A342401 Numbers k such that the number of distinct sums of distinct unitary divisors of k sets a new record.

1, 2, 6, 10, 30, 42, 60, 66, 78, 90, 110, 130, 170, 190, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 1974, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006

Offset: 1

Author

Amiram Eldar, Mar 10 2021

Keywords

• nonn

Comments

The corresponding record values are 1, 3, 12, 15, 72, 96, 108, 144, 168, 172, 183, 207, 231, 255, 576, 864, ... (see the link for more values).

The analogous sequence of records with all the divisors (A119347) is conjecturally the highly abundant numbers (A002093), excluding 3 and 10.

Examples

			The first 6 terms of A342400 are 1, 3, 3, 3, 3 and 12. The record values, 1, 3 and 12, occur are 1, 2 and 6, the first 3 terms of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..99
• Amiram Eldar, Table of n, a(n), A342400(a(n)) for n = 1..99

Crossrefs

Cf. A002093, A119347, A342400.

Programs

Mathematica

f[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], x, s, m, c}, m = Length[d]; s = Plus @@ d; c = Rest @ CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, m}], {x, 0, s}], x]; Count[c, _?(# > 0 &)]]; fmax = -1; seq = {}; Do[f1 = f[n]; If[f1 > fmax, fmax = f1; AppendTo[seq, n]], {n, 1, 10^3}]; seq