This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383301 #21 May 21 2025 17:52:27
%S A383301 4784261,338634851,433979267,713516597,829765697,1092143279,
%T A383301 1790536511,2518099229,8107348511
%N 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).
%C A383301 Here "nontrivial proper suffix" means suffix whose length is > 1, but less than the length of the string whose suffix it is.
%H A383301 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%F A383301 {k such that 1 < k' < k, and k' is equal to k modulo A002110(A235224(k')), where k' = A003415(k)}.
%e A383301 k (in primorial base, A049345) k' (in primorial base)
%e A383301 --------------------------------------------------------------------------
%e A383301 4784261 (9:6:4:1:1:1:2:1) 189671 (6:4:1:1:1:2:1)
%e A383301 338634851 (1:11:17:5:7:1:6:1:2:1) 8845391 (17:5:7:1:6:1:2:1)
%e A383301 433979267 (1:21:14:1:6:8:6:2:2:1) 7192907 (14:1:6:8:6:2:2:1)
%e A383301 713516597 (3:4:10:11:1:7:0:2:2:1) 5439227 (10:11:1:7:0:2:2:1)
%e A383301 829765697 (3:16:10:6:2:10:1:2:2:1) 5292047 (10:6:2:10:1:2:2:1)
%e A383301 1092143279 (4:20:11:5:5:3:1:4:2:1) 5777999 (11:5:5:3:1:4:2:1)
%e A383301 1790536511 (8:0:11:5:12:0:2:1:2:1) 5793551 (11:5:12:0:2:1:2:1)
%e A383301 2518099229 (11:6:11:8:10:2:4:4:2:1) 5879519 (11:8:10:2:4:4:2:1)
%e A383301 8107348511 (1:7:7:15:14:12:7:3:1:2:1) 76005191 (7:15:14:12:7:3:1:2:1)
%e A383301 Note that 4784261 = 9*A002110(7) + 189671.
%o A383301 (PARI)
%o A383301 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A383301 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));
%o A383301 (PARI)
%o A383301 A002110(n) = prod(i=1,n,prime(i));
%o A383301 A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
%o A383301 isA383301(n) = { my(ad=A003415(n)); ((ad>1) && (ad<n) && (n%A002110(A235224(ad))==ad)); };
%Y A383301 Subsequence of A048103 and of A383300.
%Y A383301 Cf. A002110, A003415, A049345, A235224.
%K A383301 nonn,base,more
%O A383301 1,1
%A A383301 _Antti Karttunen_, May 15 2025