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A356322 a(n) is the smallest number that starts a run of exactly n consecutive numbers in A126706, or -1 if no such number exists.

%I A356322 #30 Aug 27 2024 18:18:33
%S A356322 12,44,98,3174,844,22020,217070,1092747,8870024,262315467,221167422,
%T A356322 47255689915,82462576220,1043460553364,79180770078548
%N A356322 a(n) is the smallest number that starts a run of exactly n consecutive numbers in A126706, or -1 if no such number exists.
%C A356322 Term a(n) begins a run of n consecutive nonsquarefree numbers m such that omega(m) > 1.
%C A356322 The run of m must occur between successive primes.
%H A356322 Rémy Sigrist, <a href="/A356322/a356322.txt">C program</a>
%e A356322 a(n) is the first term in the sequences shown below:
%e A356322 n: a(n)..a(n)+n-1
%e A356322 -----------------
%e A356322 1: {12}
%e A356322 2: {44, 45}
%e A356322 3: {98, 99, 100}
%e A356322 4: {3174, 3175, 3176, 3177}
%e A356322 5: {844, 845, 846, 847, 848}
%e A356322 6: {22020, 22021, 22022, 22023, 22024, 22025}
%e A356322 7: {217070, 217071, 217072, 217073, 217074, 217075, 217076}
%e A356322 ...
%e A356322 There are 4 consecutive numbers m in A126706 starting from 844 and again from 2888, but since 848 and 2892, respectively, are also in A126706, these m ascribe to n = 5 instead. The range m = 3174..3177 has at most n = 4 numbers in A126706 and 3174 is the smallest number with that quality, hence a(4) = 3174.
%t A356322 j = 0; k[_] = False; Sort[Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, j++; If[! IntegerQ[c], Set[c, n]], If[j > 0, If[! k[j], Sow[{j, c}] ]; Set[{k[j], j}, {True, 0}]; Clear[c] ] ], {n, 2^16}] ][[-1, -1]] ][[All, -1]]
%o A356322 (C) // See Links section.
%Y A356322 Cf. A001221, A001222, A001223, A005250, A013929, A024619, A126706.
%K A356322 nonn,more
%O A356322 1,1
%A A356322 _Michael De Vlieger_, Oct 28 2022
%E A356322 a(10)-a(11) from _Rémy Sigrist_, Oct 29 2022
%E A356322 a(12)-a(14) from _Martin Ehrenstein_, Oct 30 2022
%E A356322 a(15) from _Martin Ehrenstein_, Nov 02 2022