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A004611
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Divisible only by primes congruent to 1 mod 3.
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28
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1, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 97, 103, 109, 127, 133, 139, 151, 157, 163, 169, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 301, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457
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OFFSET
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1,2
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COMMENTS
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In other words, if a prime p divides n, then p == 1 mod 3.
Equivalently, products of primes == 1 (mod 6), products of elements of A002476.
Positive integers n such that n+d+1 is divisible by 3 for all divisors d of n. For example, a(13)=91 since 91=7*13, 91+1+1=93=3*31, 91+7+1=99=9*11, 91+13+1=105=3*7*5, 91+91+1=183=3*61. The only prime p such that x+d+1 is divisible by p for all divisors d of x is p=3. The sequence consists of 1 and all integers whose prime divisors are of the form 6k+1. - Walter Kehowski, Aug 09 2006
Also z such that z^2 = x^2 + x*y + y^2 and gcd(x,y,z) = 1. - Frank M Jackson, Jul 30 2013
From Jean-Christophe Hervé, Nov 24 2013: (Start)
Apart from the first term (for all in this comment), this sequence is the analog of A008846 (hypotenuses of primitive Pythagorean triangles) for triangles with integer sides and a 120-degree angle: a(n), n>1, is the sequence of lengths of the longest side of the primitive triangles.
Not only the square of these numbers is equal to x^2 + xy + y^2 with x and y > 0, but the numbers themselves also are; the sequence starting at n=2 is then a subsequence of A024606.
(End)
Numbers n such that 3/n cannot be written as the sum of 2 unit fractions. - Carl Schildkraut, Jul 19 2016
a(n), n>1, is the sequence of lengths of the middle side b of the primitive triangles such that A < B < C with an angle B = 60 degrees (A335895). Compare with comment of Nov 24 2013 where a(n), n>1, is the sequence of lengths of the longest side of the primitive triangles that have an angle = 120 degrees. - Bernard Schott, Mar 29 2021
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Walter Kehowski, D Numbers.
Index entries for sequences related to A2 = hexagonal = triangular lattice
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MAPLE
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with(numtheory): for n from 1 to 1801 by 6 do it1 := ifactors(n)[2]: it2 := 1: for i from 1 to nops(it1) do if it1[i][1] mod 6 > 1 then it2 := 0; break fi: od: if it2=1 then printf(`%d, `, n) fi: od:
with(numtheory): cnt:=0: L:=[]: for w to 1 do for n from 1 while cnt<100 do dn:=divisors(n); Q:=map(z-> n+z+1, dn); if andmap(z-> z mod 3 = 0, Q) then cnt:=cnt+1; L:=[op(L), [cnt, n]]; fi; od od; L; # Walter Kehowski, Aug 09 2006
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MATHEMATICA
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ok[1]=True; ok[n_]:=And@@(Mod[#, 3]==1&)/@FactorInteger[n][[All, 1]]; Select[Range[500], ok] (* Vincenzo Librandi, Aug 21 2012 *)
lst={}; maxLen=331; Do[If[Reduce[m^2+m*n+n^2==k^2&&m>=n>=0&&GCD[k, m, n]==1, {m, n}, Integers]===False, Null[], AppendTo[lst, k]], {k, maxLen}]; lst (* Frank M Jackson, Jul 04 2013 from A034017 *)
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PROG
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(Magma) [n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 3 eq 1}]; // Vincenzo Librandi, Aug 21 2012
(PARI) is(n)=my(f=factor(n)[, 1]); for(i=1, #f, if(f[i]%3!=1, return(0))); 1 \\ Charles R Greathouse IV, Feb 06 2013
(PARI) list(lim)=my(v=List([1]), mn, mx, t); forprime(p=7, lim\=1, if(p%6==1, listput(v, p))); if(lim<49, return(Vec(v))); forprime(p=7, sqrtint(lim), if(p%6>1, next); mx=1; while(v[mx+1]*p<=lim, for(i=mn=mx+1, mx=#v, t=p*v[i]; if(t>lim, break); listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jan 11 2018
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CROSSREFS
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Cf. A004612, A034017, A120806, A024606, A008846, A335895.
Multiplicative closure of A002476.
Sequence in context: A167462 A357277 A088513 * A129904 A133290 A038590
Adjacent sequences: A004608 A004609 A004610 * A004612 A004613 A004614
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers, Oct 30 2000
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007
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STATUS
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approved
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