A381019 -id:A381019 - 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

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

A381120 Numbers k such that both A381019(k) and A381019(k+1) are composite.

Original entry on oeis.org

63, 630, 2423, 5653, 9104, 26308, 36108, 41622, 64526, 85121, 108917, 143913, 148305, 176405, 316974, 399168, 399907, 406487, 536926, 621016, 830793, 937038, 937109, 970243, 1088629, 1480545, 1895503, 3961587, 4651102, 5171081, 5487450, 6219705, 7327856, 8118740

Offset: 1

N. J. A. Sloane, Feb 15 2025

nonn

Initially, A381019 consists of runs of primes separated by single composite numbers. These indices k mark the beginning of two or more consecutive composite numbers in A381019.

Cf. A381019, A379810 (the values of A381019(k)).

lista(nn) = my(c, f, m=1, q=1, r=0, t, u=List([]), w=vector(nn)); for(n=2, nn, t=0; forprime(p=2, max(m, sqrtint(n)), c=n-1-w[p]; if(c>1&&!w[c], listput(u, c))); listsort(u, 1); for(i=1, #u, c=u[i]; f=factor(c)[, 1]; t=1; for(j=1, #f, if(n-c<=w[f[j]], t=0; break)); if(t, u=u[i+1..#u]; if(r, print1(n-1, ", ")); r=1; w[c]=1; for(j=1, #f, w[f[j]]=n); m=max(m, f[#f]); break)); if(!t, r=0; u=List([]); if(nn>q=nextprime(q+1), w[q]=n))); \\ Jinyuan Wang, Feb 16 2025

a(7)-a(8) from Michael De Vlieger, Feb 15 2025

a(9)-a(34) from Jinyuan Wang, Feb 16 2025

A379810 Composite numbers in A381019 which are immediately followed by another composite number, in order of their appearance.

Original entry on oeis.org

15, 46, 104, 305, 261, 1691, 380, 406, 508, 1175, 3281, 7729, 2827, 10877, 6289, 13289, 1737, 4829, 2945, 3205, 9673, 1940, 21253, 1970, 31921, 21127, 34861, 5457, 22219, 120983, 99893, 110843, 148613, 105029, 164107, 12905, 89279, 15245, 195617, 79909, 89827

Offset: 1

N. J. A. Sloane, Feb 15 2025

nonn

Conjecture: For any k >= 1, A381019 contains k consecutive composite terms.

When is the first instance of three consecutive composite terms?

Cf. A381019, A381120.

a(n) = A381019(A381120(n)).

a(6)-a(8) from Michael De Vlieger, Feb 15 2025

More terms from Jinyuan Wang, Feb 23 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

A381222 Smallest number missing from A381019 after A381019(n) has been found.

Original entry on oeis.org

2, 3, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 18, 18, 18

Offset: 1

N. J. A. Sloane, Feb 19 2025

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..9998

Cf. A381019, A381119.

A381223 Take the list (A381222) of successive values of the smallest number missing from A381019, and keep just the first of any run of successive equal terms.

Original entry on oeis.org

2, 3, 4, 6, 10, 14, 18, 24, 26, 30, 34, 36, 42, 44, 45, 46, 50, 52, 54, 56, 58, 60, 66, 72, 76, 78, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 122, 124, 126, 130, 136, 138, 144, 150, 156, 160, 162, 168, 170, 172, 174, 176, 180, 186, 188, 190, 192

Offset: 1

N. J. A. Sloane, Feb 20 2025

nonn

All terms except 2 and 3 are composite numbers, although it is not obvious which composite numbers appear.

			A381222 begins 2, 3, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 10, 10, 10, ... and discarding terms which have already been seen we get 2, 3, 4, 6, 10, ...

Michael De Vlieger, Table of n, a(n) for n = 1..128

Cf. A381019, A381222.

A382711 Regarding A381019 as a permutation of the natural numbers, this is the cycle with smallest term 8, read in the forward direction.

Original entry on oeis.org

8, 13, 9, 17, 41, 139, 677, 4651, 43037

Offset: 1

N. J. A. Sloane, Apr 06 2025

A382712 shows the cycle containing 8 but read in the opposite direction. If this cycle turns out to be finite, both the present sequence and A382712 will be periodic, but if the cycle is infinite, the two sequences will never meet again.

A311019 has five finite cycles involving the numbers less than 8: (1), (2), (3), (4,5,7), and (6,11,23).

Cf. A381019, A382712.

cp's OEIS Frontend

A381116 Indices of composite terms in A381019.

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A381115 Composite terms in A381019 in order of appearance.

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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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A381120 Numbers k such that both A381019(k) and A381019(k+1) are composite.

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A379810 Composite numbers in A381019 which are immediately followed by another composite number, in order of their appearance.

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A381118 Index of 2^n in A381019.

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A381222 Smallest number missing from A381019 after A381019(n) has been found.

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A381223 Take the list (A381222) of successive values of the smallest number missing from A381019, and keep just the first of any run of successive equal terms.

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A382711 Regarding A381019 as a permutation of the natural numbers, this is the cycle with smallest term 8, read in the forward direction.

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