A369963 Numbers k for which x = gcd(k, A003415(k)) = gcd(k, A276086(k)) and x > 1, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

9, 49, 99, 117, 153, 171, 207, 242, 245, 261, 279, 333, 369, 387, 423, 477, 531, 549, 603, 639, 657, 711, 747, 801, 850, 873, 909, 927, 963, 981, 1017, 1071, 1125, 1143, 1150, 1179, 1233, 1250, 1251, 1341, 1359, 1413, 1450, 1467, 1503, 1519, 1557, 1611, 1629, 1719, 1737, 1773, 1791, 1899, 1989, 2007, 2043, 2050, 2061

Offset: 1

Author

Antti Karttunen, Feb 07 2024

Keywords

• nonn

Comments

Nonsquarefree terms in A369962.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to primorial base

Crossrefs

Intersection of A013929 and A369962.

Cf. A003415, A085731, A276086, A324198, A369960 (subsequence).

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); };
isA369963(n) = (!issquarefree(n) && (gcd(n,A003415(n))==gcd(n,A276086(n))));

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); };
isA369963(n) = if(!n,0, my(u=A085731(n)); if(u<2, 0, (u == A324198(n))));

Formula

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