A381117 -id:A381117 - cp's OEIS frontend

A381116 Indices of composite terms in A381019.

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

A381115 Composite terms in A381019 in order of appearance.

Original entry on oeis.org

4, 9, 8, 6, 25, 12, 10, 49, 15, 16, 14, 27, 20, 21, 22, 18, 35, 24, 169, 28, 33, 26, 85, 32, 57, 77, 30, 34, 39, 55, 38, 51, 40, 91, 36, 121, 42, 65, 44, 45, 529, 48, 119, 46, 95, 81, 143, 50, 63, 52, 54, 115, 56, 841, 187, 69, 62, 125, 87, 64, 133, 75, 58, 221

Offset: 1

N. J. A. Sloane, Feb 14 2025

nonn

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

Cf. A381019, A381116, A381117.

nn = 500; 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[k]];
    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 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(), 64))) # Michael S. Branicky, Feb 14 2025

A381616 a(n) is the smallest prime that starts the first occurrence of exactly n consecutive primes in A381019.

Original entry on oeis.org

7643, 31, 3517, 1049, 2, 41, 173, 401, 523, 113, 337, 449, 6599, 251, 1993, 2543, 743, 593, 1481, 1301, 1069, 2357, 17657, 4079, 2797, 8219, 64123, 81299, 19289, 40129, 6709, 13999, 4271, 1669, 37579, 28793, 38039, 12413, 125711, 24907, 3181, 41597, 27253

Offset: 1

Gonzalo Martínez, Mar 01 2025

nonn

As the sequence grows, increasingly longer chains of consecutive prime numbers begin to appear.

Conjecture: a(n) always exists.

			For n = 2, we observe that 9, 31, 37, and 8 are four consecutive terms of A381019, where 31 and 37 are exactly two consecutive primes and represent the first occurrence of two consecutive terms that are prime. So, a(2) = 31.

Cf. A381019, A381117.

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A381116 Indices of composite terms in A381019.

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A381115 Composite terms in A381019 in order of appearance.

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A381616 a(n) is the smallest prime that starts the first occurrence of exactly n consecutive primes in A381019.

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