A361502 -id:A361502 - cp's OEIS frontend

A359804 a(1) = 1, a(2) = 2; thereafter let p be the smallest prime that does not divide a(n-2)*a(n-1), then a(n) is the smallest multiple of p that is not yet in the sequence.

1, 2, 3, 5, 4, 6, 10, 7, 9, 8, 15, 14, 11, 12, 20, 21, 22, 25, 18, 28, 30, 33, 35, 16, 24, 40, 42, 44, 45, 49, 26, 27, 50, 56, 36, 55, 63, 32, 60, 70, 66, 13, 65, 34, 39, 75, 38, 77, 48, 80, 84, 88, 85, 51, 46, 90, 91, 99, 52, 95, 54, 98, 100, 57, 105, 58, 110, 69, 112, 115, 72, 119, 120, 121

Offset: 1

David James Sycamore, Mar 08 2023

nonn

Let i = a(n-2), j = a(n-1). For k > 1, m >= 1, a(n) = m*prime(k) iff rad(i*j) = primorial(k-1), and this is the m-th such occurrence. This suggests the late appearance of most primes (namely those >= 7), apparent in the lowest part of scatterplot, where for example a(717126), a(63056215) = 31, 37 respectively.
As Scott R. Shannon has just observed, the following proof is incomplete, since it requires a proof that every even number appears. Even the induction step seems a little dubious. - N. J. A. Sloane, Mar 18 2023
All multiples of all primes appear in the sequence, for if not there is a least prime p such that m*p is not a term for any [some?] m >= 1. Choose any prime q < p; then every multiple of q must appear, so then p*q must be a term; contradiction since this is a multiple of p. [But what if p = 2?]
Corollary: This sequence is a permutation of the positive integers. [This question appears to be still open. - N. J. A. Sloane, Mar 18 2023]
Conjecture: The primes appear in their natural order.

			a(3) must be 3 because a(1,2) = 1,2 and 3 is the least prime which does not divide 2.
a(4) = 5 since this is the least multiple of the smallest prime which does not divide 2*3 = 6.
a(8) = 7 because a(6,7) = 6,10 and 7 is the smallest prime which does not divide 60, rad(60) = 2*3*5 = 30.
a(19,20) = 18,28, and 5 is the smallest prime not dividing rad(18*28) = 42. Since multiples of 5 have appeared 5 times already, a(20) = 6*5 = 30.

Winston de Greef, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a million terms showing primes in red.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither prime power nor squarefree in blue.

Cf. A002110, A007947, A053669.
See also A361502, A361503, A361504, A361505.
A351495 has a very similar definition.

R:= 1,2: S:= {1,2}:
for i from 3 to 100 do
  s:= R[i-2]*R[i-1]:
  p:= 2;
  while s mod p = 0 do p:= nextprime(p) od:
  for r from p by p while member(r,S) do od:
  R:= R,r; S:= S union {r}
od:
R; # Robert Israel, Mar 08 2023

nn = 2^10; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
Set[{i, j}, {a[1], a[2]}]; u = 3;
Do[(k = q[#];
      While[c[k #], k++]; k *= #;
    While[c[# q[#]], q[#]++]) &[(p = 2;
    While[Divisible[i j, p], p = NextPrime[p]]; p)];
  Set[{a[n], c[k], i, j}, {k, True, j, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 08 2023 *)

findp(n) = forprime(p=2, , if (n%p, return(p)));
lista(nn) = my(va = vector(nn, k, if (k<=2, k))); for (n=3, nn, my(vsa = vecsort(va), p=findp(va[n-1]*va[n-2]), k=p); while (vecsearch(vsa, k), k+=p); va[n] = k;); va; \\ Michel Marcus, Mar 09 2023

from itertools import count, islice
from sympy import prime, primefactors, primepi
def A359804_gen(): # generator of terms
    aset, bset, cset = set(), {1}, {1,2}
    yield from (1,2)
    while True:
        for i in count(1):
            if not (i in aset or i in bset):
                p = prime(i)
                for j in count(1):
                    if (m:=j*p) not in cset:
                        yield m
                        cset.add(m)
                        break
                break
        aset, bset = bset, set(map(primepi,primefactors(m)))
A359804_list = list(islice(A359804_gen(),30)) # Chai Wah Wu, Mar 18 2023

A361504 Index of n in A359804, or -1 if n never appears there.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 10, 9, 7, 13, 14, 42, 12, 11, 24, 347, 19, 3466, 15, 16, 17, 49012, 25, 18, 31, 32, 20, 528231, 21, 717126, 38, 22, 44, 23, 35, 63056215, 47, 45, 26, 1375559400, 27, 7038527851, 28, 29, 55

Offset: 1

N. J. A. Sloane, Mar 18 2023

Conjectured to be a permutation of the natural numbers.

Cf. A359804, A361502.

nn = 2^20; c[] = 0; q[] = 1;
Array[Set[{a[#], c[#]}, {#, #}] &, 2];
Set[{i, j}, {a[1], a[2]}]; u = 3;
Monitor[Do[
  (k = q[#]; While[c[k #] > 0, k++]; k *= #;
     While[c[# q[#]] > 0, q[#]++]) &[(p = 2;
    While[Divisible[i j, p], p = NextPrime[p]]; p)]; Sow[p];
  Set[{a[n], c[k], i, j}, {k, n, j, k}];
  If[k == u, While[c[u] > 0, u++]], {n, 3, nn}], n];
TakeWhile[Array[c, 120], # > 0 &] (* Michael De Vlieger, Mar 18 2023 *)

A361505 Index of 2^n in A359804.

Original entry on oeis.org

1, 2, 5, 10, 24, 38, 87, 172, 349, 706, 1407, 2752, 5487, 11103, 22285, 44429, 88993, 177746, 356460, 712129, 1425163, 2849424, 5701776, 11401709, 22804522, 45608572, 91219022, 182438457, 364879209, 729757797, 1459514883, 2919031155, 5838065175, 11676129412, 23352260426

Offset: 0

N. J. A. Sloane, Mar 18 2023

nonn

Rémy Sigrist, C++ program

Cf. A359804, A361502, A361504.

nn = 2^20; c[] = False; q[] = 1;
i = 1; j = 2; c[1] = c[2] = True; u = 3;
{1, 2}~Join~Reap[Monitor[Do[
     (k = q[#]; While[c[k #], k++]; k *= #;
        While[c[# q[#]], q[#]++]) &[(p = 2;
       While[Divisible[i j, p], p = NextPrime[p]]; p)];
     If[IntegerQ@ Log2[k], Sow[n]];
     Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}], n]][[-1, -1]] (* Michael De Vlieger, Mar 18 2023 *)

a(26)-a(27) from Michael De Vlieger, Mar 19 2023

More terms from Rémy Sigrist, Mar 19 2023

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A359804 a(1) = 1, a(2) = 2; thereafter let p be the smallest prime that does not divide a(n-2)*a(n-1), then a(n) is the smallest multiple of p that is not yet in the sequence.

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A361504 Index of n in A359804, or -1 if n never appears there.

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A361505 Index of 2^n in A359804.

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A361722 Index of where prime(n) first appears as a divisor of any term in A359804.

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