A086849 -id:A086849 - cp's OEIS frontend

A352738 Squares in A086849.

16, 64, 441, 729, 81796, 1320201, 2729104, 44488900, 34614230401, 209453590921, 752884200721, 5054227881921, 8106120765625, 14483961408400, 433446375390625, 530837821446724, 1270089068379481, 1383781075827264, 4819866587217081, 7032375864510896656

Offset: 1

J. M. Bergot and Robert Israel, Mar 30 2022

nonn

Squares that are partial sums of A000037.

			a(2) = 64 is a term because 64 = 8^2 = 2+3+5+6+7+8+10+11+12 is a square and the sum of the nonsquares up to 12.

Chai Wah Wu, Table of n, a(n) for n = 1..23

Cf. A000037, A086849.

R:= NULL: count:= 0:
s:= 0:
for n from 1 do
  if issqr(n) then next fi;
  s:= s+n;
  if issqr(s) then
    count:= count+1;
    R:= R,s;
    if count = 19 then break fi
  fi;
od:
R;

from itertools import islice
def A352738_gen(): # generator of terms
    c, k, ks, m, ms = 0, 1, 2, 1, 1
    while True:
        for n in range(ks,ks+2*k):
            c += n
            if c == ms:
                yield c
            elif c > ms:
                ms += 2*m+1
                m += 1
        ks += 2*k+1
        k += 1
A352738_list = list(islice(A352738_gen(),10)) # Chai Wah Wu, Mar 31 2022

a(20) from Jon E. Schoenfield, Mar 31 2022

A355371 Intersection of A000330 and A086849.

Original entry on oeis.org

5, 91, 506, 650, 11440

Offset: 1

Ivan N. Ianakiev, Jun 30 2022

Numbers that are the sum of the first r nonsquares and also the sum of the first s squares for some r and s.
If it exists, a(6) > 5.4*10^12.
If it exists, a(6) > 10^20. - Michael S. Branicky, Jul 09 2022

			1+4+9+16+25+36 = 2+3+5+6+7+8+10+11+12+13+14 = 91, so 91 is a term.

Cf. A000330, A086849.

a000330=Accumulate[Table[i^2,{i,60}]];
a086849=Accumulate[Table[i+Floor[1/2+Sqrt[i]],{i,370}]];
Intersection[a000330,a086849]

A000037 Numbers that are not squares (or, the nonsquares).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

N. J. A. Sloane, Simon Plouffe

Note the remarkable formula for the n-th term (see the FORMULA section)!
These are the natural numbers with an even number of divisors. The number of divisors is odd for the complementary sequence, the squares (sequence A000290) and the numbers for which the number of divisors is divisible by 3 is sequence A059269. - Ola Veshta (olaveshta(AT)my-deja.com), Apr 04 2001
a(n) is the largest integer m not equal to n such that n = (floor(n^2/m) + m)/2. - Alexander R. Povolotsky, Feb 10 2008
Union of A007969 and A007970; A007968(a(n)) > 0. - Reinhard Zumkeller, Jun 18 2011
Terms of even numbered rows in the triangle A199332. - Reinhard Zumkeller, Nov 23 2011
If a(n) and a(n+1) are of the same parity then (a(n)+a(n+1))/2 is a square. - Zak Seidov, Aug 13 2012
Theaetetus of Athens proved the irrationality of the square roots of these numbers in the 4th century BC. - Charles R Greathouse IV, Apr 18 2013
4*a(n) are the even members of A079896, the discriminants of indefinite binary quadratic forms. - Wolfdieter Lang, Jun 14 2013

			For example note that the squares 0, 1, 4, 9, 16 are not included.

Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 58-60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 9900 terms from N. J. A. Sloane)
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
A. J. dos Reis and D. M. Silberger, Generating nonpowers by formula, Math. Mag., 63 (1990), 53-55.
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011.
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
S. Kaji, T. Maeno, K. Nuida, and Y. Numata, Polynomial Expressions of Carries in p-ary Arithmetics, arXiv preprint arXiv:1506.02742 [math.CO], 2015-2016.
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458. doi 10.2307/2308078, see example 4 (includes the formula). [Nicolas Normand (Nicolas.Normand(AT)polytech.univ-nantes.fr), Nov 24 2009]
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343-347.
R. D. Nelson, Sequences which omit powers, The Mathematical Gazette, Number 461, 1988, pages 208-211.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Rosetta Code, Sequence of non-squares
J. Scholes, 27th Putnam 1966 Prob. A4
Aaron Snook, Augmented Integer Linear Recurrences, 2012. - From _N. J. A. Sloane_, Dec 19 2012
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Continued Fraction

