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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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%I #38 Jun 22 2023 04:08:36

%S 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,

%T 42,44,45,49,26,27,50,56,36,55,63,32,60,70,66,13,65,34,39,75,38,77,48,

%U 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

%N 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.

%C 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.

%C 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

%C 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?]

%C Corollary: This sequence is a permutation of the positive integers. [This question appears to be still open. - _N. J. A. Sloane_, Mar 18 2023]

%C Conjecture: The primes appear in their natural order.

%H Winston de Greef, <a href="/A359804/b359804.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A359804/a359804.png">Log log scatterplot of a million terms</a> showing primes in red.

%H Michael De Vlieger, <a href="/A359804/a359804_1.png">Log log scatterplot of a(n)</a>, 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.

%e a(3) must be 3 because a(1,2) = 1,2 and 3 is the least prime which does not divide 2.

%e a(4) = 5 since this is the least multiple of the smallest prime which does not divide 2*3 = 6.

%e 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.

%e 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.

%p R:= 1,2: S:= {1,2}:

%p for i from 3 to 100 do

%p s:= R[i-2]*R[i-1]:

%p p:= 2;

%p while s mod p = 0 do p:= nextprime(p) od:

%p for r from p by p while member(r,S) do od:

%p R:= R,r; S:= S union {r}

%p od:

%p R; # _Robert Israel_, Mar 08 2023

%t nn = 2^10; c[_] = False; q[_] = 1;

%t Array[Set[{a[#], c[#]}, {#, True}] &, 2];

%t Set[{i, j}, {a[1], a[2]}]; u = 3;

%t Do[(k = q[#];

%t While[c[k #], k++]; k *= #;

%t While[c[# q[#]], q[#]++]) &[(p = 2;

%t While[Divisible[i j, p], p = NextPrime[p]]; p)];

%t Set[{a[n], c[k], i, j}, {k, True, j, k}];

%t If[k == u, While[c[u], u++]], {n, 3, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Mar 08 2023 *)

%o (PARI) findp(n) = forprime(p=2, , if (n%p, return(p)));

%o 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

%o (Python)

%o from itertools import count, islice

%o from sympy import prime, primefactors, primepi

%o def A359804_gen(): # generator of terms

%o aset, bset, cset = set(), {1}, {1,2}

%o yield from (1,2)

%o while True:

%o for i in count(1):

%o if not (i in aset or i in bset):

%o p = prime(i)

%o for j in count(1):

%o if (m:=j*p) not in cset:

%o yield m

%o cset.add(m)

%o break

%o break

%o aset, bset = bset, set(map(primepi,primefactors(m)))

%o A359804_list = list(islice(A359804_gen(),30)) # _Chai Wah Wu_, Mar 18 2023

%Y Cf. A002110, A007947, A053669.

%Y See also A361502, A361503, A361504, A361505.

%Y A351495 has a very similar definition.

%K nonn

%O 1,2

%A _David James Sycamore_, Mar 08 2023