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A094783
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Numbers k such that, for all m < k, d_i(k) <= d_i(m) for i=1 to Min(d(k),d(m)), where d_i(k) denotes the i-th smallest divisor of k.
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5
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1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 720720, 1441440, 2162160, 3603600, 7207200, 10810800, 122522400, 183783600, 2327925600, 3491888400, 80313433200
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OFFSET
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1,2
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COMMENTS
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The function d(k) (A000005) is the number of divisors of k.
The defining criterion for this sequence is a sufficient, but not necessary, condition for membership in A095849.
Why is 720 not in the sequence? The divisors of 360 begin 1,2,3,4,5,6,8,9,10,12,15,18 (A018412) and the divisors of 720 begin 1,2,3,4,5,6,8,9,10,12,15,16 (A018609). - J. Lowell, Aug 23 2007 [Answer from Don Reble, Sep 11 2007: 720 is precluded by 420. (1,2,3,4,5,6,7,10,12,14,15,20,21,...) (A018444).]
Conjecture: If k is in this sequence, then so is the smallest number with k divisors. (This conjecture is definitely false for A002182 (k=840) and A019505 (k=240).) - J. Lowell, Jan 24 2008
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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EXAMPLE
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As k increases, the positive integer k=6 sets or ties the existing records for smallest first, second and third-smallest divisors (1, 2 and 3), as well as for its fourth-smallest (6). Since no smaller integer has more than three divisors, 6 is a term of this sequence.
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PROG
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(PARI) ge(va, vb) = {for(i=1, min(#va, #vb), if (va[i] > vb[i], return(0)); ); return(-1); }
isok(k) = {my(dk = divisors(k)); for (m=1, k-1, my(dm = divisors(m)); if (! ge(dk, dm), return(0)); ); return(1); } \\ Michel Marcus, Mar 16 2022
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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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