A358535 - OEIS

%I #9 Jan 28 2023 10:39:05

%S 1,2,7,8,12,27,9,18,24,64,36,152,72,128,56,84,148,80,96,316,178,88,

%T 132,206,214,104,156,622,288,344,136,204,529,228,586,343,396,484,184,

%U 276,498,384,530,225,450,469,468,549,449,511,232,348,479,545,248,372,509

%N a(1) = 1; let S(n) = Sum_{j=1..n-1} a(j) and let T(n) = Sum_{j=1..n-1} d(a(j)) where d(n) = A000005(n). a(n) = least novel k such that (T(n)+d(k)) | (S(n)+k).

%C In other words, for n > 1, a(n) = smallest k not in a(1)..a(n-1) such that the partial sums of the number of divisors of a(j) divides the partial sum of a(j), 1 <= j < n.

%C Conjecture: 3 does not appear in the sequence.

%H Michael De Vlieger, <a href="/A358535/a358535.png">Log log scatterplot of a(n)</a>, n = 1..40000.

%H Michael De Vlieger, <a href="/A358535/a358535_1.png">Annotated log log scatterplot of a(n)</a>, n = 1..5000, color coded to show primes in red, composite prime powers in gold, squarefree composites in green, products of composite prime powers in large light blue, and other numbers in dark blue.

%e a(2) = 2 since t(1) + d(2) | s(1) + 2 = 1+2 | 1+2 = 3 | 3,

%e a(3) = 7 since t(2) + d(7) | s(2) + 7 = 3+2 | 3+7 = 5 | 10,

%e a(4) = 8 since t(3) + d(8) | s(3) + 8 = 5+4 | 10+8 = 9 | 18,

%e a(5) = 12 since t(4) + d(12) | s(4) + 12 = 9+6 | 18+12 = 15 | 30,

%e a(6) = 27 since t(5) + d(27) | s(5) + 27 = 15+4 | 30+27 = 19 | 57, etc.

%t nn = 2^10; c[_] = False; a[1] = s = t = 1; c[1] = True; u = 2; Do[k = u; While[Nand[! c[k], Divisible[(s + k), (t + Set[d, DivisorSigma[0, k]])]], k++]; Set[{a[n], c[k]}, {k, True}]; s += k; t += d; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn]

%Y Cf. A000005.

%K nonn

%O 1,2

%A _Michael De Vlieger_ and _David James Sycamore_, Dec 07 2022