A284288 Numbers n such that the average of the strong divisors of n is an integer.

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 28, 29, 31, 37, 41, 43, 47, 49, 53, 54, 56, 59, 61, 64, 67, 68, 71, 73, 79, 81, 83, 89, 91, 97, 98, 99, 100, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 138, 139, 148, 149, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 188, 191, 192, 193, 197, 199

Offset: 1

Author

Ilya Gutkovskiy, Mar 24 2017

Keywords

• nonn

Comments

We say d is a strong divisor of n iff d is a divisor of n and d > 1.

Numbers n such that A032741(n) divides A039653(n).

All primes and squares of primes are in this sequence.

Positions of ones in A296082 and A296084. - Antti Karttunen, Dec 05 2017

Examples

			28 is in the sequence because 28 has 6 divisors {1, 2, 4, 7, 14, 28} therefore 5 strong divisors {2, 4, 7, 14, 28}, 2 + 4 + 7 + 14 + 28 = 55 and 5 divides 55.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000203, A000430, A003601, A023884, A023886, A032741, A039653, A296082, A296084 (characteristic function).

Programs

Maple

filter:= proc(n) local d,t;
  d:= numtheory:-divisors(n) minus {1};
  convert(d,`+`) mod nops(d) = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Mar 27 2017

Mathematica

Select[Range[2, 200], Mod[DivisorSigma[1, #1] - 1, DivisorSigma[0, #1] - 1] == 0 &]

PARI

for(n=2, 200, if((sigma(n) - 1)%(numdiv(n) - 1)==0, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

Python

from sympy.ntheory import divisor_sigma, divisor_count
print([n for n in range(2, 201) if (divisor_sigma(n) - 1)%(divisor_count(n) - 1) == 0]) # Indranil Ghosh, Mar 24 2017