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Numbers k such that the average of greatest prime factors of all positive integers <= k is an integer.
1

%I #11 May 05 2018 06:34:11

%S 2,5,6,24,27,28,79,1171,1901,1932,7845,19152,360698,649358,1731441,

%T 1979944,30658517,34337898,400114748,412626117,3738454462,7725156081,

%U 97366414806,129951504625,290788509485

%N Numbers k such that the average of greatest prime factors of all positive integers <= k is an integer.

%C Numbers k such that (k - 1) | (A046670(k) - 1).

%C Numbers k such that (k - 1) | A088822(k).

%C a(26) > 10^12. - _Giovanni Resta_, May 05 2018

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>

%e 6 is in the sequence because (gpf(2) + gpf(3) + gpf(4) + gpf(5) + gpf(6))/5 = (2 + 3 + 2 + 5 + 3)/5 = 3 is an integer.

%o (PARI) isok(n) = (n>1) && !(sum(k=2, n, vecmax(factor(k)[,1])) % (n-1)); \\ _Michel Marcus_, Apr 29 2018

%o (PARI) lista(nn) = {s = 0; for (n=2, nn, s += vecmax(factor(n)[,1]); if (! (s % (n-1)), print1(n, ", ")););} \\ _Michel Marcus_, Apr 29 2018

%Y Cf. A006530, A046670, A088822, A226647, A284755, A303480, A303482.

%K nonn,more

%O 1,1

%A _Ilya Gutkovskiy_, Apr 28 2018

%E a(13)-a(18) from _Michel Marcus_, Apr 29 2018

%E a(19)-a(25) from _Giovanni Resta_, May 05 2018