A085265 -id:A085265 - cp's OEIS frontend

A118882 Numbers which are the sum of two squares in two or more different ways.

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 325, 338, 340, 365, 370, 377, 400, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 676, 680

Offset: 1

Franklin T. Adams-Watters, May 03 2006

nonn

Numbers whose prime factorization includes at least two primes (not necessarily distinct) congruent to 1 mod 4 and any prime factor congruent to 3 mod 4 has even multiplicity. Products of two values in A004431.

Squares of distances that are the distance between two points in the square lattice in two or more nontrivially different ways. A quadrilateral with sides a,b,c,d has perpendicular diagonals iff a^2+c^2 = b^2+d^2. This sequence is the sums of the squares of opposite sides of such quadrilaterals, excluding kites (a=b,c=d), but including right triangles (the degenerate case d=0).

			50 = 7^2 + 1^2 = 5^2 + 5^2, so 50 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A004431, A009177, A085265.

Cf. A007692, A001481, A022544.

import Data.List (findIndices)
a118882 n = a118882_list !! (n-1)
a118882_list = findIndices (> 1) a000161_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range[1000], Length[PowersRepresentations[#, 2, 2]] > 1&] (* Jean-François Alcover, Mar 02 2019 *)

from itertools import count, islice
from math import prod
from sympy import factorint
def A118882_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        if 1>1):
            yield n
A118882_list = list(islice(A118882_gen(),30)) # Chai Wah Wu, Sep 09 2022

A000161(a(n)) > 1. [Reinhard Zumkeller, Aug 16 2011]

A085263 Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 2, 1, 4, 4, 2, 1, 5, 4, 3, 2, 2, 5, 2, 3, 6, 6, 3, 2, 6, 4, 3, 2, 5, 6, 3, 2, 5, 6, 3, 2, 4, 6, 4, 3, 4, 6, 4, 1, 7, 5, 3, 3, 7, 6, 4, 4, 6, 8, 3, 3, 6, 7, 2, 4, 8, 5, 4, 3, 7, 9, 4, 2, 8, 9, 4, 3, 6, 6, 5, 4, 7, 9, 5, 3, 8, 4, 3, 5, 9

Offset: 1

Reinhard Zumkeller, Jun 23 2003

a(A085265(n))>0; a(A085266(n))=1; a(A085267(n))>1.
a(A085264(n))=n and a(i)<>n for i < A085264(n).
First occurrence of k: 2, 6, 11, 23, 30, 38, 62, 71, 83, 110, 138, 155, 182, 203, 227, 263, 302, 327, 383, 435, 447, 503, 542, 602, 635, ..., . Conjecture: For each k above, there is a finite number of terms; for example, only the two numbers 1 and 13 cannot be represented as the sum of a squarefree number and a square. The number of k terms beginning with 0: 2, 9, 19, 27, 38, 36, 57, 63, 62, 74, 94, ..., . - Robert G. Wilson v, May 16 2014

			a(11)=3:
11 = 1 + 10 = A000290(1) + A005117(7)
   = 4 + 7  = A000290(2) + A005117(6)
   = 9 + 2  = A000290(3) + A005117(2).

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Numbers.
Eric Weisstein's World of Mathematics, Squarefree
Robert G. Wilson v, Plot of first 100000 terms

Cf. A000290, A002636, A005117, A115288, A160324, A160325, A160326, A240088, A242442, A242443.

f[n_] := Count[ SquareFreeQ@# & /@ (n - Range[1, Floor[ Sqrt[ n]]]^2), True]; Array[f, 105] (* Robert G. Wilson v, May 16 2014 *)

a(n) = sum(k=1, n-1, issquare(k) * issquarefree(n-k)); \\ Michel Marcus, Oct 30 2020

a(n+1) = Sum_{k=1..n} A008966(k)*A010052(n-k+1). - Reinhard Zumkeller, Nov 04 2009
a(n) < sqrt(n). - Robert G. Wilson v, May 17 2014
G.f.: (Sum_{i>=1} x^(i^2))*(Sum_{j>=1} mu(j)^2*x^j). - Ilya Gutkovskiy, Feb 06 2017

A011760 Elevator buttons in U.S.A.: Positive integers except 13.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Juan-Carlos Lerman (jclerm(AT)aztec.asu.edu), Dec 11 1996

Eric Weisstein's World of Mathematics, Triskaidekaphobia
Wikipedia, Triskaidekaphobia
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A085265.

Cf. A076805.

Drop[Range[70],{13}] (* Harvey P. Dale, Dec 11 2012 *)

a(n)=if(n<13,n,n+1) \\ Charles R Greathouse IV, Sep 02 2011

a(n) = n + sign( floor( n/13 ) ). - Wesley Ivan Hurt, Jan 12 2013

G.f.: -x*(x^13-x^12-1) / (x-1)^2. - Colin Barker, May 04 2014

A085266 Numbers having a unique representation as sum of a squarefree number and a square.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 25, 29, 61

Offset: 1

Reinhard Zumkeller, Jun 23 2003

A085263(a(n))=1; are there more?

No more terms through 625000000. - Ryan Propper, Jan 05 2008

Eric Weisstein's World of Mathematics, Square Numbers.
Eric Weisstein's World of Mathematics, Squarefree

Cf. A085265, A085267.

A085267 Numbers having at least two representations as sum of a squarefree number and a nonzero square.

Original entry on oeis.org

6, 7, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

Numbers n such that A085263(a(n)) > 1.

Conjecture: for n > 50, a(n) = n + 11. - Charles R Greathouse IV, Oct 26 2011

Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Squarefree

Cf. A085265, A085266.

is(n)=my(t);forstep(k=sqrtint(n-1),1,-1,if(issquarefree(n-k^2),if(t++>1,return(1))));0 \\ Charles R Greathouse IV, Mar 12 2012

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A118882 Numbers which are the sum of two squares in two or more different ways.

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A085263 Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).

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A011760 Elevator buttons in U.S.A.: Positive integers except 13.

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A085266 Numbers having a unique representation as sum of a squarefree number and a square.

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A085267 Numbers having at least two representations as sum of a squarefree number and a nonzero square.

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