A362803 Indices of records in A362802.
1, 6, 24, 48, 60, 84
Offset: 1
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.
The first 4 terms of A339665 are 1, 2, 2 and 3. The record values, 1, 2 and 3, occur at 1, 2 and 4, the first 3 terms of this sequence.
c[n_] := Count[Subsets[Divisors[n]], _?(Length[#]>0 && IntegerQ[HarmonicMean[#]] &)]; cm = -1; s = {}; Do[If[(c1 = c[n]) > cm, cm = c1; AppendTo[s, n]], {n, 1, 240}]; s
6 is the smallest number whose set of divisors can be partitioned into two disjoint sets whose arithmetic means are both integers: {1, 3} and {2, 6}.
12 is the smallest number whose set of divisors can be partitioned into two disjoint sets whose arithmetic means are both integers in three ways: ({1, 3}, {2, 4, 6, 12}), ({2, 6}, {1, 3, 4, 12}) and ({4, 12}, {1, 2, 3, 6}).
q[d_] := Length[d] > 1 && IntegerQ @ Mean[d]; c[n_] := Count[Subsets[(d = Divisors[n])], _?(q[#] && q[Complement[d, #]] &)]/2; cm = -1; s = {}; Do[If[(c1 = c[n]) > cm, cm = c1; AppendTo[s, n]], {n, 1, 240}]; s
from itertools import count, islice, combinations from sympy import divisors def A348719_gen(): # generator of terms c = 0 yield 1 for n in count(2): divs = tuple(divisors(n, generator=True)) l, b = len(divs), sum(divs) if l>=4 and 2**(l-1)-l>c: m = sum(1 for k in range(2,(l-1>>1)+1) for p in combinations(divs,k) if not ((s:=sum(p))%k or (b-s)%(l-k))) if l&1 == 0: k = l>>1 m += sum(1 for p in combinations(divs,k) if 1 in p and not ((s:=sum(p))%k or (b-s)%k)) if m > c: yield n c = m A348719_list = list(islice(A348719_gen(),10)) # Chai Wah Wu, Sep 24 2023
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