A139544 -id:A139544 - cp's OEIS frontend

A024352 Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c.

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

Offset: 1

David W. Wilson

These are the solutions to the equation x^2 + xy = n where y mod 2 = 0, y is positive and x is any positive integer. - Andrew S. Plewe, Oct 19 2007
Ordered different terms of A120070 = 3, 8, 5, 15, 12, 7, ... (which contains two 15's, two 40's, and two 48's). Complement: A139544. (See A139491.) - Paul Curtz, Sep 01 2009
A024359(a(n)) > 0. - Reinhard Zumkeller, Nov 09 2012
If a(n) mod 6 = 3, n > 1, then a(n) = c^2 - f(a(n))^2 where f(n) = (floor(4*n/3) - 3 - n)/2. For example, 171 = 30^2 - 27^2 and f(171) = 27. - Gary Detlefs, Jul 15 2014

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Ron Knott, Pythagorean Triples and Online Calculators
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Same as A042965 except for initial terms. - Michael Somos, Jun 08 2000
Different from A020884.
Cf. A009005, A020884, A120070, A139544, A139491.

a024352 n = a024352_list !! (n-1)
a024352_list = 3 : drop 4 a042965_list
-- Reinhard Zumkeller, Nov 09 2012

[3] cat [4 +Floor((4*n-3)/3): n in [2..100]]; // G. C. Greubel, Apr 22 2023

Union[Flatten[Table[Select[Table[b^2 - c^2, {c, b-1}], # < 100 &], {b, 100}]]] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{1,0,1,-1},{3,5,7,8,9},70] (* Harvey P. Dale, Dec 20 2021 *)

is(n)=(n%4!=2 && n>4) || n==3 \\ Charles R Greathouse IV, May 31 2013

def A024352(n): return 3 if n==1 else 3+(n<<2)//3 # Chai Wah Wu, Feb 10 2025

def A024352(n): return 4 + ((4*n-3)//3) - int(n==1)
[A024352(n) for n in range(1,101)] # G. C. Greubel, Apr 22 2023

Consists of all positive integers except 1, 4 and numbers == 2 (mod 4).
a(n) = a(n-3) + 4, n > 4.
G.f.: (3 + 2*x + 2*x^2 - 2*x^3 - x^4)/(1 - x - x^3 + x^4). - Ralf Stephan, before May 13 2008
a(n) = a(n-1) + a(n-3) - a(n-4), for n > 5. - Ant King, Oct 03 2011
a(n) = 4 + floor((4*n-3)/3), n > 1. - Gary Detlefs, Jul 15 2014

Edited by N. J. A. Sloane, Sep 19 2008

A024359 Number of primitive Pythagorean triangles with short leg n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 2, 2, 1, 0, 1, 1, 2, 0, 1, 3, 1, 0, 1, 1, 2, 0, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 1, 3, 2, 0, 2

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times A takes value n.
Number of times n occurs in A020884.
a(A139544(n)) = 0; a(A024352(n)) > 0. - Reinhard Zumkeller, Nov 09 2012

T. D. Noe, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020884, A024352, A024360, A024361, A132404 (where records occur), A139544.

a024359_list = f 0 1 a020884_list where
   f c u vs'@(v:vs) | u == v = f (c + 1) u vs
                    | u /= v = c : f 0 (u + 1) vs'
-- Reinhard Zumkeller, Nov 09 2012

solns[a_] := Module[{b, c, soln}, soln = Reduce[a^2 + b^2 == c^2 && a < b && c > 0 && GCD[a, b, c] == 1, {b, c}, Integers]; If[soln === False, 0, If[soln[[1, 1]] === b, 1, Length[soln]]]]; Table[solns[n], {n, 100}]
(* Second program: *)
a[n_] := Module[{s = 0, b, c, d, g}, Do[g = Quotient[n^2, d]; If[d <= g && Mod[d+g, 2] == 0, c = Quotient[d+g, 2]; b = g-c; If[n < b && GCD[b, c] == 1, s++]], {d, Divisors[n^2]}]; s]; Array[a, 100] (* Jean-François Alcover, Apr 27 2019, from PARI *)

nppt(a) = {
  my(s=0, b, c, d, g);
  fordiv(a^2, d,
    g=a^2\d;
    if(d<=g && (d+g)%2==0,
      c=(d+g)\2; b=g-c;
      if(aColin Barker, Oct 25 2015

Formula

a(n) = A024361(n) - A024360(n). - Ray Chandler, Feb 03 2020

A322489 Numbers k such that k^k ends with 4.

