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.

135, 24, 120, 27, 96, 88, 32, 40, 328, 168, 136, 104, 1288, 1161, 352, 488, 8, 783, 189, 952, 4520, 56, 11576, 67384, 5088, 1336, 35768, 16173, 53768, 80328, 128169, 28576, 247375, 208552, 2556192, 1486568, 3099368, 1653032, 910568, 7864008, 34242976, 14484152

Offset: 0

Author

Amiram Eldar, Jul 11 2024

Keywords

• nonn

Examples

			a(0) = 135 because 135 and 136 are successive nonsquarefree exponentially odd numbers with no squarefree number between them.
a(1) = 24 because 24 and 27 are successive nonsquarefree exponentially odd numbers with one squarefree number between them, 26.
a(2) = 120 because 120 and 125 are successive nonsquarefree exponentially odd numbers with two squarefree number between them, 122 and 123.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..53

Crossrefs

Cf. A005117, A045882, A268335, A270344, A374459, A374460.

Programs

Mathematica

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]

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);}
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; }

Formula

a(n) = A268335(A374460(k)), where k is the least number such that A374459(k+1) - A374459(k) = n + 1.