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%I #12 Jul 19 2024 14:31:33
%S 2,8,44,83,4475,75093,164903,59480,1342805
%N a(n) is the start of the least run of exactly n consecutive positive numbers with an equal value of A177329, or -1 if no such run exists.
%C For n > 1, a(n)! is the start of the least run of successive factorials of positive numbers (i.e., ignoring 0!) with an equal number of infinitary divisors (A037445).
%C a(9) > 320000, if it exists.
%e n | a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
%e --|--------|------------------------------------------------
%e 1 | 2 | A177329(2) = 1
%e 2 | 8 | A177329(8) = A177329(9) = 6
%e 3 | 44 | A177329(44) = A177329(45) = A177329(46) = 21
%e 4 | 83 | A177329(83) = ... = A177329(86) = 35
%e 5 | 4475 | A177329(4475) = ... A177329(4479) = 923
%e 6 | 75093 | A177329(75093) = ... = A177329(75098) = 10857
%e 7 | 164903 | A177329(164903) = ... = A177329(164909) = 22038
%e 8 | 59480 | A177329(59480) = ... = A177329(59487) = 8814
%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 = {0}, 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 = [0], 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 A374673(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. A177329, A318166, A374671, A374672, A374674.
%K nonn,hard,more
%O 1,1
%A _Amiram Eldar_, Jul 16 2024
%E a(9) from _Chai Wah Wu_, Jul 18 2024
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