A156787 Composite integers n such that 2^{n-1}=1 mod s(n), where s(n) is the sum of the distinct prime factors of n.

9, 10, 25, 27, 40, 49, 81, 100, 105, 116, 121, 125, 160, 169, 243, 250, 289, 343, 361, 400, 525, 529, 561, 568, 625, 640, 729, 805, 841, 945, 961, 1000, 1001, 1018, 1045, 1105, 1309, 1331, 1369, 1596, 1600, 1681, 1729, 1849, 1856, 1881, 2001, 2187, 2197, 2205

Offset: 1

Author

Florian Luca (fluca(AT)matmor.unam.mx), Feb 15 2009

Keywords

• nonn

Examples

			For n=2, the second number is a(2)=10 because s(10)=2+5=7 divides 2^{10-1}-1=2^9-1=511.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• F. Luca and V. Tipu, On positive integers n with a certain divisibility property, J. Integer Sequences 12 (2009), 11 pp.

Crossrefs

Cf. A006145.

Programs

Maple

B := {}; for n from 2 to 1000 do A := (numtheory[factorset])(n); b := add(a, `in`(a, A)); if `and`(b < n, `mod`(2^(n-1), b) = 1) then B := [op(B), n] else end if end do; print(c := 2);

Mathematica

Select[Range[2, 2300], CompositeQ[#] && PowerMod[2, #-1, Total[First /@ FactorInteger[#]]] == 1 &] (* Amiram Eldar, Nov 20 2019 *)

PARI

is(n)=if(isprime(n),0,my(f=factor(n)[,1]);Mod(2, sum(i=1, #f, f[i]))^(n-1)==1) \\ Charles R Greathouse IV, Feb 01 2013

Formula

n log n << a(n) << n^(1+e) for any e > 0. See Luca & Tipu for more precise results. - Charles R Greathouse IV, Feb 01 2013

Extensions

More terms from Amiram Eldar, Nov 20 2019