

A004611


Divisible only by primes congruent to 1 mod 3.


28



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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 JeanChristophe 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 120degree 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


LINKS

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), 13001306 (Abstract, pdf, ps).
Walter Kehowski, D Numbers.
Index entries for sequences related to A2 = hexagonal = triangular lattice


MAPLE

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


MATHEMATICA

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 *)


PROG

(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


CROSSREFS

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


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



