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%I #15 Jul 19 2024 14:31:43
%S 2,5,68,33,709,2313,13251,17961,231881,525323,4172904,7163595
%N a(n) is the start of the least run of exactly n consecutive positive numbers with strictly decreasing values of A177329, or -1 if no such run exists.
%C For n > 1, a(n)! is the start the least run of successive factorials with strictly decreasing number of infinitary divisors (A037445).
%C a(9) > 170000, if it exists.
%e n | a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
%e --|-------|------------------------------------------------------
%e 1 | 2 | 1
%e 2 | 5 | 4 > 3
%e 3 | 68 | 31 > 28 > 27
%e 4 | 33 | 21 > 17 > 16 > 15
%e 5 | 709 | 199 > 197 > 195 > 193 > 190
%e 6 | 2313 | 528 > 523 > 519 > 518 > 513 > 508
%e 7 | 13251 | 2355 > 2354 > 2353 > 2351 > 2350 > 2345 > 2343
%e 8 | 17961 | 3060 > 3056 > 3051 > 3049 > 3048 > 3047 > 3044 > 3041
%t s[n_] := Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; seq[len_] := Module[{v = Table[0, {len}], w = {s[2]}, c = 0, k = 3, m, s1}, While[c < len, s1 = s[k]; m = Length[w]; If[s1 < w[[m]], AppendTo[w, s1], If[m <= len && v[[m]] == 0, v[[m]] = k-m; c++]; w = {s1}]; k++]; v]; seq[5]
%o (PARI) s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
%o lista(len) = {my(v = vector(len), w = [s(2)], c = 0, k = 3, m, s1); while(c < len, s1 = s(k); m = #w; if(s1 < w[m], w = concat(w, s1), if(m < = len && v[m] == 0, v[m] = k-m; c++); w = [s1]); k++); v;}
%o (Python)
%o from itertools import count
%o from collections import Counter
%o from sympy import factorint
%o def A374674(n):
%o if n==1: return 2
%o c, a, l = Counter(), 0, 0
%o for m in count(2):
%o c += Counter(factorint(m))
%o b = sum(map(int.bit_count,c.values()))
%o if b<a:
%o l += 1
%o else:
%o if l==n-1:
%o return m-n
%o l = 0
%o a = b # _Chai Wah Wu_, Jul 18 2024
%Y Cf. A037445, A177329, A374671, A374672, A374673.
%K nonn,hard,more,new
%O 1,1
%A _Amiram Eldar_, Jul 16 2024
%E a(9)-a(12) from _Chai Wah Wu_, Jul 18 2024
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