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.

72, 1, 9, 289, 702464, 7827111875, 1321223037317, 1795433547131287

Offset

1,1

Comments

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.

a(9) > 10^16, if it exists.

Links

Table of n, a(n) for n=1..8.

Example

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.
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.
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.

Mathematica

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];
pows[lim_] := Union[Flatten[Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]];
seq[nmax_, lim_] := Module[{v = Table[0, {nmax}], s = {}, len = 0, init = 0, c = 0}, Do[len = Length[s];
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]

Prog

(PARI) iscoprime(s) = {for(i = 1, #s, for(j = 1, i-1, if(gcd(s[i], s[j]) > 1, return(0)))); 1; }
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); }
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; }
lista(6, 10^10)

Crossrefs

Cf. A001694, A373704.

Sequences related to successive powerful numbers: A349062, A363191, A363192.

Sequence in context: A240254 A375942 A228095 * A036186 A036206 A249700

Adjacent sequences: A373702 A373703 A373704 * A373706 A373707 A373708

Keyword

nonn,hard,more

Author

Amiram Eldar, Jun 14 2024

Status

approved