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A355578 Numbers whose sum of 3-smooth divisors sets a new record.

%I A355578 #18 Jul 08 2022 15:56:00
%S A355578 1,2,3,4,6,8,12,16,18,24,32,36,48,64,72,96,108,144,192,216,288,324,
%T A355578 384,432,576,648,768,864,972,1152,1296,1536,1728,1944,2304,2592,2916,
%U A355578 3072,3456,3888,4608,5184,5832,6912,7776,8748,9216,10368,11664,13824,15552,17496
%N A355578 Numbers whose sum of 3-smooth divisors sets a new record.
%C A355578 Numbers m such that A072079(m) > A072079(k) for all k < m.
%C A355578 All the terms are 3-smooth numbers (A003586).
%C A355578 Equivalently, 3-smooth numbers k such that A000203(k) sets a new record.
%C A355578 Analogous to highly abundant numbers (A002093) with 3-smooth numbers only.
%H A355578 Michael S. Branicky, <a href="/A355578/b355578.txt">Table of n, a(n) for n = 1..11233</a> (all < 10^60)
%e A355578 The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The record values, 1, 3, 4 and 12, occur at 1, 2, 3, 4 and 6, the first 5 terms of this sequence.
%t A355578 s[n_] := Module[{e = IntegerExponent[n, {2, 3}], p}, p = {2, 3}^e; If[Times @@ p == n, (2^(e[[1]] + 1) - 1)*(3^(e[[2]] + 1) - 1)/2, 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 18000}]; seq
%o A355578 (PARI) lista(nmax) = {my(list = List(), smax = 0, e2, e3, s); for(n = 1, nmax, e2 = valuation(n, 2); e3 = valuation(n, 3); s = if(2^e2 * 3^e3 == n, (2^(e2 + 1) - 1)*(3^(e3 + 1) - 1)/2, 0); if(s > smax, smax = s;  listput(list, n))); Vec(list)};
%o A355578 (Python)
%o A355578 from sympy import multiplicity as v
%o A355578 from itertools import count, takewhile
%o A355578 def f(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2
%o A355578 def smooth3(lim):
%o A355578     pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(0))))
%o A355578     pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(0))))
%o A355578     return sorted(c*d for c in pows2 for d in pows3 if c*d <= lim)
%o A355578 def aupto(lim):
%o A355578     data, records, record = smooth3(lim), [], -1
%o A355578     for argv, v in zip(data, map(f, data)):
%o A355578         if v > record: record = v; records.append(argv)
%o A355578     return records
%o A355578 print(aupto(10**5)) # _Michael S. Branicky_, Jul 08 2022
%Y A355578 Subsequence of A003586.
%Y A355578 A355579 is a subsequence.
%Y A355578 Cf. A000203, A002093, A072079, A309015.
%K A355578 nonn
%O A355578 1,2
%A A355578 _Amiram Eldar_, Jul 08 2022