A138244 - OEIS

%I #23 Jan 28 2023 12:11:42

%S 1,2,5,7,11,17,25,37,47,61,79,101,127,151,167,197,229,257,277,311,347,

%T 389,433,479,541,593,631,691,739,797,853,907,947,997,1061,1129,1201,

%U 1277,1361,1427,1517,1597,1667,1741,1823,1901,1987,2081,2179,2267,2371

%N a(1)=1; for n>1, a(n) = smallest integer > a(n-1) such that a(n) is coprime to (a(k) - a(k-1)) for all k, 2 <= k <= n and such that (a(n) - a(n-1)) doesn't equal (a(k) - a(k-1)) for any k, 2 <= k <= n-1.

%C a(n+1) - a(n) = A138245(n).

%H Michel Marcus, <a href="/A138244/b138244.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Joshua Zucker, Mar 09 2008)

%t a[1] = 1; a[n_] := a[n] = Module[{m = a[n - 1] + 1}, While[MemberQ[(t = Table[a[k] - a[k - 1], {k, 2, n - 1}]), m - a[n - 1]] || ! CoprimeQ[m, m - a[n - 1]] || !AllTrue[t, CoprimeQ[m, #] &], m++]; m]; Array[a, 50] (* _Amiram Eldar_, Nov 16 2021 *)

%o (PARI) isok(dlist, last, k) = {if (gcd(k, k-last) != 1, return (0)); for (i=1, #dlist, if (gcd(k, dlist[i]) != 1, return(0)); if ((k-last) == dlist[i], return(0));); return(1);}

%o findnext(dlist, last, n) = {my(k=last+1); while (!isok(dlist, last, k), k++); k;}

%o lista(nn) = {my(list=List(), dlist=List()); listput(list, 1); for (n=2, nn, my(x=findnext(dlist, list[n-1], n)); listput(dlist, x-list[n-1]); listput(list, x);); Vec(list);} \\ _Michel Marcus_, Nov 16 2021

%o (Python)

%o from math import gcd

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     yield from [1, 2]

%o     anm1, an, diffs, mind = 1, 2, {1}, 2

%o     while True:

%o         anm1, an = an, max(an+1, mind)

%o         while an-anm1 in diffs or gcd(an, an-anm1) != 1 or any(gcd(an, dk) != 1 for dk in diffs):

%o             an += 1

%o         diffs.add(an - anm1)

%o         while mind in diffs: mind += 1

%o         yield an

%o print(list(islice(agen(), 51))) # _Michael S. Branicky_, Jan 28 2023

%Y Cf. A138245.

%K nonn

%O 1,2

%A _Leroy Quet_, Mar 08 2008

%E More terms from _Joshua Zucker_, Mar 09 2008