cp's OEIS Frontend

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).

%I A192035 #22 Jun 27 2024 22:18:24
%S A192035 6,14,28,51,120,260,270,496,672,679,752,924,1260,1320,1540,1960,2055,
%T A192035 2262,2651,3808,3948,4381,6413,6435,6944,7900,7980,8010,8128,9809,
%U A192035 9945,10242,10920,12690,15456,16830,18018,21728,21970,22320,25296,27930,29190,29792
%N 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).
%C A192035 The even perfect numbers (A000396) are a subsequence.
%C A192035 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
%H A192035 Amiram Eldar, <a href="/A192035/b192035.txt">Table of n, a(n) for n = 1..600</a>
%F A192035 { k : A187680(k) = A191906(k) }.
%e A192035 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.
%t A192035 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 *)
%t A192035 Select[Range[2, 30000], Mod[(p = #^(DivisorSigma[0, #]/2)), (s = DivisorSigma[1, #])] == Mod[p/#, s - #] &] (* _Amiram Eldar_, Jul 21 2019 *)
%Y A192035 Cf. A001065, A007956, A187680, A191906.
%K A192035 nonn
%O A192035 1,1
%A A192035 _Juri-Stepan Gerasimov_, Jun 21 2011
%E A192035 Values from a(4) onwards from _R. J. Mathar_, Jul 05 2011