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A004144
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Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).
(Formerly M0542)
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43
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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
(list;
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refs;
listen;
history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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Evan M. Bailey, Table of n, a(n) for n = 1..20000 (Terms 1..1000 from T. D. Noe)
Evan M. Bailey, a004114.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.
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Complement of A009003.
Cf. A000290, A002145, A005089, A072437, A244659, A244662.
The subsequence of primes is A045326.
Sequence in context: A209921 A268377 A201010 * A124391 A200381 A050118
Adjacent sequences: A004141 A004142 A004143 * A004145 A004146 A004147
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KEYWORD
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nonn
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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 Reinhard Zumkeller, Jun 17 2002
Name clarified by Evan M. Bailey, Sep 17 2019
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STATUS
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approved
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