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A342401
Numbers k such that the number of distinct sums of distinct unitary divisors of k sets a new record.
1
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
1,2
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.
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 10 2021
STATUS
approved