A358535 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).

1, 2, 7, 8, 12, 27, 9, 18, 24, 64, 36, 152, 72, 128, 56, 84, 148, 80, 96, 316, 178, 88, 132, 206, 214, 104, 156, 622, 288, 344, 136, 204, 529, 228, 586, 343, 396, 484, 184, 276, 498, 384, 530, 225, 450, 469, 468, 549, 449, 511, 232, 348, 479, 545, 248, 372, 509

Offset

1,2

Comments

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.

Conjecture: 3 does not appear in the sequence.

Links

Table of n, a(n) for n=1..57.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..40000.

Michael De Vlieger, Annotated log log scatterplot of a(n), 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.

Example

a(2) = 2 since t(1) + d(2) | s(1) + 2 = 1+2 | 1+2 = 3 | 3,
a(3) = 7 since t(2) + d(7) | s(2) + 7 = 3+2 | 3+7 = 5 | 10,
a(4) = 8 since t(3) + d(8) | s(3) + 8 = 5+4 | 10+8 = 9 | 18,
a(5) = 12 since t(4) + d(12) | s(4) + 12 = 9+6 | 18+12 = 15 | 30,
a(6) = 27 since t(5) + d(27) | s(5) + 27 = 15+4 | 30+27 = 19 | 57, etc.

Mathematica

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]

Crossrefs

Cf. A000005.

Sequence in context: A070044 A129850 A037073 * A329407 A329408 A047239

Adjacent sequences: A358532 A358533 A358534 * A358536 A358537 A358538

Keyword

nonn

Author

Michael De Vlieger and David James Sycamore, Dec 07 2022

Status

approved