A381115 -id:A381115 - cp's OEIS frontend

A381019 a(n) is the smallest positive integer not yet in the sequence such that a(n) is relatively prime to a(n-i) for all 1 <= i <= min(a(n), n-1).

1, 2, 3, 5, 7, 11, 4, 13, 17, 19, 23, 29, 9, 31, 37, 8, 41, 43, 47, 53, 59, 61, 6, 67, 71, 73, 79, 83, 89, 25, 97, 101, 103, 107, 109, 12, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 10, 173, 179, 181, 191, 193, 197, 199, 49, 211, 223, 227, 229, 233

Offset: 1

Ali Sada and Allan C. Wechsler, Feb 12 2025

Theorem (Russ Cox, Feb 14-16, 2025): Every positive number will eventually appear. For proof see link.

Jinyuan Wang (Feb 16, 2025) has informed us that he also proved that every number appears.

			After a(2)=2, the next term that shares a common factor with 2 is a(7)=4, which is permitted since the difference 7-2 = 5 is greater than 4.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Michael De Vlieger)
Michael S. Branicky, Python program
Russ Cox, Proof that every number appears
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..20000, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers. Composite terms are enlarged for visibility.

Cf. A381115-A381120, A379810.

A381167 is a different but closely related sequence.

N:= 1000: # for terms before the first term > N
Cands:= [$2..N]: R:= [1]: x:= 1:
for n from 2 do
  found:= false;
  for j from 1 to N - n do
    if andmap(t -> igcd(t, Cands[j]) = 1, [seq(R[n-i],i=1 .. min(Cands[j],n-1))]) then
      found:= true; x:= Cands[j]; R:= [op(R),x]; Cands:= subsop(j=NULL,Cands); break
    fi od:
  if not found then break fi
od:
R; # Robert Israel, Feb 14 2025

nn = 120; c[_] = False; u = v = 2; a[1] = 1;
Do[k = u;
  While[Or[c[k],
    ! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ],
    If[k > n - 1, k = v, k++]];
  Set[{a[n], c[k]}, {k, True}];
  If[k == u, While[c[u], u++]];
  If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Feb 14 2025 *)

# see link for faster version
from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, an, m = [1], {1}, 1, 2
    for n in count(2):
        yield an
        an = next(k for k in count(m) if k not in aset and all(gcd(alst[-j], k) == 1 for j in range(1, min(k, n-1)+1)))
        alst.append(an)
        aset.add(an)
        while m in aset: m += 1
print(list(islice(agen(), 61))) # Michael S. Branicky, Feb 13 2025

More terms from Michael S. Branicky, Feb 13 2025

A381116 Indices of composite terms in A381019.

Original entry on oeis.org

7, 13, 16, 23, 30, 36, 47, 55, 63, 64, 79, 91, 100, 113, 123, 142, 149, 167, 178, 196, 201, 223, 235, 256, 259, 279, 290, 325, 330, 346, 364, 382, 405, 422, 442, 468, 485, 488, 530, 534, 541, 583, 605, 630, 631, 665, 674, 682, 729, 735, 790, 798, 847, 854, 862

Offset: 1

N. J. A. Sloane, Feb 14 2025

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..336

Cf. A381019, A381115, A381117.

nn = 1000; c[_] = False; u = v = 2; a[1] = 1;
Monitor[Reap[
  Do[k = u;
    While[Or[c[k],
      ! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ],
      If[k > n - 1, k = v, k++]];
    Set[{a[n], c[k]}, {k, True}];
    If[CompositeQ[k], Sow[n]];
    If[k == u, While[c[u], u++]];
If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Feb 14 2025 *)

from math import gcd
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, an, m = [1], {1}, 1, 2
    for n in count(2):
        if an > 3 and not isprime(an):
            yield n-1
        an = next(k for k in count(m) if k not in aset and all(gcd(alst[-j], k) == 1 for j in range(1, min(k, n-1)+1)))
        alst.append(an)
        aset.add(an)
        while m in aset: m += 1
print(list(islice(agen(), 55))) # Michael S. Branicky, Feb 14 2025

A381117 Lengths of runs of consecutive primes in A381019.

Original entry on oeis.org

5, 5, 2, 6, 6, 5, 10, 7, 7, 14, 11, 8, 12, 9, 18, 6, 17, 10, 17, 4, 21, 11, 20, 2, 19, 10, 34, 4, 15, 17, 17, 22, 16, 19, 25, 16, 2, 41, 3, 6, 41, 21, 24, 33, 8, 7, 46, 5, 54, 7, 48, 6, 7, 5, 41, 13, 31, 18, 5, 50, 1, 49, 10, 26, 41, 24, 45, 53, 20, 21, 44, 3

Offset: 1

N. J. A. Sloane, Feb 14 2025

nonn

For n > 1, a(n) = A381116(n) - A381116(n-1) - 1. (This is a trivial consequence of the definitions.)

Cf. A381019, A381115, A381116.

nn = 500; c[_] = False; i = 0; u = v = 2; a[1] = 1;
Monitor[Reap[
  Do[k = u;
    While[Or[c[k],
      ! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ],
    If[k > n - 1, k = v, k++]];
    Set[{a[n], c[k]}, {k, True}];
    If[CompositeQ[k], Sow[i]; i = 0, i++];
    If[k == u, While[c[u], u++]];
If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Feb 14 2025 *)

A381119 Index of n in A381019.

Original entry on oeis.org

1, 2, 3, 7, 4, 23, 5, 16, 13, 47, 6, 36, 8, 79, 63, 64, 9, 142, 10, 100, 113, 123, 11, 167, 30, 223, 91, 196, 12, 290, 14, 256, 201, 325, 149, 442, 15, 364, 330, 405, 17, 485, 18, 530, 534, 630, 19, 583, 55, 682, 382, 735, 20, 790, 346, 847, 259, 1034, 21, 1095

Offset: 1

N. J. A. Sloane, Feb 14 2025

nonn

Every number does eventually appear in A381019 (see that sequence for proof).

Jinyuan Wang, Table of n, a(n) for n = 1..10000 (terms 1..257 from Michael S. Branicky)

Cf. A381019, A381115, A381116, A381118.

More terms from Alois P. Heinz, Feb 14 2025

A381118 Index of 2^n in A381019.

Original entry on oeis.org

1, 2, 7, 16, 64, 256, 975, 3856, 16647, 65039, 260112, 1044504, 4177980, 16777224

Offset: 0

N. J. A. Sloane, Feb 14 2025

Every power of 2 appears in A381019 (see that entry for proof).

Cf. A381019, A381115, A381116.

a(7) from Michael S. Branicky, Feb 14 2025
a(8) from Michael S. Branicky, Feb 15 2025
a(9)-a(13) from Jinyuan Wang, Feb 16 2025

cp's OEIS Frontend

A381019 a(n) is the smallest positive integer not yet in the sequence such that a(n) is relatively prime to a(n-i) for all 1 <= i <= min(a(n), n-1).

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A381116 Indices of composite terms in A381019.

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A381117 Lengths of runs of consecutive primes in A381019.

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A381119 Index of n in A381019.

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A381118 Index of 2^n in A381019.

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