cp's OEIS Frontend

A384354 Numbers k such that the arithmetic mean of the divisors of k evenly divides k+1.

%I A384354 #24 Jun 04 2025 17:58:56
%S A384354 1,2,3,5,7,11,13,17,19,20,23,29,31,35,37,41,43,47,53,59,61,67,71,73,
%T A384354 79,83,89,97,101,103,104,107,109,113,127,131,137,139,149,151,157,163,
%U A384354 167,173,179,181,191,193,197,199,207,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293
%N A384354 Numbers k such that the arithmetic mean of the divisors of k evenly divides k+1.
%C A384354 A term k with a fractional arithmetic mean of divisors is allowed as long as that arithmetic mean evenly divides k+1.
%C A384354 There exist triples such as (19,20,21) and quadruples such as (1,2,3,4) of consecutive numbers where the arithmetic mean of the divisors of every earlier number evenly divides the immediately following number. Are there similar quintuples?
%C A384354 Contains every prime p since (1+p)/2 evenly divides 1+p. - _Michael S. Branicky_, May 29 2025
%e A384354 2 is a term since (1+2)/2 = 3/2 and 3/2 evenly divides 3.
%e A384354 19 is a term since (1+19)/2 is 10 and 10 evenly divides 20.
%e A384354 20 is a term since (1+2+4+5+10+20)/6 = 7 and 7 evenly divides 21.
%t A384354 fQ[n_]:=Divisible[n+1,Mean[Divisors[n]]]; Select[Range[300],fQ]
%o A384354 (Python)
%o A384354 from sympy import divisors
%o A384354 def ok(n): return n and (n+1)*len(d:=divisors(n))%sum(d) == 0
%o A384354 print([k for k in range(300) if ok(k)]) # _Michael S. Branicky_, May 29 2025
%o A384354 (PARI) isok(k) = my(f=factor(k)); denominator((k+1)/(sigma(f)/numdiv(f))) == 1; \\ _Michel Marcus_, May 31 2025
%Y A384354 Cf. A000005, A000040 (subsequence), A000203.
%K A384354 nonn
%O A384354 1,2
%A A384354 _Ivan N. Ianakiev_, May 27 2025