A002828 -id:A002828 - cp's OEIS frontend

A077773 Number of integers between n^2 and (n+1)^2 that are the sum of two squares; multiple representations are counted once.

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 9, 8, 8, 10, 10, 11, 11, 12, 11, 14, 12, 13, 15, 16, 15, 15, 17, 16, 17, 19, 18, 19, 20, 19, 20, 21, 20, 22, 22, 24, 22, 25, 23, 26, 26, 24, 29, 26, 27, 28, 27, 29, 26, 31, 32, 30, 29, 33, 33, 31, 31, 35, 34, 35, 35, 35, 36, 37, 37, 33, 42, 37, 38

Offset: 0

T. D. Noe, Nov 20 2002

nonn

Related to the circle problem, cf. A077770. See A077774 for a more restrictive case. A077768 counts the representations multiply.
Number of integers k in range [n^2, ((n+1)^2)-1] for which 2 = the least number of squares that add up to k (A002828). Because of this interpretation a(0)=0 was prepended to the beginning. - Antti Karttunen, Oct 04 2016
This sequence is not surjective, since, for instance, there is no n such that a(n) = 46. This follows from a bound observed by Jon E. Schoenfield, that if a(n) = m then n < ((m+1)^2)/2, and the fact that a(n) != 46 for all n < 1105. - Rainer Rosenthal, Jul 25 2023

			a(8)=6 because 65=64+1=49+16, 68=64+4, 72=36+36, 73=64+9, 74=49+25 and 80=64+16 are between squares 64 and 81. Note that 65 is counted only once.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000 (terms 0..1024 from T. D. Noe and Antti Karttunen).
Hugo Pfoertner, Table of n, a(n) for n = 0..500000
Rainer Rosenthal, Illustrating A077773

Cf. A000290, A000404, A002828, A010052, A077768, A077770, A077774, A229062, A277192, A277193, A277194.

Cf. A363762 (terms not occurring in this sequence), A363763.

maxN=100; lst={}; For[n=1, n<=maxN, n++, sqrs={}; i=n; j=0; While[i>=j, j=1; While[i^2+j^2<(n+1)^2, If[i>=j&&i^2+j^2>n^2, AppendTo[sqrs, i^2+j^2]]; j++ ]; i--; j-- ]; AppendTo[lst, Length[Union[sqrs]]]]; lst

a(N)=s=0;for(n=N^2+1,(N+1)^2-1,f=0;r=sqrtint(n);forstep(i=r,1,-1,if(issquare(n-i*i),f=1;s=s+1;break)));s /* Ralf Stephan, Sep 17 2013 */

from sympy import factorint
def A077773(n): return sum(1 for m in range(n**2+1,(n+1)**2) if all(p==2 or p&3==1 or e&1^1 for p, e in factorint(m).items())) # Chai Wah Wu, Jun 20 2023

(define (A077773 n) (add (lambda (i) (* (- 1 (A010052 i)) (A229062 i))) (A000290 n) (+ -1 (A000290 (+ 1 n)))))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Antti Karttunen, Oct 04 2016

a(n) = Sum_{i=n^2+1..(n+1)^2-1} A229062(i). - Ralf Stephan, Sep 17 2013
From Antti Karttunen, Oct 04 2016: (Start)
For n >= 0, a(n) + A277193(n) + A277194(n) = 2n.
For n >= 1, A277192(n) = a(n) + A277194(n). (End)

Term a(0)=0 prepended by Antti Karttunen, Oct 04 2016

A000415 Numbers that are the sum of 2 but no fewer nonzero squares.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 170, 173, 178, 180, 181

Offset: 1

N. J. A. Sloane and J. H. Conway

Only these numbers can occur as discriminants of quintic polynomials with solvable Galois group F20. - Artur Jasinski, Oct 25 2007
Complement of A022544 in the nonsquare positive integers A000037. - Max Alekseyev, Jan 21 2010
Nonsquare positive integers D such that Pell equation y^2 - D*x^2 = -1 has rational solutions. - Max Alekseyev, Mar 09 2010
Nonsquares for which all 4k+3 primes in the integer's canonical form occur with even multiplicity. - Ant King, Nov 02 2010

E. Grosswald, Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), p.15. - Ant King, Nov 02 2010

T. D. Noe, Table of n, a(n) for n = 1..10000
R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172. - _Ant King_, Nov 02 2010
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A000404, A000419, A001481, A002828, A009003, A134422.