Cf. A007412, A000005, A000290, A059269, A134986, A087153, A172151, A000196, A049068 (subsequence).
Cf. A242401 (subsequence).
Cf. A086849 (partial sums), A048395.

a000037 n = n + a000196 (n + a000196 n)
-- Reinhard Zumkeller, Nov 23 2011

[n : n in [1..1000] | not IsSquare(n) ];

at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;

Maple
```
A000037 := n->n+floor(1/2+sqrt(n));
```

a[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[a[n], {n, 71}] (* Robert G. Wilson v, Sep 24 2004 *)
With[{upto=100},Complement[Range[upto],Range[Floor[Sqrt[upto]]]^2]] (* Harvey P. Dale, Dec 02 2011 *)
a[ n_] :=  If[ n < 0, 0, n + Round @ Sqrt @ n]; (* Michael Somos, May 28 2014 *)

A000037(n):=n + floor(1/2 + sqrt(n))$ makelist(A000037(n),n,1,50); /* Martin Ettl, Nov 15 2012 */

{a(n) = if( n<0, 0, n + (1 + sqrtint(4*n)) \ 2)};

from math import isqrt
def A000037(n): return n+isqrt(n+isqrt(n)) # Chai Wah Wu, Mar 31 2022

from math import isqrt
def A000037(n): return n+(k:=isqrt(n))+int(n>=k*(k+1)+1) # Chai Wah Wu, Jun 17 2024

a(n) = n + floor(1/2 + sqrt(n)).
a(n) = n + floor(sqrt( n + floor(sqrt n))).
A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 26 2010
A173517(a(n)) = n; a(n)^2 = A030140(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = A000194(n) + n. - Jaroslav Krizek, Jun 14 2009
a(A002061(n)) = a(n^2-n+1) = A002522(n) = n^2 + 1. - Jaroslav Krizek, Jun 21 2009

Edited by Charles R Greathouse IV, Oct 30 2009

A048395 Sum of consecutive nonsquares.

Original entry on oeis.org

0, 5, 26, 75, 164, 305, 510, 791, 1160, 1629, 2210, 2915, 3756, 4745, 5894, 7215, 8720, 10421, 12330, 14459, 16820, 19425, 22286, 25415, 28824, 32525, 36530, 40851, 45500, 50489, 55830, 61535, 67616, 74085, 80954, 88235, 95940, 104081

Offset: 0

Patrick De Geest, Mar 15 1999

Relationship with natural numbers: a(4) = (first term + last term)*n = (10+15)*3 = (25)*3 = 75; a(5) = (17+24)*4 = (41)*4 = 164; ...
Also (X*Y*Z)/(X+Y+Z) of primitive Pythagorean triples (X,Y,Z=Y+1) as described in A046092 and A001844. - Lambert Herrgesell (zero815(AT)googlemail.com), Dec 13 2005
First differences are in A201279. - J. M. Bergot, Jun 22 2013 [Corrected by Omar E. Pol, Dec 26 2021]

			Between 3^2 and 4^2 we have 10+11+12+13+14+15 which is 75 or a(4).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for sequences related to sums of squares

Cf. A048396, A048397, A046092, A001844.

Cf. A001844, A046092, A086849, A199771, A201279.

a048395 0 = 0
a048395 n = a199771 (2 * n)  -- Reinhard Zumkeller, Oct 26 2015

Table[n(1+2*n(1+n)),{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,5,26,75},40] (* Harvey P. Dale, Nov 01 2013 *)

v0=[1,0,1]; M=[1, 2, 2; -2, -1, -2; 2, 2, 3];
g(v)=v[1]*v[2]*v[3]/(v[1]+v[2]+v[3]);
a(n)=g(v0*M^n);
for(i=0,50,print1(a(i),", ")) \\ Lambert Herrgesell (zero815(AT)googlemail.com), Dec 13 2005

a(n) = 2*n^3 + 2*n^2 + n.
a(n) = Sum_{j=0..n} ((n+j+2)^2 - j^2 + 1). - Zerinvary Lajos, Sep 13 2006
O.g.f.: x(x+5)(1+x)/(1-x)^4. - R. J. Mathar, Jun 12 2008
a(n) = A199771(2*n) for n > 0. - Reinhard Zumkeller, Nov 23 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=5, a(2)=26, a(3)=75. - Harvey P. Dale, Nov 01 2013
E.g.f.: exp(x)*x*(5 + 8*x + 2*x^2). - Stefano Spezia, Jun 25 2022

A109470 Sum of first n noncubes.

