A361330 -id:A361330 - cp's OEIS frontend

A361331 Smallest index of n-th prime in A361330, or -1 if it does not appear.

Original entry on oeis.org

1, 2, 5, 22, 160, 1770, 23022, 391390, 7436428

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 18 2023

It appears that a(n) is growing as constant*primorial(n). - Scott R. Shannon, Mar 20 2023

Cf. A002110 (primorials), A351495, A361330.

a(7)-a(8) from Winston de Greef, Mar 18 2023

a(9) from Scott R. Shannon, Mar 20 2023

A351495 a(1) = 1, for n > 1, a(n) is the smallest positive number that has not yet appeared that is a multiple of the smallest prime that does not divide a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 15, 14, 18, 20, 21, 16, 24, 25, 22, 27, 26, 30, 7, 28, 33, 32, 36, 35, 34, 39, 38, 42, 40, 45, 44, 48, 50, 51, 46, 54, 55, 52, 57, 56, 60, 49, 58, 63, 62, 66, 65, 64, 69, 68, 72, 70, 75, 74, 78, 80, 81, 76, 84, 85, 82, 87, 86, 90, 77, 88, 93, 92, 96, 95, 94

Offset: 1

Scott R. Shannon, May 03 2022

nonn

The sequence is conjectured to be a permutation of the positive integers.

The k-th prime appears as the next term after A002110(k-1) appears.

			a(5) = 6 as a(4) = 2 = 2*2 which does not contain 3 as a prime factor, and 6 is the smallest unused number that is a multiple of 3.
a(6) = 5 as a(5) = 6 = 2*3 which does not contain 5 as a prime factor, and 5 is the smallest unused number that is a multiple of 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16, showing m with lpf(m) = 7 in red, lpf(m) = 11 in gold, lpf(m) = 13 in green, and lpf(m) = 17 in blue.
Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.

Cf. A353026, A352793, A002110, A000040, A064413, A352588, A359804,

See also A361330, A361331.

nn = 75; c[] = 0; a[1] = c[1] = 1; u = 2; Do[q = 2; While[! CoprimeQ[q, a[i - 1]], Set[q, NextPrime[q]]]; m = Ceiling[u/q]; While[c[m*q] != 0, m++]; Set[{a[i], c[m*q]}, {m*q, i}]; If[m*q == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 03 2022 *)

from itertools import count, islice
from sympy import prime, primepi, primefactors
def A351495_gen(): # generator of terms
    aset, cset = set(), {1}
    yield 1
    while True:
        for i in count(1):
            if not i in aset:
                p = prime(i)
                for j in count(1):
                    if (m:=j*p) not in cset:
                        yield m
                        cset.add(m)
                        break
                break
        aset = set(map(primepi,primefactors(m)))
A351495_list = list(islice(A351495_gen(),30)) # Chai Wah Wu, Mar 18 2023

A361693 Index of where prime(n) first appears as a divisor of any term in A351495.

Original entry on oeis.org

2, 3, 6, 12, 19, 21, 29, 31, 39, 47, 49, 58, 65, 67, 75, 85, 93, 95, 104, 111, 113, 123, 131, 139, 150, 157, 159, 167, 169, 177, 196, 203, 213, 215, 231, 233, 242, 251, 259, 269, 277, 279, 295, 297, 305, 307, 325, 343, 351, 353, 361, 369, 371, 387, 397, 407, 415, 417, 426, 433, 435, 453, 472, 479

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 20 2023

nonn

Cf. A351495, A361333, A361330, A361332.

cp's OEIS Frontend

A361331 Smallest index of n-th prime in A361330, or -1 if it does not appear.

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A351495 a(1) = 1, for n > 1, a(n) is the smallest positive number that has not yet appeared that is a multiple of the smallest prime that does not divide a(n-1).

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A361693 Index of where prime(n) first appears as a divisor of any term in A351495.

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