c = {}; Do[Do[k = a^2 + b^2; If[IntegerQ[Sqrt[k]], Null, AppendTo[c,k]], {a, 1, 100}], {b, 1, 100}]; Union[c] (* Artur Jasinski, Oct 25 2007 *)
Select[Range[181],Length[PowersRepresentations[ #,2,2]]>0 && !IntegerQ[Sqrt[ # ]] &] (* Ant King, Nov 02 2010 *)

is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); !issquare(n) \\ Charles R Greathouse IV, Feb 07 2017

from itertools import count, islice
from sympy import factorint
def A000415_gen(startvalue=2): # generator of terms >= startvalue
    for n in count(max(startvalue,2)):
        f = factorint(n).items()
        if any(e&1 for p,e in f if p&3<3) and not any(e&1 for p,e in f if p&3==3):
            yield n
A000415_list = list(islice(A000415_gen(),20)) # Chai Wah Wu, Aug 01 2023

{ A000404 } minus { A134422 }. - Artur Jasinski, Oct 25 2007

More terms from Arlin Anderson (starship1(AT)gmail.com)

A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

Original entry on oeis.org

0, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 26

Offset: 0

Labos Elemer, Jul 31 2002

nonn

a(n-1) = A018252(n) - n. a(n-1) = inverse (frequency distribution) sequence of A014689(n), i.e. number of terms of sequence A014689(n) less than n. a(n) = A073169(n+1) - 1, for n >= 1. For n >= 1: a(n) + 1 = A073169(n) = the number of set {1, primes}, i.e. (A008578) less than (n)-th composite numbers (A002828(n)). a(n-1) = The number of primes (A000040(n)) less than n-th nonprime (A018252(n)). - Jaroslav Krizek, Jun 27 2009

			n=100: composite[100]=133,Pi[133]=32=a(100)

Cf. A065890, A073426, A000720, A002808, A000040, A018252, A158611, A073169.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[PrimePi[c[w]], {w, 1, 128}]
With[{nn=150},PrimePi/@Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, Jun 26 2013 *)

from sympy import composite
def A073425(n): return composite(n)-n-1 if n else 0 # Chai Wah Wu, Oct 11 2024

a(n) = A000720(A002808(n)).
a(n) ~ n. - Charles R Greathouse IV, Sep 02 2015
a(n) = A002808(n)-n-1 for n > 0. - Chai Wah Wu, Oct 11 2024

Edited by N. J. A. Sloane, Jul 04 2009 at the suggestion of R. J. Mathar

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A000419 Numbers that are the sum of 3 but no fewer nonzero squares.

Original entry on oeis.org

3, 6, 11, 12, 14, 19, 21, 22, 24, 27, 30, 33, 35, 38, 42, 43, 44, 46, 48, 51, 54, 56, 57, 59, 62, 66, 67, 69, 70, 75, 76, 77, 78, 83, 84, 86, 88, 91, 93, 94, 96, 99, 102, 105, 107, 108, 110, 114, 115, 118, 120, 123, 126, 129, 131, 132, 133, 134, 138, 139, 140, 141, 142

Offset: 1

N. J. A. Sloane and J. H. Conway

A002828(a(n)) = 3; A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000378, A000408, A000415, A002828, A004215, A025427.

a000419 n = a000419_list !! (n-1)
a000419_list = filter ((== 3) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015

Select[Range[150],SquaresR[3,#]>0&&SquaresR[2,#]==0&] (* Harvey P. Dale, Nov 01 2011 *)

is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return( n/4^valuation(n,4)%8 !=7 ))); 0 \\ Charles R Greathouse IV, Feb 07 2017

def aupto(lim):
  squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
  sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
  sum3sqs = set(a+b for a in sum2sqs for b in squares)
  return sorted(set(range(lim+1)) & (sum3sqs - sum2sqs - set(squares)))
print(aupto(142)) # Michael S. Branicky, Mar 06 2021

Legendre: a nonnegative integer is a sum of three (or fewer) squares iff it is not of the form 4^k m with m == 7 (mod 8).

More terms from Arlin Anderson (starship1(AT)gmail.com)

A100878 Smallest number of pentagonal numbers which sum to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 3, 3, 2, 3, 3, 4, 1, 2, 3, 4, 5, 2, 2, 3, 3, 4, 3, 3, 2, 3, 4, 3, 4, 3, 3, 1, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 4, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 4, 2, 3, 2

Offset: 0

Franz Vrabec, Jan 09 2005

nonn

From Bernard Schott, Jul 15 2022: (Start)
In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers, four squares, five pentagonal numbers, and so on.
The square case was proved by Lagrange in 1770; it is known as Lagrange's four squares theorem (see A002828). Then Gauss proved the triangular case in 1796 (see A061336).
In 1813, Cauchy proved this polygonal number theorem: for m >= 3, every positive integer N can be represented as a sum of m+2 (m+2)-gonal numbers, at most four of which are different from 0 and 1 (Deza reference). Hence every number is expressible as the sum of at most five positive pentagonal numbers (A000326). (End)

			a(5)=1 since 5=5, a(6)=2 since 6=1+5, a(7)=3 since 7=1+1+5, a(10)=2 since 10=5+5 with 1 and 5 pentagonal numbers.

Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Augustin-Louis Cauchy, Démonstration du théorème général de Fermat sur les nombres polygones, Extrait des Mémoires de l'Institut, 1813-15.
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem.
Wikipedia, Fermat polygonal number theorem.

Cf. A002828, A061336, A355717.

Cf. A000326 (a(n) = 1), A003679 (a(n) = 4 or 5), A355660 (a(n) = 4), A133929 (a(n) = 5).

a(n) = my(nb=oo); forpart(vp=n, if (vecsum(apply(x->ispolygonal(x, 5), Vec(vp))) == #vp, nb = min(nb, #vp)),,5); nb; \\ Michel Marcus, Jul 15 2022

a(n) = for(i = 1, oo, p = partitions(n, , [i,i]); for(j = 1, #p, if(sum(k = 1, i, ispolygonal(p[j][k],5)) == i, return(i)))) \\ David A. Corneth, Jul 15 2022

a(n) <= 5 (inequality proposed by Fermat and proved by Cauchy). - Bernard Schott, Jul 13 2022

More terms from David Wasserman, Mar 04 2008

A277015 Numbers whose squares are present in A276573 (the infinite trunk of least squares beanstalk).

Original entry on oeis.org

0, 4, 8, 12, 16, 19, 20, 24, 36, 40, 45, 48, 56, 60, 68, 72, 80, 83, 84, 92, 96, 104, 109, 112, 120, 124, 132, 136, 140, 144, 147, 148, 156, 160, 164, 168, 173, 176, 180, 192, 204, 208, 211, 216, 220, 224, 228, 232, 237, 240, 252, 264, 272, 275, 276, 280, 284

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Indexing starts from zero because a(0)=0 is a special case in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..130

Cf. A000196, A002828, A276573, A277014, A277016 (the corresponding squares), A277025 (multiples of four present / 4).

Positions of squares in A277023, zeros in A277024.

(define (A277015 n) (A000196 (A277016 n)))

a(n) = A000196(A277016(n)) = A000196(A276573(A277014(n))).

A278216 Number of children that node n has in the tree defined by the edge relation A255131(child) = parent, "the least squares beanstalk".

Original entry on oeis.org

4, 0, 0, 4, 0, 0, 1, 0, 3, 1, 0, 3, 0, 0, 0, 2, 2, 0, 2, 2, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 0, 3, 0, 1, 3, 0, 1, 1, 0, 3, 0, 0, 3, 0, 0, 0, 3, 1, 0, 2, 2, 0, 0, 1, 1, 2, 1, 1, 2, 0, 0, 1, 0, 3, 1, 0, 3, 0, 1, 0, 1, 3, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 4, 0, 0, 2, 0, 2, 1, 0, 3, 1, 0, 0, 2, 1, 0, 1, 3, 0, 1, 0, 0, 4

Offset: 0

Antti Karttunen, Nov 25 2016

nonn

			a(0) = 4 as 0 - A002828(0) = 0, 1 - A002828(1) = 0, 2 - A002828(2) = 0 and 3 - A002828(3) = 0. (But 4 - A002828(4) = 3.) Note that 0 is the only number which is its own child as 0 - A002828(0) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002828, A255131, A276573.

Cf. A278490 (positions of zeros), A278489 (positions of nonzeros), A278491 (positions of 4's).

(define (A278216 n) (let loop ((s 0) (k (+ 4 n))) (if (< k n) s (loop (+ s (if (= n (A255131 k)) 1 0)) (- k 1)))))

a(n) = Sum_{i=0..4} [A002828(n+i) = i]. (Here [ ] is the Iverson bracket, giving as its result 1 only if A002828(n+i) is i, otherwise zero.)

A286366 Compound filter: a(n) = 2*A286365(n) + floor(A072400(n)/4).

