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%I #9 Sep 17 2023 01:50:04
%S 1,2,3,4,6,5,12,10,9,20,18,15,8,30,7,60,14,45,28,90,21,40,42,25,84,50,
%T 63,100,126,75,56,150,35,36,70,27,200,189,400,378,125,756,250,567,800,
%U 1134,375,112,750,49,1600,1701,3200,3402,500,5103,6400,10206,625,20412
%N a(n) = n for n <= 2. Thereafter a(n) is the least novel multiple of the greatest prior term which is coprime to a(n-1).
%C The only way a prime can enter the sequence is consequent to a primorial term, thus prime(k) follows A002110(k). However it seems that only the first 4 primorial numbers (1,2,6,30) appear in the sequence, and that consequently primes > 7 do not become terms.
%C Conjecture: All terms are 7-smooth numbers. Is this a permutation of A002473?
%C From _Michael De Vlieger_, Sep 11 2023: (Start)
%C Sequence enters a multiplicative recurrence for n > 110 such that a(73(m+1)+j)/a(73m+j) is a prime power p(k)^e(k) in S = {2, 2^6, 2^12, 3^10, 5^4, 7}.
%C Since a(1..110) are all 7-smooth and so is S, the sequence is in A002473 but is not a permutation, since we are missing 105, 210, and 2^k or 3^k for k > 3, and there is no way to produce these given S.
%C There are 4 primes, 9 squarefree composites in the sequence; together with a(1) = 1, these are 14 of the 16 divisors of 210. {2^k : k=0..3}, {3^k : k=0..3}, A000351 and A000420 are each contained in this sequence. a(73m+50) = 7^(m+2), m >= 0. Aside from these, the sequence is in A126706 (numbers neither prime powers nor squarefree). (End)
%t nn = 2^12; c[_] := False; m[_] := 1;
%t Array[Set[{a[#], c[#]}, {#, True}] &, 2]; w = {1}; j = 2;
%t Do[k = SelectFirst[w, CoprimeQ[j, #] &];
%t While[c[k*m[k]], m[k]++]; k *= m[k];
%t w = Insert[w, j, LengthWhile[w, # > j &] + 1];
%t Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
%t Array[a, nn] (* _Michael De Vlieger_, Sep 11 2023 *)
%o (PARI) findgcd(v, k) = v = vecsort(v,, 4); for (i=1, #v, if (gcd(v[i], k) == 1, return(v[i])););
%o findmult(k, v) = v = vecsort(v); for (i=1, oo, if (!vecsearch(v, i*k), return (i*k)));
%o lista(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 2; for (n=3, nn, my(v = Vec(va, n-1)); my(x = findgcd(v, va[n-1])); my(y = findmult(x, v)); va[n] = y;); va; \\ _Michel Marcus_, Sep 08 2023
%Y Cf. A000351, A000420, A002473, A126706.
%K nonn
%O 1,2
%A _David James Sycamore_, Sep 08 2023
%E More terms from _Michel Marcus_, Sep 08 2023
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