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Numbers k such that Sum_{j=1..k} tau(j)^j == 0 (mod k), where tau(j) = A000005(j), the number of divisors of j.
5

%I #27 Feb 28 2024 01:34:37

%S 1,46,135,600,1165,1649,5733,6788,6828,9734,29686,363141,1542049

%N Numbers k such that Sum_{j=1..k} tau(j)^j == 0 (mod k), where tau(j) = A000005(j), the number of divisors of j.

%C a(12) > 200000. - _Michel Marcus_, Feb 25 2016

%C a(13) > 500000. - _Harvey P. Dale_, Dec 13 2018

%C a(14) > 3000000. - _Jason Yuen_, Feb 27 2024

%e tau(1)^1 + tau(2)^2 + ... + tau(45)^45 + tau(46)^46 = 1^1 + 2^2 + ... + 6^45 + 4^46 = 86543618042218910328339719795268200166 and 86543618042218910328339719795268200166 / 46 = 1881383000917802398442167821636265221.

%p with(numtheory); P:=proc(q) local n, t; t:=0;

%p for n from 1 to q do t:=t+tau(n)^n; if t mod n=0 then print(n);

%p fi; od; end: P(10^6);

%t Module[{nn=30000,ac},ac=Accumulate[Table[DivisorSigma[0,i]^i,{i,nn}]];Select[ Thread[{ac,Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]](* _Harvey P. Dale_, Dec 13 2018 *)

%o (PARI) isok(n) = sum(i=1, n, Mod(numdiv(i), n)^i) == 0; \\ _Michel Marcus_, Feb 25 2016

%Y Cf. A000005, A227427, A227429, A227502, A227848, A229095, A229208, A229209, A229210, A229211.

%K nonn,more

%O 1,2

%A _Paolo P. Lava_, Sep 16 2013

%E a(12) added by _Harvey P. Dale_, Dec 13 2018

%E a(13) added by _Jason Yuen_, Feb 27 2024