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A382607 Natural numbers ordered by the probability (lowest to highest) to occur in the sum of repeated rolls of a fair 6-sided die.

%I A382607 #28 May 06 2025 16:15:10
%S A382607 1,2,3,7,4,8,13,9,14,19,18,24,25,20,30,29,31,35,36,41,40,34,46,42,47,
%T A382607 45,51,52,57,56,58,62,63,68,67,69,73,74,79,78,84,80,85,83,89,90,95,94,
%U A382607 96,100,101,91,106,105,107,111,112,117,116,122,118,123,128,127
%N A382607 Natural numbers ordered by the probability (lowest to highest) to occur in the sum of repeated rolls of a fair 6-sided die.
%C A382607 The asymptotic probability for large n is 2/7 since the average roll of a die is 7/2.
%C A382607 Only terms with probability < 2/7 occur. - _Michael S. Branicky_, Apr 01 2025
%C A382607 Of any six consecutive integers, at least one is present and gives a maximum in the sequence (i.e., all terms preceding it are smaller). - _Javier Múgica_, May 01 2025
%H A382607 Alois P. Heinz, <a href="/A382607/b382607.txt">Table of n, a(n) for n = 1..10000</a>
%e A382607 The probability of achieving a '6' in n>=6 rolls is 1/6 + 5/36 + 10/216 + 10/1296 + 5/7776 + 1/46656 which is about 36.02%.
%e A382607 The probability of achieving a '1' is just 1/6 (about 16.67%). 1 is the lowest of all, so a(1)=1.
%o A382607 (Python)
%o A382607 from fractions import Fraction
%o A382607 from math import factorial, prod
%o A382607 from itertools import count, islice
%o A382607 from sympy.utilities.iterables import partitions
%o A382607 def prob(n): return sum(factorial(N:=sum(p.values()))//prod(factorial(v) for v in p.values())*Fraction(1, 6**N) for p in partitions(n, k=6))
%o A382607 def agen(): # generator of terms
%o A382607     n, vdict = 1, dict()
%o A382607     for k in count(1):
%o A382607         vdict[prob(k)] = k
%o A382607         if k%6 == 0:
%o A382607             s = [vdict[v] for v in sorted(vdict) if v < Fraction(2, 7)]
%o A382607             yield from (s[i-1] for i in range(n, len(s)-1))
%o A382607             n = len(s) - 1
%o A382607 print(list(islice(agen(), 20))) # _Michael S. Branicky_, Apr 01 2025
%Y A382607 Complement of A382606. Cf. A365443.
%K A382607 nonn
%O A382607 1,2
%A A382607 _Sergio Pimentel_, Mar 31 2025