This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295265 #46 Aug 17 2025 01:06:46
%S A295265 4,8,10,13,14,16,19,20,21,22,26,28,30,32,34,38,39,40,43,44,46,50,52,
%T A295265 53,56,58,60,62,63,64,68,70,72,74,76,80,82,86,88,89,90,92,94,98,99,
%U A295265 100,103,104,106,110,111,112,116,117,118,122,124,128,130,132,134,135
%N A295265 Numbers m such that sum of its i first divisors equals the sum of its j first non-divisors for some i, j.
%C A295265 Or numbers m such that Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) for some i, j where d(k) are the i first divisors and nd(k) the j non-divisors of m.
%C A295265 The corresponding sums are 3, 3, 3, 14, 3, 3, 20, 3, 11, 3, 3, (3 or 14), 11, 3, 3, 3, 17, 3, 44, 3, 3, 3, 3, 54, 3, 3, 15, 3, 11, 3, 3, 3, 33, 3, 3, 3, ... containing the set of primes {3, 11, 17, 23, 29, 37, 41, 43, 53, 59, 61, 71, 79, ...}.
%C A295265 The equality Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) is not always unique, for instance for a(12) = 28, we find 1 + 2 = 3 and 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14.
%C A295265 The primes of the sequence are 13, 19, 43, 53, 89, 103, 151, 229, 251, 349, 433, ... (primes of the form k(k+1)/2 - 2; see A124199).
%C A295265 +-----+-----+-----+------+-----------------------------------------+
%C A295265 | n | i | j | a(n) | Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) |
%C A295265 +-----+-----+-----+------+-----------------------------------------+
%C A295265 | 1 | 2 | 1 | 4 | 1 + 2 = 3 |
%C A295265 | 2 | 2 | 1 | 8 | 1 + 2 = 3 |
%C A295265 | 3 | 2 | 1 | 10 | 1 + 2 = 3 |
%C A295265 | 4 | 2 | 4 | 13 | 1 + 13 = 2 + 3 + 4 + 5 = 14 |
%C A295265 | 5 | 2 | 1 | 14 | 1 + 2 = 3 |
%C A295265 | 6 | 2 | 1 | 16 | 1 + 2 = 3 |
%C A295265 | 7 | 2 | 5 | 19 | 1 + 19 = 2 + 3 + 4 + 5 + 6 = 20 |
%C A295265 | 8 | 2 | 1 | 20 | 1 + 2 = 3 |
%C A295265 | 9 | 3 | 3 | 21 | 1 + 3 + 7 = 2 + 4 + 5 = 11 |
%C A295265 | 10 | 2 | 1 | 22 | 1 + 2 = 3 |
%C A295265 | 11 | 2 | 1 | 26 | 1 + 2 = 3 |
%C A295265 | 12 | 2 | 1 | 28 | 1 + 2 = 3 |
%C A295265 | | 4 | 3 | 28 | 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14 |
%C A295265 | 13 | 4 | 2 | 30 | 1 + 2 + 3 + 5 = 4 + 7 = 11 |
%C A295265 | 14 | 2 | 1 | 32 | 1 + 2 = 3 |
%H A295265 Amiram Eldar, <a href="/A295265/b295265.txt">Table of n, a(n) for n = 1..10000</a>
%e A295265 30 is in the sequence because d(1) + d(2) + d(3) + d(4) = 1 + 2 + 3 + 5 = 11 and nd(1) + nd(2) = 4 + 7 = 11.
%p A295265 with(numtheory):nn:=300:
%p A295265 for n from 1 to nn do:
%p A295265 d:=divisors(n):n0:=nops(d):lst:={}:ii:=0:
%p A295265 for i from 1 to n do:
%p A295265 lst:=lst union {i}:
%p A295265 od:
%p A295265 lst:=lst minus d:n1:=nops(lst):
%p A295265 for m from 1 to n0 while(ii=0) do:
%p A295265 s1:=sum(‘d[i]’, ‘i’=1..m):
%p A295265 for j from 1 to n1 while(ii=0) do:
%p A295265 s2:=sum(‘lst[i]’, ‘i’=1..j):
%p A295265 if s1=s2
%p A295265 then
%p A295265 ii:=1:printf(`%d, `,n):
%p A295265 else
%p A295265 fi:
%p A295265 od:
%p A295265 od:
%p A295265 od:
%t A295265 fQ[n_] := Block[{d = Divisors@ n}, nd = nd = Complement[Range@ n, d]; Intersection[Accumulate@ d, Accumulate@ nd] != {}]; Select[ Range@135, fQ] (* _Robert G. Wilson v_, Mar 06 2018 *)
%o A295265 (PARI) isok(n) = {d = divisors(n); psd = vector(#d, k, sum(j=1, k, d[j])); nd = setminus([1..n], d); psnd = vector(#nd, k, sum(j=1, k, nd[j])); #setintersect(psd, psnd) != 0;} \\ _Michel Marcus_, May 05 2018
%Y A295265 Cf. A064510, A124199, A185729, A240698.
%K A295265 nonn
%O A295265 1,1
%A A295265 _Michel Lagneau_, Feb 22 2018