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a(n) is the least start of a run of exactly n consecutive integers with an even number of Fermi-Dirac factors (A064547).
1

%I #6 Dec 24 2023 02:44:34

%S 1,14,20,62,32,115,1464,141,1343,3090,1135,13382,47752,126658,246406,

%T 68420,1277700,205737,4457044,382196,2211385,21274837,9444464,

%U 29186922,52699949,140918773,185359148,108907632,218549045,2655616337,7466369825,2879711373,20145833217,34980297750,42278164437

%N a(n) is the least start of a run of exactly n consecutive integers with an even number of Fermi-Dirac factors (A064547).

%t q[n_] := EvenQ[Sum[DigitCount[e, 2, 1], {e, FactorInteger[n][[;;, 2]]}]]; q[1] = True; seq[len_] := Module[{s = Table[0, {len}], n = 1, count = 0, n1, d}, While[count < len, n1 = n; If[q[n], While[q[++n1]]; d = n1 - n; If[d <= len && s[[d]] == 0, count++; s[[d]] = n]]; n = n1 + 1]; s]; seq[16]

%o (PARI) is(n) = {my(e = factor(n)[, 2]); !(sum(i = 1, #e, hammingweight(e[i])) % 2);}

%o lista(len) = {my(s = vector(len), n = 1, count = 0, n1, d); while(count < len, n1 = n; if(is(n), n1++; while(is(n1), n1++); d = n1 - n; if(d <= len && s[d] == 0, count++; s[d] = n)); n = n1 + 1); s};

%Y Cf. A000379, A064547, A368407.

%Y Analogous to A275508.

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Dec 23 2023