Original entry on oeis.org

4, 6, 8, 4, 13, 11, 9, 6, 28, 14, 8, 8, 13, 11, 21, 4, 12, 30, 8, 13, 65, 11, 9, 11, 40, 14, 116, 9, 13, 23, 9, 6, 64, 14, 20, 28, 13, 11, 21, 14, 12, 66, 8, 8, 49, 11, 9, 8, 28, 42, 20, 13, 13, 119, 21, 11, 64, 14, 8, 21, 13, 11, 269, 4, 84, 66, 8, 12, 65, 23, 9, 30, 12, 14, 56, 8, 65, 23, 9, 13, 484, 14, 8, 65, 85, 11, 21, 11, 12, 50, 20, 9, 65, 11, 21, 11

Offset: 1

Antti Karttunen, May 08 2017

nonn

Each term of this sequence contains, in addition to the information contained in A286365 (which packs the values of A286361(n) and A286363(n) and parity of the exponent of the highest power of 2 dividing n) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). Note that the whole A072400(n) can be recovered based on the other information contained in a(n). Together all this information is enough - by Lagrange's "Four Squares theorem" - to determine what is the least number of squares that add up to n. Thus it follows that for all i, j: a(i) = a(j) => A002828(i) = A002828(j).

A286369 is similar, but without the parity of the 2-adic value present.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000035, A002828, A007814, A072400, A286361, A286363, A286364, A286365, A286368, A286369, A286372, A286373.

from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a007814(n): return 1 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a286365(n): return 2*a286364(n) + a007814(n)%2
def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8
def a(n): return 2*a286365(n) + int(a072400(n)/4) # Indranil Ghosh, May 09 2017

(define (A286366 n) (+ (* 2 (A286365 n)) (floor->exact (/ (A072400 n) 4))))

a(n) = 2*A286365(n) + floor(A072400(n)/4).

A101412 Least number of odd squares that sum to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 1

N. J. A. Sloane, Aug 08 2009

			a(13) = 5: 13 = 1+1+1+1+9.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002828, A151925, A097269.

A101412 := proc(n) local lsq; lsq := [seq((2*j+1)^2,j=0..floor((sqrt(n)-1)/2))] ; lsq := convert(lsq,set) ; a := n ; for p in combinat[partition](n) do if convert(p,set) minus lsq = {} then a := min(a,nops(p)) ; fi; od: a ; end: for n from 1 do printf("%d,\n",A101412(n)) ; od: # R. J. Mathar, Aug 08 2009
# problem has optimal substructure:
a:= proc(n) option remember; local r; r:= isqrt(n);
      `if`(r^2=n and irem(r, 2)=1, 1,
       min(seq(a(i)+a(n-i), i=1..n/2)))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 31 2011

a[n_] := a[n] = Module[{r}, r = Sqrt[n]; If[IntegerQ[r] && OddQ[r], 1, Min[Table[a[i]+a[n-i], {i, 1, Floor[n/2]}]]]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)

a(n)={x=n-1;if(x%8>1,k=1+x%8);if(n%8==1,k=9;if(issquare(n)&&n%2==1,k=1));if(x%8==1,k=10;y=1;while(x>0,if(issquare(x)&&x%2==1,k=2);y=y+2;x=n-y^2));k;} \\ Jinyuan Wang, Jan 29 2019

From Jinyuan Wang, Jan 29 2019: (Start)
For n == 1 (mod 8), if n is a perfect square, a(n) = 1, otherwise a(n) = 9.
For n == 2 (mod 8), if n is a term in A097269, a(n) = 2, otherwise a(n) = 10.
For n == k (mod 8), k = 3,4,...,8, a(n) = k.
For positive integer x, a(72*x+42) = a(72*x+66) = 10. (End)

More terms from R. J. Mathar, Aug 08 2009

More terms from Alois P. Heinz, Jan 30 2011

A234533 Smallest number of hex numbers summing to n.

Original entry on oeis.org

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

Offset: 1

Allan C. Wechsler, Dec 27 2013

nonn

This sequence is to A003215 as A002828 is to A000290, and as A061336 is to A000217.

			a(21) = 3, because 21 = 19 + 1 + 1 = 7 + 7 + 7, so there are two ways to express 21 as the sum of 3 hex numbers, but no way to express 21 as the sum of 2 hex numbers.

Lars Blomberg, Table of n, a(n) for n = 1..10000

a(27)-a(87) from Lars Blomberg, Jan 07 2014

cp's OEIS Frontend

A077773 Number of integers between n^2 and (n+1)^2 that are the sum of two squares; multiple representations are counted once.

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A000415 Numbers that are the sum of 2 but no fewer nonzero squares.

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A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

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A000419 Numbers that are the sum of 3 but no fewer nonzero squares.

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A100878 Smallest number of pentagonal numbers which sum to n.

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A277015 Numbers whose squares are present in A276573 (the infinite trunk of least squares beanstalk).

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