Original entry on oeis.org

2, 5, 9, 14, 20, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 370, 399, 429, 460, 492, 525, 559, 594, 630, 667, 705, 744, 784, 825, 867, 910, 954, 999, 1045, 1092, 1140, 1189, 1239, 1290, 1342, 1395, 1449, 1504, 1560

Offset: 1

Jonathan Vos Post, Aug 28 2005

			a(6) = 2 + 3 + 4 + 5 + 6 + 7 = 27.
a(7) = 2 + 3 + 4 + 5 + 6 + 7 + 9 = 36.

Cf. A000537, A007412, A048766, A064524, A086849.

Accumulate[With[{no=60},Complement[Range[no],Range[Floor[Power[no, (3)^-1]]]^3]]]  (* Harvey P. Dale, Feb 14 2011 *)

a(n) = sum(i=1, n, i + sqrtnint(i + sqrtnint(i, 3), 3)); \\ Michel Marcus, Jun 20 2024

from sympy import integer_nthroot
def A109470(n): return ((m:=n+(k:=integer_nthroot(n,3)[0])+int(n>=(k+1)**3-k))*(m+1)>>1)-((r:=integer_nthroot(m,3)[0])*(r+1)>>1)**2 # Chai Wah Wu, Jun 17 2024

a(n) = Sum_{i=1..n} A007412(i).
a(n) = A000217(A007412(n)) - Sum_{i=1..floor((A007412(n)^(1/3)))} i^3.
a(n) = A000217(A007412(n)) - A000217(floor(A007412(n)^(1/3)))^2.
Let R = A007412(n) and S = floor(R^(1/3)); then a(n) = (R*(R+1))/2 - ((S*(S+1))/2)^2. - Gerald Hillier, Dec 21 2008

A132295 Sum of the nonsquare numbers not larger than n.

Original entry on oeis.org

0, 2, 5, 5, 10, 16, 23, 31, 31, 41, 52, 64, 77, 91, 106, 106, 123, 141, 160, 180, 201, 223, 246, 270, 270, 296, 323, 351, 380, 410, 441, 473, 506, 540, 575, 575, 612, 650, 689, 729, 770, 812, 855, 899, 944, 990, 1037, 1085, 1085, 1135, 1186, 1238, 1291, 1345

Offset: 1

Cino Hilliard, Nov 07 2007

			Let n=5. The sum of the nonsquare numbers <= 5 is 2+3+5 = 10, the 5th entry in the sequence.

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A000037, A000330, A086849.

Table[Total[Select[Range[n],!IntegerQ[Sqrt[#]]&]],{n,54}] (* James C. McMahon, Mar 04 2025 *)

sumNsq(n)= { for(x=1,n,r=floor(sqrt(x));sq=r*(r+1)*(2*r+1)/6;sn=x*(x+1)/2; print1(sn-sq",")) }

from math import isqrt
def A000330(n): return n*(n+1)*(2*n+1)//6
def A132295(n): return n*(n+1)//2-A000330(isqrt(n)) # Jason Yuen, Jan 30 2024

Let r = floor(sqrt(n)). Then a(n) = n(n+1)/2 - r(r+1)(2r+1)/6.

Definition corrected by R. J. Mathar, Sep 10 2016

A270439 Alternating sum of nonsquares (A000037).

Original entry on oeis.org

2, -1, 4, -2, 5, -3, 7, -4, 8, -5, 9, -6, 11, -7, 12, -8, 13, -9, 14, -10, 16, -11, 17, -12, 18, -13, 19, -14, 20, -15, 22, -16, 23, -17, 24, -18, 25, -19, 26, -20, 27, -21, 29, -22, 30, -23, 31, -24, 32, -25, 33, -26, 34, -27, 35, -28, 37, -29, 38, -30, 39, -31, 40, -32, 41, -33, 42, -34, 43, -35, 44, -36, 46, -37, 47

Offset: 1

Ilya Gutkovskiy, Jul 12 2016

Interleaving of nontriangular numbers (A014132) and negative integers (A001478).

			a(1) = a(2*1-1) = 1 + floor(1/2 + sqrt(2*1)) = 2;
a(2) = a(2*1) = -1;
a(3) = a(2*2-1) = 2 + floor(1/2 + sqrt(2*2)) = 4;
a(4) = a(2*2) = -2;
a(5) = a(2*3-1) = 3 + floor(1/2 + sqrt(2*3)) = 5;
a(6) = a(2*3) = -3, etc.
or
a(1) = 2;
a(2) = 2 - 3 = -1;
a(3) = 2 - 3 + 5 = 4;
a(4) = 2 - 3 + 5 - 6 = -2;
a(5) = 2 - 3 + 5 - 6 + 7 = 5;
a(6) = 2 - 3 + 5 - 6 + 7 - 8 = -3, etc.
(2, 3, 5, 6, 7, 8, ... is the nonsquares).

Cf. A000037, A001478, A014132, A086849.

