Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Oct 24 2024 09:25:56
%S 6,28,36720,222768,12646368,5154170112,34725010231296
%N 4-infinitary perfect numbers: numbers k such that 4-infinitary-sigma(k) = 2*k.
%C Here 4-infinitary-sigma(k) means sum of 4-infinitary-divisors of k. If k = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a <= b in its 4-ary expansion everywhere that the corresponding r(i) has a digit b, then d is a 4-infinitary-divisor of k.
%F {k: A074847(k) = 2*k}. - _R. J. Mathar_, Mar 13 2024
%e Factorizations: 2*3, 2^2*7, 2^4*3^3*5*17, 2^4*3^2*7*13*17, 2^5*3^4*7*17*41, 2^8*3^2*7*13^2*31*61, 2^12*3^5*7*11*41*43*257.
%t f[p_, e_] := Module[{d = IntegerDigits[e, 4]}, m = Length[d]; Product[(p^((d[[j]] + 1)*4^(m - j)) - 1)/(p^(4^(m - j)) - 1), {j, 1, m}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300000], s[#] == 2*# &] (* _Amiram Eldar_, Oct 24 2024 *)
%Y Cf. A007357, A038182, A074847, A097464.
%K nonn,more
%O 1,1
%A _Yasutoshi Kohmoto_, Sep 10 2002