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A374536 a(n) is the least exponentially odd number that is nonsquarefree and is followed by exactly n successive exponentially odd numbers that are squarefree, or -1 if no such number exists.

%I A374536 #10 Apr 26 2025 20:13:14
%S A374536 135,24,120,27,96,88,32,40,328,168,136,104,1288,1161,352,488,8,783,
%T A374536 189,952,4520,56,11576,67384,5088,1336,35768,16173,53768,80328,128169,
%U A374536 28576,247375,208552,2556192,1486568,3099368,1653032,910568,7864008,34242976,14484152
%N A374536 a(n) is the least exponentially odd number that is nonsquarefree and is followed by exactly n successive exponentially odd numbers that are squarefree, or -1 if no such number exists.
%H A374536 Amiram Eldar, <a href="/A374536/b374536.txt">Table of n, a(n) for n = 0..53</a>
%F A374536 a(n) = A268335(A374460(k)), where k is the least number such that A374459(k+1) - A374459(k) = n + 1.
%e A374536 a(0) = 135 because 135 and 136 are successive nonsquarefree exponentially odd numbers with no squarefree number between them.
%e A374536 a(1) = 24 because 24 and 27 are successive nonsquarefree exponentially odd numbers with one squarefree number between them, 26.
%e A374536 a(2) = 120 because 120 and 125 are successive nonsquarefree exponentially odd numbers with two squarefree number between them, 122 and 123.
%t A374536 sq[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, If[AnyTrue[e, EvenQ], 0, If[k == 1 || Max[e] == 1, 2, 1]]]; seq[len_, kmax_ : Infinity] := Module[{v = Table[0, {len}], c = 0, k = 1, k0 = 0, m, i = 1}, While[c < len && k < kmax, m = sq[k]; If[m > 0, If[m == 2, i++, If[k0 > 0, If[i <= len && v[[i]] == 0, c++; v[[i]] = k0]; i = 1];	k0 = k]]; k++]; v]; seq[10]
%o A374536 (PARI) issq(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] % 2), return(0))); if(k == 1 || vecmax(e) == 1, 2, 1);}
%o A374536 lista(len, kmax = oo) = {my(v = vector(len), c = 0, k = 1, k0 = 0, m, i = 1); while(c < len && k < kmax, m = issq(k); if(m > 0, if(m == 2, i++, if(k0 > 0, if(i <= len && v[i] == 0, c++; v[i] = k0); i = 1); k0 = k)); k++); v; }
%Y A374536 Cf. A005117, A045882, A268335, A270344, A374459, A374460.
%K A374536 nonn
%O A374536 0,1
%A A374536 _Amiram Eldar_, Jul 11 2024