A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares). (Formerly M0542)

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

1,2

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.

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

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 *)

Prog

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

Crossrefs

Complement of A009003.

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

The subsequence of primes is A045326.

Sequence in context: A209921 A268377 A201010 * A356930 A124391 A200381

Adjacent sequences: A004141 A004142 A004143 * A004145 A004146 A004147

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Reinhard Zumkeller, Jun 17 2002

Name clarified by Evan M. Bailey, Sep 17 2019

Status

approved