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A362855
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a(n) = n for n <= 3; for n > 3, a(n) is the least novel multiple of k, the product of all distinct prime factors of a(n-2) that do not divide a(n-1).
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4
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1, 2, 3, 4, 6, 5, 12, 10, 9, 20, 15, 8, 30, 7, 60, 14, 45, 28, 75, 42, 25, 84, 35, 18, 70, 21, 40, 63, 50, 105, 16, 210, 11, 420, 22, 315, 44, 525, 66, 140, 33, 280, 99, 350, 132, 175, 198, 245, 264, 385, 24, 770, 27, 1540, 36, 1155, 26, 2310, 13, 4620, 39, 3080, 78, 1925, 156, 2695, 234, 3465, 52, 5775
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OFFSET
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1,2
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COMMENTS
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Motivated by A362631, but instead of using one prime divisor of a(n-2) which does not divide a(n-1), the product of all such primes is used to compute a(n). - David James Sycamore, May 07 2023
Primes p(k) enter the sequence in order and fairly regularly through a(20543) = p(15) = 47, immediately following primorial A002110(k-1). However, a(87723) = p(17) = 59 is the next prime to appear, following a(87722) = A002110(16).
Conjecture: all primes appear eventually, but not in order. (End)
Conjecture: permutation of natural numbers.
Conjecture: the smallest missing number is always either prime or a powerful number.
Primes do not appear in order; a(87723) = 59 but a(91307) = 53.
Powerful numbers appear in clusters, e.g., for n roughly between 91200 and 91320.
Though it appears primorials are always followed by primes, it is logically possible but rare that primorials can be followed by a composite number. (End)
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LINKS
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Michael De Vlieger, Log log scatterplot of a(n), n = 1..701, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Powerful numbers are labeled in gold (if a prime power) or blue. Primorials P(i) = A002110(i) are labeled in green.
Michael De Vlieger, Plot of k = pi(p) | a(n) at (x, y) = (n, k), n = 1..4581, with a color function representing multiplicity where black indicates 1, red = 2, etc. The bar of color at the bottom indicates primes in red, composite prime powers in gold, composite squarefree in green, and other numbers in blue.
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FORMULA
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EXAMPLE
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Let S = { prime p : p | a(n-2) } and let T = { prime p : p | a(n-1) }. Then k = Product_{p in S\T} p = rad(a(n-2)*a(n-1))/rad(a(n-1)).
a(3) = 3 since rad(1*2)/rad(2) = 1; a(1) = 1, a(2) = 2, therefore a(3) = 3*1.
a(4) = 4 since rad(2*3)/rad(3) = 2; a(2) = 2, thus a(4) = 2*2.
a(5) = 6 since rad(3*4)/rad(4) = 6/2 = 3; a(3) = 3, thus a(5) = 2*3.
a(91305) = 108 and a(91306) = P(17), therefore k = 1 since rad(108) | P(17). The smallest missing number is 53, therefore a(91307) = 53*1. Related sequence A368133 = b is such that it is coincident with this sequence until b(91307) = 61, since prime(18) = 61 is the smallest prime that does not divide b(91306) = P(17). (End)
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MATHEMATICA
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nn = 1000; c[_] := False; m[_] := 1;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; i = 1; j = r = 2;
Do[(While[c[Set[k, # m[#]]], m[#]++]) &[r/f[j]];
Set[{a[n], c[k], i, j, r}, {k, True, j, k, f[j*k]}], {n, 3, nn}], n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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