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.

2313, 30033, 30069, 30249, 30282, 32350, 32553, 60093, 60273, 510550, 510561, 510579, 510633, 510723, 510741, 513063, 540963, 542853, 570573, 572910, 1021023, 1021062, 1021239, 1023363, 1531539, 1561563, 9699741, 9699746, 9699759, 9699903, 9699942, 9699957, 9699965, 9700150, 9700353, 9702009, 9702027, 9702049, 9702121

Offset: 1

Author

Antti Karttunen, Feb 07 2024

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..350
• Index entries for sequences related to primorial base

Crossrefs

Intersection of A351228 and A369963.

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

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

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369960(n) = { my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((t >= u) && (g==h) && (g>1)); };
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)));

PARI

A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };
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); };
isA369960(n) = if(!n || issquarefree(n),0,((A085731(n) == A324198(n)) && (A003415(n) >= A276086(n))));

Formula

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