A192035 Numbers k with equal remainders of (product of divisors of k) mod (sum of divisors of k) and (product of proper divisors of k) mod (sum of proper divisors of k).

6, 14, 28, 51, 120, 260, 270, 496, 672, 679, 752, 924, 1260, 1320, 1540, 1960, 2055, 2262, 2651, 3808, 3948, 4381, 6413, 6435, 6944, 7900, 7980, 8010, 8128, 9809, 9945, 10242, 10920, 12690, 15456, 16830, 18018, 21728, 21970, 22320, 25296, 27930, 29190, 29792

Offset: 1

Author

Juri-Stepan Gerasimov, Jun 21 2011

Keywords

• nonn

Comments

The even perfect numbers (A000396) are a subsequence.

The deficient numbers (A005100) in the sequence are 14, 51, 679, 752, 2055, 2651, 4381, 6413, 9809, 9945, 21970, ... - Juri-Stepan Gerasimov, Jul 07 2011

Examples

			14 is in this sequence because (1*2*7*14) mod (1+2+7+14) = 196 mod 24 = 4 and (1*2*7) mod (1+2+7) = 14 mod 10 = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..600

Crossrefs

Cf. A001065, A007956, A187680, A191906.

Programs

Mathematica

erQ[n_]:=Module[{divs=Divisors[n],ds=DivisorSigma[1,n]},Mod[ Times@@ divs,ds] == Mod[ Times@@Most[divs],ds-n]]; Select[Range[2,30000],erQ] (* Harvey P. Dale, Jun 13 2015 *)
Select[Range[2, 30000], Mod[(p = #^(DivisorSigma[0, #]/2)), (s = DivisorSigma[1, #])] == Mod[p/#, s - #] &] (* Amiram Eldar, Jul 21 2019 *)

Formula

{ k : A187680(k) = A191906(k) }.

Extensions

Values from a(4) onwards from R. J. Mathar, Jul 05 2011