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Numbers k such that A308485(k) is a multiple of k.
0

%I #12 Mar 07 2024 05:42:19

%S 1,2,8,9,10,12,65,142,2539,9027,9905,23634,27638428

%N Numbers k such that A308485(k) is a multiple of k.

%C Numbers k such that the sum of integer logs of the composite numbers between prime(k) and prime(k+1) is divisible by k.

%e a(3) = 8 is a term because prime(8) = 19 and prime(9) = 23, and the sum of integer logs of 20, 21 and 22 is 2*2+5+3+7+2+11 = 32 which is a multiple of 8.

%p f:= proc(m) local t; add(t[1]*t[2],t=ifactors(m)[2]) end proc:

%p t:= 0: n:= 0: p:= 2: count:= 0: R:= NULL:

%p for m from 3 to 10^6 do

%p if isprime(m) then

%p n:= n+1;

%p if t mod n = 0 then R:= R, n; count:= count+1 fi;

%p p:= m; t:= 0

%p else

%p t:= t+f(m)

%p fi

%p od:

%p R;

%t f[m_] := Sum[t[[1]]*t[[2]], {t, FactorInteger[m]}];

%t t = 0; n = 0; p = 2; count = 0; R = {};

%t For[m = 3, m <= 10^6, m++, If[PrimeQ[m], n++; If[Mod[t, n] == 0, AppendTo[R, n]; count++]; p = m; t = 0, t = t+f[m]]];

%t R (* _Jean-François Alcover_, Mar 07 2024, translated from Maple code *)

%Y Cf. A001414, A308485.

%K nonn,more

%O 1,2

%A _J. M. Bergot_ and _Robert Israel_, Mar 29 2022