A374673 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.

2, 8, 44, 83, 4475, 75093, 164903, 59480, 1342805

Offset: 1

Author

Amiram Eldar, Jul 16 2024

Keywords

• nonn
• hard
• more

Comments

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).

a(9) > 320000, if it exists.

Examples

			  n |   a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
  --|--------|------------------------------------------------
  1 |      2 | A177329(2) = 1
  2 |      8 | A177329(8) = A177329(9) = 6
  3 |     44 | A177329(44) = A177329(45) = A177329(46) = 21
  4 |     83 | A177329(83) = ... = A177329(86) = 35
  5 |   4475 | A177329(4475) = ... A177329(4479) = 923
  6 |  75093 | A177329(75093) = ... = A177329(75098) = 10857
  7 | 164903 | A177329(164903) = ... = A177329(164909) = 22038
  8 |  59480 | A177329(59480) = ... = A177329(59487) = 8814
		

Crossrefs

Cf. A177329, A318166, A374671, A374672, A374674.

Programs

Mathematica

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]

PARI

s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
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;}

Python

from itertools import count
from collections import Counter
from sympy import factorint
def A374673(n):
    if n==1: return 2
    c, a, l = Counter(), 0, 0
    for m in count(2):
        c += Counter(factorint(m))
        b = sum(map(int.bit_count,c.values()))
        if b==a:
            l += 1
        else:
            if l==n-1:
                return m-n
            l = 0
        a = b # Chai Wah Wu, Jul 18 2024

Extensions

a(9) from Chai Wah Wu, Jul 18 2024