A383300 -id:A383300 - cp's OEIS frontend

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.

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;
```

A348283 Numbers that are multiples of their arithmetic derivative, A003415.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Wesley Ivan Hurt, Oct 09 2021

nonn

Here, m' denotes the arithmetic derivative of m (A003415).
Not the same as A211781 since this sequence does not contain 225, 252, etc.
All prime numbers p are in the sequence since p' = 1 | p.
Numbers k such that k' | k. - The original definition of the sequence.
Sequence consists of 0, primes, and the prime powers of the form p^p (A051674, that together with 0 give the only fixed points of A003415). This can be seen from theorems 4-6 given in the Ufnarovski & Åhlander paper. - Antti Karttunen, May 17 2025

			0 is in the sequence as A003415(0) = 0 and 0 is a multiple 0.
27 is in the sequence as A003415(27) = 27' = 27, and 27 is a multiple of 27.
127 (like any prime) is in the sequence since 127' = 1 | 127.

Antti Karttunen & R. J. Mathar, Table of n, a(n) for n = 1..20001
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A003415, A211781.
After the initial zero, gives the indices of 0's in A369049.
Disjoint union of {0}, A000040 and A051674.
Apart from term 2, a subsequence of A383300.

q:= n-> is(irem(n, n*add(i[2]/i[1], i=ifactors(n)[2]))=0):
select(q, [$2..300])[];  # Alois P. Heinz, Oct 11 2021

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = (0==k) || ((k>1) && !(k % ad(k))); \\ Michel Marcus, Oct 10 2021

a(1) = 0 inserted because of a new, more inclusive definition. - Antti Karttunen, May 17 2025

A383301 Numbers k whose primorial base expansion has the primorial base expansion of k' as its nontrivial proper suffix, where k' stands for the arithmetic derivative of k (A003415).

Original entry on oeis.org

4784261, 338634851, 433979267, 713516597, 829765697, 1092143279, 1790536511, 2518099229, 8107348511

Offset: 1

Antti Karttunen, May 15 2025

Here "nontrivial proper suffix" means suffix whose length is > 1, but less than the length of the string whose suffix it is.

			k          (in primorial base, A049345)   k'       (in primorial base)
--------------------------------------------------------------------------
4784261    (9:6:4:1:1:1:2:1)              189671   (6:4:1:1:1:2:1)
338634851  (1:11:17:5:7:1:6:1:2:1)        8845391  (17:5:7:1:6:1:2:1)
433979267  (1:21:14:1:6:8:6:2:2:1)        7192907  (14:1:6:8:6:2:2:1)
713516597  (3:4:10:11:1:7:0:2:2:1)        5439227  (10:11:1:7:0:2:2:1)
829765697  (3:16:10:6:2:10:1:2:2:1)       5292047  (10:6:2:10:1:2:2:1)
1092143279 (4:20:11:5:5:3:1:4:2:1)        5777999  (11:5:5:3:1:4:2:1)
1790536511 (8:0:11:5:12:0:2:1:2:1)        5793551  (11:5:12:0:2:1:2:1)
2518099229 (11:6:11:8:10:2:4:4:2:1)       5879519  (11:8:10:2:4:4:2:1)
8107348511 (1:7:7:15:14:12:7:3:1:2:1)     76005191 (7:15:14:12:7:3:1:2:1)
Note that 4784261 = 9*A002110(7) + 189671.

Index entries for sequences related to primorial base.

Subsequence of A048103 and of A383300.

Cf. A002110, A003415, A049345, A235224.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA383301(n) = if(n<2, 0, my(p=2, k=A003415(n), i=0); while(k, if((k%p)!=(n%p), return(0)); n = n\p; k = k\p; p = nextprime(1+p); i++); (n>0)&&(i>1));

A002110(n) = prod(i=1,n,prime(i));
A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
isA383301(n) = { my(ad=A003415(n)); ((ad>1) && (adA002110(A235224(ad))==ad)); };

{k such that 1 < k' < k, and k' is equal to k modulo A002110(A235224(k')), where k' = A003415(k)}.

A383933 Numbers k such that primorial base expansion of A276086(k) has the primorial base expansion of A003415(k) as its suffix, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 6, 26, 95, 122, 185, 206, 1382, 1919, 2006, 2285, 2306, 2966, 4681, 4841, 5909, 13961, 14269, 21446, 30026, 34249, 37231, 54589, 54611, 61459, 90065, 135229, 145309, 204566, 217621, 262099, 266950, 289621, 306302, 310939, 341699, 350099, 353779, 356809, 358091, 364361, 496751, 501289, 503669, 510506, 515059

Offset: 1

Antti Karttunen, May 15 2025

			0 and 1 are terms as A003415(0) = A003415(1) = 0, whose primorial base expansion is here understood as an empty sequence of digits, thus occurring as a suffix of all representations.
2 is a term as A003415(2) = 1, with A049345(1) = 1, which is a suffix of A049345(A276086(2)) = 11.
6 is a term as A003415(6) = 5, with A049345(5) = 21, which is a suffix of A049345(A276086(6)) = 21.
95 is a term as A003415(95) = 24, with A049345(24) = 400, which is a suffix of A049345(A276086(95)) = 272400.

Index entries for sequences related to primorial base.

Subsequence of A383303.
Cf. A003415, A049345, A276086, A276087.
Cf. also A383300.

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); };
isA383933(n) = { my(p=2, k=A003415(n)); n = A276086(n); while(k, if((k%p)!=(n%p), return(0)); n = n\p; k = k\p; p = nextprime(1+p)); (1); };

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A348283 Numbers that are multiples of their arithmetic derivative, A003415.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A383301 Numbers k whose primorial base expansion has the primorial base expansion of k' as its nontrivial proper suffix, where k' stands for the arithmetic derivative of k (A003415).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A383933 Numbers k such that primorial base expansion of A276086(k) has the primorial base expansion of A003415(k) as its suffix, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI