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A369960 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(k)) = gcd(k, A276086(k)) > 1, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

%I A369960 #10 Feb 07 2024 20:39:34
%S A369960 2313,30033,30069,30249,30282,32350,32553,60093,60273,510550,510561,
%T A369960 510579,510633,510723,510741,513063,540963,542853,570573,572910,
%U A369960 1021023,1021062,1021239,1023363,1531539,1561563,9699741,9699746,9699759,9699903,9699942,9699957,9699965,9700150,9700353,9702009,9702027,9702049,9702121
%N A369960 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(k)) = gcd(k, A276086(k)) > 1, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.
%H A369960 Antti Karttunen, <a href="/A369960/b369960.txt">Table of n, a(n) for n = 1..350</a>
%H A369960 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A369960 {k | A085731(n) > 1 and A085731(n) == A324198(n) and A083345(k) >= A351251(k)}.
%o A369960 (PARI)
%o A369960 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369960 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A369960 isA369960(n) = { my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((t >= u) && (g==h) && (g>1)); };
%o A369960 isA369960(n) = if(!n || issquarefree(n), 0, my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((t >= u) && (g==h)));
%o A369960 (PARI)
%o A369960 A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };
%o A369960 A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
%o A369960 isA369960(n) = if(!n || issquarefree(n),0,((A085731(n) == A324198(n)) && (A003415(n) >= A276086(n))));
%Y A369960 Intersection of A351228 and A369963.
%Y A369960 Subsequence of the following sequences: A013929, A369958, A369959, A369962.
%Y A369960 Cf. A003415, A060735, A083345, A085731, A276086, A324198, A351251.
%K A369960 nonn
%O A369960 1,1
%A A369960 _Antti Karttunen_, Feb 07 2024