341518 (k where a(k) is squarefree), A369641 (composite k where a(k) is squarefree), A369642.

Cf. A370114 (where a(k) is a multiple of k), A370115 (where k is a multiple of a(k)), 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); };
A327859(n) = A276086(A003415(n));

a(n) = A276086(A003415(n)).

a(p) = 2 for all primes p.

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

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));

A328110 Fixed points of A327860: numbers k for which A003415(A276086(k)) = k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 7, 8, 2556

Offset: 1

Antti Karttunen, Oct 08 2019

Applying A276086 to these terms gives the fixed points of A327859: 1, 2, 10, 15, 5005, ..., i.e., A369650 without any of the terms of A100716.
No more terms below <= 2550136832.
From Antti Karttunen, Feb 09 2024: (Start)
The known five terms are all members of A276156, which is equal to the claim that the intersection of A048103 and A369650 is squarefree. See the example, and also comments in A351088 and in A380527.
Even terms here must be multiples of 4, see comment in A327860.
No terms of A047257 may occur in this sequence, which is equal to the claim that A276086(a(n)) is never a multiple of 9. See comment in A327859.
(End)

			Computing A327860(2556) is easy, because it is a member of A276156, as 2556 = 6 + 30 + 210 + 2310. Therefore A327860(2556) = A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556, and 2556 is included in this sequence. - _Antti Karttunen_, Feb 04 2024

Cf. A002110, A002620, A003415, A047257, A048103, A100716, A276085, A276086, A327859, A327860, A369650.
After 0, the intersection of A351087 and A380527, thus like the latter, also this is conjectured to be a subsequence of A276156.
After two initial terms (0 & 1), a subsequence of A328118. Subsequence of A351088.

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA328110(n) = (A327860(n) == n);

A373144 Numbers k such that both A003415(k) and A276085(k) are multiples of 3, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

1, 8, 27, 35, 36, 64, 65, 77, 95, 119, 125, 135, 143, 155, 161, 162, 180, 185, 189, 203, 209, 215, 216, 221, 252, 275, 280, 287, 288, 297, 299, 305, 323, 329, 335, 341, 343, 351, 365, 371, 377, 395, 396, 407, 413, 425, 437, 459, 468, 473, 485, 497, 512, 513, 515, 520, 527, 533, 545, 551, 575, 581, 605, 611, 612, 616

Offset: 1

Antti Karttunen, May 26 2024

nonn

This is a multiplicative semigroup; if m and n are in the sequence then so is m*n.

			65 is present as A003415(65) = 18 = 3*6 and A276085(65) = 2316 = 3*772.
77 is present as A003415(77) = 18 = 3*6 and A276085(77) = 240 = 3*80.
5005 (= 65*77) is present as A003415(5005) = A276085(5005) = 2556 = 3*852. (See A369650).

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

Cf. A003415, A276085, A373143 (characteristic function).
Intersection of A327863 and A339746.
Positions of multiples of 3 in A373145.
Cf. also A369650.

PARI
```
\\ See A373143.
```

A370114 Numbers k for which A276086(A003415(k)) is a multiple of k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 6, 10, 15, 42, 50, 70, 98, 105, 245, 294, 330, 343, 375, 726, 770, 825, 1029, 1078, 1155, 1210, 1375, 1617, 1694, 1815, 1925, 2310, 2450, 2541, 2662, 2695, 3025, 3822, 3850, 4290, 4550, 4719, 5005, 5145, 5390, 6125, 6655, 7098, 7150, 7546, 7865, 8450, 8470, 8575, 8918, 9317, 9438, 9555, 10010, 11319, 11375

Offset: 1

Antti Karttunen, Feb 11 2024

nonn

Numbers k that divide A327859(k). Sequence contains only terms of A048103 and does not contain any multiples of 9.

Antti Karttunen, Table of n, a(n) for n = 1..7501

Fixed points of A369964. Positions of 1's in A370116.
Intersection of A048103 and A369650 is a subsequence of this sequence. See the comments in latter.
Cf. A003415, A276086, A327859, A370115.

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); };
isA370114(n) = ((n>0) && !(A276086(A003415(n))%n));

A369964 a(n) = gcd(n, A276086(A003415(n))), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 10, 1, 3, 1, 2, 15, 1, 1, 6, 1, 5, 3, 2, 1, 3, 5, 2, 3, 7, 1, 2, 1, 1, 3, 2, 5, 1, 1, 2, 3, 5, 1, 42, 1, 1, 15, 2, 1, 3, 1, 50, 3, 1, 1, 6, 5, 7, 3, 2, 1, 3, 1, 2, 21, 1, 5, 2, 1, 1, 3, 70, 1, 1, 1, 2, 25, 1, 1, 6, 1, 5, 1, 2, 1, 21, 5, 2, 3, 1, 1, 6, 1, 1, 3, 2, 5, 3, 1, 98, 3, 25

Offset: 0

Antti Karttunen, Feb 11 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A276086, A327859, A370114 (fixed points, see also A369650), A370116, A370117.

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); };
A369964(n) = gcd(n, A276086(A003415(n)));

a(n) = gcd(n, A327859(n)) = gcd(n, A276086(A003415(n))).
For n >= 1, a(n) = n / A370116(n).
For n >= 0, a(n) = A327859(n) / A370117(n).

A373487 Numbers k such that A276085(k) is a multiple of A003415(k), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 15, 17, 19, 23, 28, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 215, 221, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

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

Cf. A003415, A276085, A369650 (subsequence), A373486 (characteristic function).
After the initial 1, gives the positions of 0's in A373148.
Disjoint union of A000040 and A373488 (nonprime terms).
Cf. also A373489.

PARI
```
isA373487 = A373486;
```

A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 10, 15, 161, 2189, 5005, 27030, 29861, 510221, 223092341

Offset: 1

Antti Karttunen, Feb 11 2024

nonn

Question: Is the squarefreeness a necessary condition for the nonzero terms of this sequence?
Many of the terms occur also in A368703, because the arithmetic derivative of those terms is one of the primorial numbers, A002110.
If it exists, a(13) > 1241513984.

Positions of 1's in A370117, positions of 0's in A370120.
Intersection of A048103 and A369650 is a subsequence of this sequence. See the comments in latter.
Cf. A002110, A003415, A276086, A327859, A368703.
Cf. also A369970, A370114.

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); };
isA370115(n) = !(n%A276086(A003415(n)));

A373488 Nonprime natural numbers k such that A276085(k) is a multiple of A003415(k), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

1, 10, 15, 28, 35, 91, 143, 155, 161, 187, 203, 209, 215, 221, 295, 299, 319, 323, 329, 341, 371, 377, 391, 403, 437, 515, 538, 559, 581, 611, 649, 667, 689, 703, 707, 731, 779, 803, 851, 869, 893, 899, 901, 923, 979, 989, 1037, 1073, 1081, 1121, 1139, 1147, 1189, 1199, 1207, 1219, 1261, 1363, 1391, 1403, 1457, 1498

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

Nonsquarefree terms are rare: only 3 less than 2^20: 28, 33212, 855523.

Antti Karttunen, Table of n, a(n) for n = 1..25127

Setwise difference A373487 \ A000040.

Cf. A003415, A276085, A369650, A373486.

isA373488(n) = (!isprime(n) && A373486(n));

A383299 Numbers k such that A276086(k) is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 119, 127, 131, 137, 139, 141, 147, 149, 151, 157, 161, 163, 165, 167, 171, 173, 177, 179, 181, 191, 193, 197, 199, 203, 207, 209, 211, 223, 227, 229, 233, 239

Offset: 1

Antti Karttunen, May 15 2025

nonn

The sequence contains the intersection of A048103, A369650, and A328387. That is, {1, 15, 5005}, at least.

			5 is a term as A003415(5) = 1, and A276086(5) = 18 is a multiple of A276086(1) = 2, and ditto for all odd primes.
9 is a term as A003415(9) = 6, and A276086(9) = 30 is a multiple of A276086(6) = 5.
15 is a term as A003415(15) = 8, and A276086(15) = 150 is a multiple of A276086(8) = 15.
5005 is a term as A003415(5005) = 2556, and A276086(5005) = 39055266250 = 7803250 * A276086(2556) = 7803250 * 5005. See also A369650.
See also examples in A383300.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A003415, A048103, A276086, A327859, A328387, A383298 (characteristic function).
Cf. A006005, A051674, A383300, A383301 (subsequences).
Cf. also A369650.

PARI
```
isA383299 = A383298;
```

cp's OEIS Frontend

A327859 a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A328110 Fixed points of A327860: numbers k for which A003415(A276086(k)) = k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A373144 Numbers k such that both A003415(k) and A276085(k) are multiples of 3, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A370114 Numbers k for which A276086(A003415(k)) is a multiple of k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A369964 a(n) = gcd(n, A276086(A003415(n))), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A373487 Numbers k such that A276085(k) is a multiple of A003415(k), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A373488 Nonprime natural numbers k such that A276085(k) is a multiple of A003415(k), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Views

Author

Keywords

Comments

Links

Crossrefs