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).

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

Offset: 1

Author

Antti Karttunen, May 15 2025

Keywords

• nonn
• base
• more

Comments

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

Examples

			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.
		

Links

• Index entries for sequences related to primorial base.

Crossrefs

Subsequence of A048103 and of A383300.

Cf. A002110, A003415, A049345, A235224.

Programs

PARI

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));

PARI

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)); };

Formula

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