A097464 5-infinitary perfect numbers: numbers k such that 5-infinitary-sigma(k) = 2*k.

6, 28, 496, 47520, 288288, 308474880

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• more

Comments

Here 5-infinitary-sigma(k) means sum of 5-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 5-ary expansion everywhere that the corresponding r_i has a digit b, then d is a 5-infinitary-divisor of k.

Is it certain that 308474880 is the 6th term? M. F. Hasler, Nov 20 2010

Data is verified. a(7) > 10^11, if it exists. - Amiram Eldar, Oct 24 2024

Examples

			Factorizations: 2*3, 2^2*7, 2^4*31, 2^5*3^3*5*11, 2^5*3^2*7*11*13, 2^10*3*5*7*19*151.
		

Crossrefs

Cf. A007357, A038182, A074849, A097863.

Programs

Mathematica

f[p_, e_] := Module[{d = IntegerDigits[e, 5]}, m = Length[d]; Product[(p^((d[[j]] + 1)*5^(m - j)) - 1)/(p^(5^(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: A097863(k) = 2*k}.

Extensions

Missing a(4) inserted by R. J. Mathar, Nov 20 2010