A074849 4-infinitary perfect numbers: numbers k such that 4-infinitary-sigma(k) = 2*k.

6, 28, 36720, 222768, 12646368, 5154170112, 34725010231296

Offset: 1

Author

Yasutoshi Kohmoto, Sep 10 2002

Keywords

• nonn
• more

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A007357, A038182, A074847, A097464.

Programs

Mathematica

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 *)

Formula

{k: A074847(k) = 2*k}. - R. J. Mathar, Mar 13 2024