A280387 Composite numbers n such that sum of proper divisors of n divides sum of proper divisors of n^n.

4, 8, 9, 16, 21, 25, 27, 32, 36, 45, 49, 64, 81, 87, 91, 99, 121, 125, 128, 144, 169, 196, 217, 243, 256, 289, 325, 343, 361, 400, 417, 481, 512, 529, 559, 625, 685, 697, 703, 721, 729, 745, 749, 775, 801, 841, 925, 931, 961, 1024, 1156, 1157, 1261, 1331

Offset: 1

Author

Altug Alkan, Jan 01 2017

Keywords

• nonn
• less

Comments

Terms are 2^2, 2^3, 3^2, 2^4, 3*7, 5^2, 3^3, 2^5, 2^2*3^2, 3^2*5, 7^2, 2^6, 3^4, 3*29, 7*13, 3^2*11, 11^2, 5^3, ...

Terms that are not Duffinian numbers are 45, 87, 91, 99, 196, 703, 745, 775, 801, 931, ...

Examples

			Composite number 21 is a term because (sigma(21) - 21) = 11 divides (sigma(21^21) - 21^21) = 4381940263463668467705506011
		

Crossrefs

Cf. A000203, A001065, A003624, A275123.

Programs

Mathematica

Select[Range[10^3], And[CompositeQ@ #, Divisible @@ Map[DivisorSigma[1, #] - # &, {#^#, #}]] &] (* Michael De Vlieger, Jan 02 2017 *)

PARI

is(n) = !isprime(n) && (sigma(n^n)-n^n)%(sigma(n)-n)==0;

Extensions

More terms from Amiram Eldar, Feb 19 2019