A369963 - OEIS

%I #13 Apr 29 2024 10:52:37

%S 9,49,99,117,153,171,207,242,245,261,279,333,369,387,423,477,531,549,

%T 603,639,657,711,747,801,850,873,909,927,963,981,1017,1071,1125,1143,

%U 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

%N 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.

%C Nonsquarefree terms in A369962.

%H Antti Karttunen, <a href="/A369963/b369963.txt">Table of n, a(n) for n = 1..20000</a>

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

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

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

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

%Y Intersection of A013929 and A369962.

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

%K nonn

%O 1,1

%A _Antti Karttunen_, Feb 07 2024