A206708 Numbers k such that sigma(k) = sigma(sigma(k)-k).

6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, 122265, 122368, 123152, 124155, 139815, 141664, 142310, 153176

Offset: 1

Author

Michel Lagneau, Feb 11 2012

Keywords

• nonn

Comments

For all k, let s(k) = sigma(k) - k, the aliquot sum function A001065; then this sequence is the set of k such that s(s(k)) = k. - Jeppe Stig Nielsen, Jan 12 2020

Examples

			220 is in the sequence because sigma(220) = 504, sigma(504 - 220) = sigma(284) = 504.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Amiram Eldar)
• Eric Weisstein's World of Mathematics, Perfect Number
• Eric Weisstein's World of Mathematics, Amicable Pair
• Wikipedia, Perfect number
• Wikipedia, Amicable number

Crossrefs

Cf. A000396 (perfect numbers), A063990 (amicable numbers).

Cf. A000203 (sum of divisors), A001065 (sum of proper divisors).

Programs

Magma

[k:k in [2..154000]|s eq DivisorSigma(1,s-k) where s is DivisorSigma(1,k)]; // Marius A. Burtea, Jan 13 2020

Maple

q:= n-> (s-> s(n)=s(s(n)-n))(numtheory[sigma]):
select(q, [$1..100000])[];  # Alois P. Heinz, Jan 31 2023

Mathematica

Select[Range[10^6],DivisorSigma[1,#]==DivisorSigma[1, DivisorSigma[1,#]-#]&]

PARI

isok(k) = if (k != 1, my(sk=sigma(k)); sk == sigma(sk-k)); \\ Michel Marcus, Jun 24 2019

Formula

Equals {A063990} union {A000396} = (amicable numbers) union (perfect numbers).