Original entry on oeis.org

2, 18, 22, 38, 42, 58, 62, 78, 82, 98, 102, 118, 122, 138, 142, 158, 162, 178, 182, 198, 202, 218, 222, 238, 242, 258, 262, 278, 282, 298, 302, 318, 322, 338, 342, 358, 362, 378, 382, 398, 402, 418, 422, 438, 442, 458, 462, 478, 482, 498, 502, 518, 522, 538, 542, 558

Offset: 1

Bruno Berselli, Dec 12 2018

Also numbers k == 2 (mod 4) such that 2^k and k^2 end with the same digit.

Numbers congruent to {2, 18} mod 20. - Amiram Eldar, Feb 27 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A056849.
Subsequence of A139544, A235700.
Numbers k such that k^k ends with d: A008592 (d=0), A017281 (d=1), A067870 (d=3), this sequence (d=4), A017329 (d=5), A271346 (d=6), A322490 (d=7), A017377 (d=9).

GAP
```
List([1..70], n -> 10*n+3*(-1)^n-5);
```

[10*n+3*(-1)^n-5 for n in 1:70] |> println

Magma
```
[10*n+3*(-1)^n-5: n in [1..70]];
```

select(n->n^n mod 10=4,[$1..558]); # Paolo P. Lava, Dec 18 2018

Mathematica
```
Table[10 n + 3 (-1)^n - 5, {n, 1, 60}]
```
Maxima
```
makelist(10*n+3*(-1)^n-5, n, 1, 70);
```

apply(A322489(n)=10*n+3*(-1)^n-5, [1..70]) \\ M. F. Hasler, Dec 14 2018

Vec(2*x*(1 + 8*x + x^2) / ((1 - x)^2*(1 + x)) + O(x^70)) \\ Colin Barker, Dec 13 2018

[10*n+3*(-1)**n-5 for n in range(1, 70)]

Sage
```
[10*n+3*(-1)^n-5 for n in (1..70)]
```

O.g.f.: 2*x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
E.g.f.: 2 + 3*exp(-x) + 5*(2*x - 1)*exp(x).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 10*n + 3*(-1)^n - 5. Therefore:
a(n) = 10*n - 8 for odd n;
a(n) = 10*n - 2 for even n.
a(n+2*k) = a(n) + 20*k.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(2*Pi/5)*Pi/20 = sqrt(5+2*sqrt(5))*Pi/20. - Amiram Eldar, Feb 27 2023

A270783 Numbers of the form p^2 + q^2 + r^2 + s^2 = a^2 + b^2 + c^2 for some primes p, q, r, and s and some integers a, b, and c.

Original entry on oeis.org

16, 21, 26, 36, 37, 42, 52, 58, 61, 66, 68, 76, 82, 84, 100, 106, 108, 116, 132, 133, 138, 148, 154, 164, 172, 178, 180, 181, 186, 196, 202, 204, 212, 226, 228, 236, 244, 250, 260, 268, 276, 292, 298, 300, 301, 306, 308, 322, 324, 332, 340

Offset: 1

Griffin N. Macris, Mar 23 2016

nonn

This sequence is infinite since 4p^2 = 0^2 + 0^2 + (2p)^2 is in the sequence for all primes p.
A069262 is a subsequence.
It appears at first that the squares of A139544(n) for n >= 3 are a subsequence. n=22 is the first counterexample, where A139544(22)^2 = 6084 is not an element of this sequence.

			a(1) = 16 = 2^2 + 2^2 + 2^2 + 2^2 = 0^2 + 0^2 + 4^2.

Griffin N. Macris, Table of n, a(n) for n = 1..500

Cf. A069262, A139544.
Difference of A214515 and A270781.
Difference of A214515 and A004215.

n=340 #change for more terms
P=prime_range(1,ceil(sqrt(n)))
S=cartesian_product_iterator([P,P,P,P])
A=list(Set([sum(i^2 for i in y) for y in S if sum(i^2 for i in y)<=n]))
A.sort()
T=[sum(i^2 for i in y) for y in cartesian_product_iterator([[0..ceil(sqrt(n))],[0..ceil(sqrt(n))],[0..ceil(sqrt(n))]])]
[x for x in A if x in T] # Tom Edgar, Mar 24 2016

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A024352 Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c.

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A024359 Number of primitive Pythagorean triangles with short leg n.

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A322489 Numbers k such that k^k ends with 4.

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A270783 Numbers of the form p^2 + q^2 + r^2 + s^2 = a^2 + b^2 + c^2 for some primes p, q, r, and s and some integers a, b, and c.

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