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A365536
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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).
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0
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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, 63, 100, 126, 75, 56, 150, 35, 36, 70, 27, 200, 189, 400, 378, 125, 756, 250, 567, 800, 1134, 375, 112, 750, 49, 1600, 1701, 3200, 3402, 500, 5103, 6400, 10206, 625, 20412
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OFFSET
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1,2
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COMMENTS
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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.
Conjecture: All terms are 7-smooth numbers. Is this a permutation of A002473?
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}.
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.
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)
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LINKS
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MATHEMATICA
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nn = 2^12; c[_] := False; m[_] := 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; w = {1}; j = 2;
Do[k = SelectFirst[w, CoprimeQ[j, #] &];
While[c[k*m[k]], m[k]++]; k *= m[k];
w = Insert[w, j, LengthWhile[w, # > j &] + 1];
Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
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PROG
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(PARI) findgcd(v, k) = v = vecsort(v, , 4); for (i=1, #v, if (gcd(v[i], k) == 1, return(v[i])); );
findmult(k, v) = v = vecsort(v); for (i=1, oo, if (!vecsearch(v, i*k), return (i*k)));
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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