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A373705 a(n) is the least start of a run of exactly n successive powerful numbers that are pairwise coprime, or -1 if no such run exists.

%I A373705 #12 Jun 16 2024 02:18:57
%S A373705 72,1,9,289,702464,7827111875,1321223037317,1795433547131287
%N A373705 a(n) is the least start of a run of exactly n successive powerful numbers that are pairwise coprime, or -1 if no such run exists.
%C A373705 A run of exactly n successive powerful numbers is composed of n successive powerful numbers such that the powerful number that precedes the start of the run (if it does not start with 1) and the powerful number that follows the endpoint of the run are both not coprime to one of the members of the run.
%C A373705 a(9) > 10^16, if it exists.
%e A373705 a(1) = 72 because 72 is powerful, it is preceded by the powerful number 64 and gcd(64, 72) = 8 > 1, it is followed by the powerful number 81 and gcd(72, 81) = 9 > 1, and 72 is the least number with this property.
%e A373705 a(2) = 1 because 1 and 4 are successive powerful numbers that are coprime. 8, the powerful number that follows 4, is not coprime to 4 since gcd(4, 8) = 4 > 1.
%e A373705 a(3) = 9 because 9, 16 and 25 are 3 successive powerful numbers that are pairwise coprime: gcd(9, 16) = gcd(16, 25) = gcd(9, 25) = 1. They are not a part of a longer run since the powerful number that precedes 9 is 8 and gcd(8, 16) = 8 > 1, and the powerful number that follows 25 is 27 and gcd(9, 27) = 9 > 1. (9, 16, 25) is the run with the least start, 9, that has this property.
%t A373705 pairCoprimeQ[s_] := Module[{ans = True}, Do[Do[If[! CoprimeQ[s[[i]], s[[j]]], ans = False; Break[]], {j, 1, i - 1}], {i, 1, Length[s]}]; ans];
%t A373705 pows[lim_] := Union[Flatten[Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]];
%t A373705 seq[nmax_, lim_] := Module[{v = Table[0, {nmax}], s = {}, len = 0, init = 0, c = 0}, Do[len = Length[s];
%t A373705 AppendTo[s, k]; While[!pairCoprimeQ[s], s = Drop[s, 1]]; If[Length[s] <= len, If[len <= nmax && v[[len]] == 0, c++; v[[len]] = init]]; init = s[[1]]; If[c == nmax, Break[]], {k, pows[lim]}]; v]; seq[6, 10^10]
%o A373705 (PARI) iscoprime(s) = {for(i = 1, #s, for(j = 1, i-1, if(gcd(s[i], s[j]) > 1, return(0)))); 1;}
%o A373705 pows(lim) = {my(pows = List()); for(j = 1, sqrtnint(lim, 3), for(i = 1, sqrtint(lim \ j^3), listput(pows, i^2 * j^3))); Set(pows);}
%o A373705 lista(nmax, lim) = {my(pws = pows(lim), v = vector(nmax), s = List(), len = 0, init = 0); for(k = 1, #pws, len = #s; listput(s, pws[k]); while(!iscoprime(s), listpop(s, 1)); if(#s <= len, if(len <= nmax && v[len] == 0, v[len] = init)); init = s[1]); v;}
%o A373705 lista(6, 10^10)
%Y A373705 Cf. A001694, A373704.
%Y A373705 Sequences related to successive powerful numbers: A349062, A363191, A363192.
%K A373705 nonn,hard,more
%O A373705 1,1
%A A373705 _Amiram Eldar_, Jun 14 2024