A109473
Let m = n-th squarefree number = A005117(n), and consider the smallest pair of consecutive squarefree numbers (r,s) with gcd(r,s) = m; sequence gives values of r.
Original entry on oeis.org
1, 422, 174, 22830, 9216772051242, 234374
Offset: 1
n | m | a(n) = r
---+----+---------------
1 | 1 | 1
2 | 2 | 422
3 | 3 | 174
4 | 5 | 22830
5 | 6 | 9216772051242
6 | 7 | 234374
7 | 10 | ?
8 | 11 | 21971536246
9 | 13 | 8678016978774
10 | 14 | ?
11 | 15 | 36442589727570
Specifically, 174 is squarefree, 177 is the next squarefree integer, and gcd(174,177) = 3; this is the first pair of consecutive squarefree numbers whose GCD is 3, so a(3)=174. - _Jud McCranie_, Nov 25 2019
Don Reble pointed out that the value of a(5), 9216772051254, given in the DATA section should have been 9216772051242, as in the EXAMPLE section. Revised definition to clarify the difference between n and m. -
N. J. A. Sloane, Nov 25 2019
A337914
Squarefree multiples of 11 such that a(n)+11 is also squarefree and all integers between are not squarefree.
Original entry on oeis.org
21971536246, 28062102871, 35322768074, 70470712246, 80352468746, 81659032818, 91738254619, 92802362774, 99529917146, 112184349167, 114294175743, 143264113674, 183904912070, 199569905722, 219349795115, 223879398215, 242719744322, 261356598415
Offset: 1
21971536246 and 21971536246+11 are successive squarefree integers, and their GCD is 11.
A337915
Squarefree multiples of 15 such that a(n)+15 is also squarefree and all integers between are not squarefree.
Original entry on oeis.org
36442589727570, 59068995605115, 70458333341415, 93553802716515, 99321073884915, 104575423448670, 225934088816670, 241427108122215, 245679971474670, 257496300981015, 263154806220570, 275458942061070, 312686748592170, 425667633054315, 448345051339515
Offset: 1
A373704
a(n) is the least start of a run of exactly n successive squarefree numbers that are pairwise coprime, or -1 if no such run exists.
Original entry on oeis.org
2526, 173, 5, 1, 17, 11, 89, 1049, 111539, 213341, 54848527
Offset: 1
a(1) = 2526 because 2526 is squarefree, it is preceded by the squarefree number 2522 and gcd(2522, 2526) = 2 > 1, it is followed by the squarefree number 2530 and gcd(2526, 2530) = 2 > 1, and 2526 is the least number with this property.
a(2) = 173 because 173 and 174 are successive squarefree numbers that are coprime. 170, the squarefree number that precedes 173, and 177, the squarefree number that follows 174, are both not coprime to 174.
a(2) does not equal 170 because although 170 and 173 are two successive squarefree numbers that are coprime, they are a part of a longer run of 3 successive squarefree numbers that are pairwise coprime: 167, 170 and 173.
a(3) = 5 because 5, 6 and 7 are 3 successive squarefree numbers that are pairwise coprime: gcd(5, 6) = gcd(6, 7) = gcd(5, 7) = 1. They are not a part of a longer run since the squarefree number that precedes 5 is 3 and gcd(3, 6) = 3 > 1, and the squarefree number that follows 7 is 10 and gcd(5, 10) = 5 > 1. (5, 6, 7) is the run with the least start, 5, that has this property.
-
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];
seq[nmax_, kmax_] := Module[{v = Table[0, {nmax}], s = {}, len = 0, init = 0, c = 0}, Do[If[SquareFreeQ[k], 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 || k > kmax, Break[]]], {k, 1, kmax}]; v]; seq[10, 250000]
-
iscoprime(s) = {for(i = 1, #s, for(j = 1, i-1, if(gcd(s[i], s[j]) > 1, return(0)))); 1;}
seq(nmax, kmax) = {my(v = vector(nmax), s = List(), len = 0, init = 0, c = 0); forsquarefree(k = 1, kmax, len = #s; listput(s, k[1]); while(!iscoprime(s), listpop(s, 1)); if(#s <= len, if(len <= nmax && v[len] == 0, c++; v[len] = init)); init = s[1]; if(c == nmax || k[1] > kmax, break)); v;}
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