A369960 - OEIS

%I #10 Feb 07 2024 20:39:34

%S 2313,30033,30069,30249,30282,32350,32553,60093,60273,510550,510561,

%T 510579,510633,510723,510741,513063,540963,542853,570573,572910,

%U 1021023,1021062,1021239,1023363,1531539,1561563,9699741,9699746,9699759,9699903,9699942,9699957,9699965,9700150,9700353,9702009,9702027,9702049,9702121

%N 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 Antti Karttunen, <a href="/A369960/b369960.txt">Table of n, a(n) for n = 1..350</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F {k | A085731(n) > 1 and A085731(n) == A324198(n) and A083345(k) >= A351251(k)}.

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o isA369960(n) = { my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((t >= u) && (g==h) && (g>1)); };

%o 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 (PARI)

%o A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };

%o 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 isA369960(n) = if(!n || issquarefree(n),0,((A085731(n) == A324198(n)) && (A003415(n) >= A276086(n))));

%Y Intersection of A351228 and A369963.

%Y Subsequence of the following sequences: A013929, A369958, A369959, A369962.

%Y Cf. A003415, A060735, A083345, A085731, A276086, A324198, A351251.

%K nonn

%O 1,1

%A _Antti Karttunen_, Feb 07 2024