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A059267
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Numbers n with 2 divisors d1 and d2 having difference 2: d2 - d1 = 2; equivalently, numbers that are 0 (mod 4) or have a divisor d of the form d = m^2 - 1.
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5
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3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128
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OFFSET
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1,1
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COMMENTS
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Complement of A099477; A008586, A008585 and A037074 are subsequences - Reinhard Zumkeller, Oct 18 2004
These numbers have an asymptotic density of ~ 0.522. This corresponds to all numbers which are multiples of 4 (25%), or of 3 (having 1 & 3 as divisors: + (1-1/4)*1/3 = 1/4), or of 5*7, or of 11*13, etc. (Generally, multiples of lcm(k,k+2), but multiples of 3 and 4 are already taken into account in the 50% covered by the first 2 terms.) - M. F. Hasler, Jun 02 2012
By considering divisors of the form m^2-1 with m <= 200 it is possible to prove that the density of this sequence is in the interval (0.5218, 0.5226). The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 52, 521, 5219, 52206, 522146, 5221524, 52215473, 522155386, 5221555813, ..., so the asymptotic density of this sequence can be estimated empirically by 0.522155... . - Amiram Eldar, Sep 25 2022
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LINKS
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M. F. Hasler, Table of n, a(n) for n = 1..3131
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FORMULA
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A099475(a(n)) > 0. - Reinhard Zumkeller, Oct 18 2004
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EXAMPLE
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a(18) = 35 because 5 and 7 divide 35 and 7 - 5 = 2.
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MAPLE
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with(numtheory): for n from 1 to 1000 do flag := 1: if n mod 4 = 0 then printf(`%d, `, n):flag := 0 fi: for m from 2 to ceil(sqrt(n)) do if n mod (m^2-1) = 0 and flag=1 then printf(`%d, `, n); break fi: od: od:
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MATHEMATICA
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d1d2Q[n_]:=Mod[n, 4]==0||AnyTrue[Sqrt[#+1]&/@Divisors[n], IntegerQ]; Select[ Range[ 200], d1d2Q] (* Harvey P. Dale, May 31 2020 *)
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PROG
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(PARI) isA059267(n)={ n%4==0 || fordiv( n, d, issquare(d+1) && return(1))} \\ M. F. Hasler, Aug 29 2008
(PARI) is_A059267(n) = fordiv( n, d, n%(d+2)||return(1)) \\ M. F. Hasler, Jun 02 2012
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CROSSREFS
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Cf. A099475, A099477, A008585, A008586, A037074.
Sequence in context: A192519 A036446 A284469 * A355200 A049433 A250984
Adjacent sequences: A059264 A059265 A059266 * A059268 A059269 A059270
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KEYWORD
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nonn
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AUTHOR
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Avi Peretz (njk(AT)netvision.net.il), Jan 23 2001
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EXTENSIONS
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More terms from James A. Sellers, Jan 24 2001
Removed comments linking to A143714, which seem wrong, as observed by Ignat Soroko, M. F. Hasler, Jun 02 2012
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STATUS
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approved
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