A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 107, 108, 112, 114, 118, 121, 124, 126, 127

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also numbers with no prime factors of form 4*k+1.

m is a term iff A072438(m) = m.

Density 0. - Charles R Greathouse IV, Apr 16 2012

Closed under multiplication. Primitive elements are A045326, 2 and the primes of form 4*k+3. - Jean-Christophe Hervé, Nov 17 2013

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Evan M. Bailey, Table of n, a(n) for n = 1..20000 (Terms 1..1000 from T. D. Noe)
• Evan M. Bailey, a004144.cpp.
• Steven R. Finch, Landau-Ramanujan Constant [Broken link]
• Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]
• Daniel Shanks, Non-hypotenuse numbers, Fib. Quart., Vol. 13, No. 4 (1975), pp. 319-321.
• Eric Weisstein's World of Mathematics, Pythagorean Triple.
• Index entries for sequences related to sums of squares.

Crossrefs

Complement of A009003.

Cf. A000290, A002145, A005089, A072437, A244659, A244662.

The subsequence of primes is A045326.

Programs

Haskell

import Data.List (elemIndices)
a004144 n = a004144_list !! (n-1)
a004144_list = map (+ 1) $ elemIndices 0 a005089_list
-- Reinhard Zumkeller, Jan 07 2013

Mathematica

fQ[n_] := If[n > 1, First@ Union@ Mod[ First@# & /@ FactorInteger@ n, 4] != 1, True]; Select[ Range@ 127, fQ]
A004144 = Select[Range[127],Length@Reduce[s^2 + t^2 == s # && s > t > 0, Integers] == 0 &] (* Gerry Martens, Jun 09 2020 *)

PARI

is(n)=n==1||vecmin(factor(n)[,1]%4)>1 \\ Charles R Greathouse IV, Apr 16 2012

PARI

list(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); for(i=1,lim, if(u[i]==0, listput(v,i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

Formula

A005089(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013

The number of terms below x is ~ (A * x / sqrt(log(x))) * (1 + C/log(x) + O(1/log(x)^2)), where A = A244659 and C = A244662 (Shanks, 1975). - Amiram Eldar, Jan 29 2022

Extensions

More terms from Reinhard Zumkeller, Jun 17 2002

Name clarified by Evan M. Bailey, Sep 17 2019