A318170 Composite numbers k such that A008480(k) = k.

1781030694643200, 2671546041964800, 2968384491072000, 558162298053360000, 1953568043186760000

Offset: 1

Author

Amiram Eldar, Aug 20 2018

Keywords

• nonn
• more
• fini

Comments

Knopfmacher and Luca proved that this sequence is finite.

These numbers are named "prime-factor-perfect numbers" by Knopfmacher and Mays and "prime-perfect numbers" by Knopfmacher and Luca.

Examples

			1781030694643200 = 2^9 * 3^5 * 5^2 * 7^2 * 11^2 * 13 * 17 * 19 * 23 is in the sequence since multinomial(9+5+2+2+2+1+1+1+1,9,5,2,2,2,1,1,1,1) = 1781030694643200.
		

Links

• Arnold Knopfmacher and Florian Luca, On prime-perfect numbers, International Journal of Number Theory, Vol. 7, No. 7 (2011), pp. 1705-1716
• Arnold Knopfmacher and M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, Vol. 1, No. 4 (2005), pp. 563-581, DOI: 10.1142/S1793042105000315.

Crossrefs

Cf. A008480.

Programs

Mathematica

mul[w_] := Total[w]!/Times @@ (w!); f[n_] := Select[ IntegerPartitions@ n, # == Reverse@ Sort[ Last /@ FactorInteger[mul[#]]] &]; Sort[mul /@ Flatten[f /@ Range[2, 30], 1]] (* terms with sum of exponents in prime factorization <= 30, Giovanni Resta, Aug 20 2018 *)