A383301 - OEIS

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

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OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..9.

Index entries for sequences related to primorial base.

FORMULA

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

EXAMPLE

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.

PROG

(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) && (ad<n) && (n%A002110(A235224(ad))==ad)); };

CROSSREFS

Subsequence of A048103 and of A383300.

Cf. A002110, A003415, A049345, A235224.

Sequence in context: A017503 A017635 A203655 * A248587 A204053 A176772

Adjacent sequences: A383298 A383299 A383300 * A383302 A383303 A383304

KEYWORD

nonn,base,more

AUTHOR

Antti Karttunen, May 15 2025

STATUS

approved