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A383300 Numbers k such that primorial base expansion of k has the primorial base expansion of k' as its suffix, where k' stands for the arithmetic derivative of k (A003415).

%I A383300 #33 May 19 2025 04:26:11
%S A383300 0,1,3,4,5,7,11,13,17,19,23,27,29,31,37,41,43,47,53,59,61,67,71,73,79,
%T A383300 83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,
%U A383300 179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331
%N A383300 Numbers k such that primorial base expansion of k has the primorial base expansion of k' as its suffix, where k' stands for the arithmetic derivative of k (A003415).
%C A383300 a(n) = A348283(n) for n=1 and n=3..334432. a(334433) = 4784261, which is not present in A348283 (see examples). - _R. J. Mathar_ and _Antti Karttunen_, May 16 2025
%H A383300 Antti Karttunen, <a href="/A383300/b383300.txt">Table of n, a(n) for n = 1..20000</a>
%H A383300 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%F A383300 {k such that A003415(k) is equal to k modulo A002110(A235224(A003415(k)))}.
%e A383300 0 is a term as A003415(0) = 0.
%e A383300 1 is a term as A003415(1) = 0, whose primorial base expansion is here understood as an empty sequence of digits, thus it is a suffix of A049345(1) = 1.
%e A383300 3, like all odd primes, is a term as A003415(3) = 1, with A049345(3) = 11 and A049345(1) = 1.
%e A383300 4 and 27 are terms as they are in A051674 (the nonzero fixed points of A003415).
%e A383300 4784261 is a term as A003415(4784261) = 189671, with A049345(4784261) = 96411121 and A049345(189671) = 6411121. 4784261 is the first term > 1 of this sequence that is not in A348283. See more examples in A383301.
%o A383300 (PARI)
%o A383300 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A383300 isA383300(n) = if(n<2, 1, my(p=2, k=A003415(n)); while(k, if((k%p)!=(n%p), return(0)); n = n\p; k = k\p; p = nextprime(1+p)); (1));
%o A383300 (PARI)
%o A383300 A002110(n) = prod(i=1,n,prime(i));
%o A383300 A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
%o A383300 isA383300(n) = { my(ad=A003415(n)); (n%A002110(A235224(ad))==ad); };
%Y A383300 Cf. A003415, A049345, A235224.
%Y A383300 Disjoint union of {1}, A348283\{2} and A383301.
%Y A383300 Cf. A006005, A051674 (other subsequences).
%Y A383300 Subsequence of A383299.
%Y A383300 Cf. also A383933.
%K A383300 nonn
%O A383300 1,3
%A A383300 _Antti Karttunen_, May 15 2025