Table[Sum[(-1)^(k + 1) (k + Floor[1/2 + Sqrt[k]]), {k, n}], {n, 75}]

a(n)=if(n%2, sqrtint(4*n-3)+n+2, -n)\2 \\ Charles R Greathouse IV, Aug 03 2016

from math import isqrt
def A270439(n): return (n>>1)+1+(m:=isqrt(n+1))+int(n-m*(m+1)>=0) if n&1 else -(n>>1) # Chai Wah Wu, Nov 14 2022

a(n) = Sum_{k = 1..n} (-1)^(k+1)*(k + floor(1/2 + sqrt(k))).
a(n) = Sum_{k = 1..n} (-1)^(k+1)*A000037(k).
a(2^k*m) = -2^(k-1) * m, k > 0.
a(2k - 1) = k + floor(1/2 + sqrt(2*k)), a(2k) = -k, k > 0.
a(2k - 1) = A014132(k), a(2k) = A001478(k).

A329598 Partial sums of the nontriangular numbers (A014132).

Original entry on oeis.org

2, 6, 11, 18, 26, 35, 46, 58, 71, 85, 101, 118, 136, 155, 175, 197, 220, 244, 269, 295, 322, 351, 381, 412, 444, 477, 511, 546, 583, 621, 660, 700, 741, 783, 826, 870, 916, 963, 1011, 1060, 1110, 1161, 1213, 1266, 1320, 1376, 1433, 1491, 1550, 1610, 1671, 1733

Offset: 1

Metin Sariyar and Robert G. Wilson v, Nov 17 2019

nonn

Terms which are triangular: 6, 136, 351, 741, 2415, 3916, 5995, 12561, 17391, 23436, ..., .

			The nontriangular numbers begin 2, 4, 5, 7, ..., so their partial sums begin 2, 6, 11, 18, etc.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Cf. A329472, A014132, A086849.

Cf. A000217, A060432.

triQ[n_] := IntegerQ @ Sqrt[8n + 1]; Accumulate@ Select[ Range@ 70, !triQ@# &]

from math import isqrt
def A329598(n): return (k:=(r:=isqrt(m:=n+1<<1))+int((m<<2)>(r<<2)*(r+1)+1)-1)*(k*(-k - 3) + 6*n - 2)//6 + (n*(n+3)>>1) # Chai Wah Wu, Jun 18 2024

a(n) = Sum_{i=1..n} A014132(i).

a(n) = A000217(n) + A060432(n). [corrected by Gerald Hillier, Jul 31 2022]

A351855 Partial sums of nonsquares that are partial sums of nonprimes.

Original entry on oeis.org

5, 64, 506, 64325, 268723, 480129, 6282620, 64548862, 9657523883, 13480852825, 29766135708, 105223301080, 519861666225, 851245744041, 1378216791896, 581522966976875, 583298551668358, 885441628670251, 1651966084813205, 16868988672306046, 17170433482837259

Offset: 1

J. M. Bergot and Robert Israel, Mar 31 2022

nonn

			a(2) = 64 is a term because 64 = 1+4+6+8+9+10+12+14 = 2+3+5+6+7+8+10+11+12 is the sum of the first 8 nonprimes and the sum of the first 9 nonsquares.

Chai Wah Wu, Table of n, a(n) for n = 1..26

Intersection of A051349 and A086849.

i:= 0: j:= 0: s:= 0: t:= 0:
R:= NULL: count:= 0:
while count < 13 do
  if s <= t then
     i:= i+1;
     if not issqr(i) then
       s:= s+i;
       if s=t then R:= R,s; count:= count+1 fi;
     fi
  else
     j:= j+1;
     if not isprime(j) then
       t:= t+j;
       if s=t then R:= R,t; count:= count+1 fi;
     fi
  fi
od:
R;

from itertools import islice
from sympy import nextprime
def A351855_gen(): # generator of terms
    c, k, ks, m, p, q = 0, 1, 2, 1, 4, 5
    while True:
        for n in range(ks,ks+2*k):
            c += n
            if c == m:
                yield c
            else:
                while c > m:
                    m += p
                    p += 1
                    if p == q:
                        q = nextprime(q)
                        p += 1
        ks += 2*k+1
        k += 1
A351855_list = list(islice(A351855_gen(),20)) # Chai Wah Wu, Apr 04 2022

a(20)-a(21) from Jon E. Schoenfield, Mar 31 2022

cp's OEIS Frontend

A352738 Squares in A086849.

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A355371 Intersection of A000330 and A086849.

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A000037 Numbers that are not squares (or, the nonsquares).

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A048395 Sum of consecutive nonsquares.

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A109470 Sum of first n noncubes.

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A132295 Sum of the nonsquare numbers not larger than n.

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