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A346447
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Positive integers k such that the positive divisors of k <= sqrt(k) form a linear recurrence of order at most 2.
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0
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73
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OFFSET
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1,2
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COMMENTS
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A complete classification of all elements of this list has been obtained. In particular, all elements of this list have at most 3 distinct prime divisors.
Consequently, the above list has asymptotic density zero in the natural numbers.
Any prime belongs to this sequence since the tuple (1) vacuously satisfies some second-order linear recurrence.
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LINKS
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EXAMPLE
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36, whose divisors are (1,2,3,4,6,9,12,28,36), is the smallest non-recurrent number as (1,2,3,4,6) do not satisfy a second-order linear recurrence.
60 is in the sequence as its divisors <= sqrt(60) are (1, 2, 3, 4, 5, 6). These numbers satisfy b(n) = 2*b(n-1) - b(n-2), b(1) = 1, b(2) = 2. - David A. Corneth, Aug 14 2021
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PROG
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(PARI) is(n) = { my(f = factor(n), d, m, linrec, nd); if(#f~ > 3, return(0)); if(numdiv(f) <= 6, return(1) ); d = divisors(n); d = vector((#d + 1)\2, i, d[i]); m = [d[2], d[1]; d[3], d[2]]; if(matdet(m) != 0, linrec = matsolve([d[2], d[1]; d[3], d[2]], [d[3]; d[4]]); nd = vector(#d); nd[1] = d[1]; nd[2] = d[2]; for(i = 3, #nd, nd[i] = nd[i-1] * linrec[1, 1] + nd[i-2]*linrec[2, 1]; ); return(nd==d) , my(r = d[2]/d[1]); for(i = 3, #d, if(d[i-1]*r != d[i], return(0)); ); return(1) ) } \\ David A. Corneth, Aug 14 2021
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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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