A354330 -id:A354330 - cp's OEIS frontend

A351830 Distance from the n-th square pyramidal number (sum of the first n positive squares) to the nearest square.

0, 0, 1, 2, 5, 6, 9, 4, 8, 4, 15, 22, 25, 22, 9, 15, 25, 21, 7, 30, 46, 53, 49, 32, 0, 49, 40, 41, 30, 91, 46, 12, 9, 15, 4, 26, 77, 114, 25, 91, 61, 105, 15, 122, 129, 66, 22, 1, 1, 24, 76, 157, 170, 37, 131, 141, 91, 139, 165, 15, 174, 247, 150, 80, 39, 29

Offset: 0

Paolo Xausa, Feb 21 2022

As noted by Conway and Sloane (1999), the only zero terms appear at n = 0, n = 1 and n = 24, and the n = 24 case allows for the Lorentzian construction of the Leech lattice through the A351831 vector.

The zero terms are equivalently the subject of the "pile of cannonballs" problem posed by Lucas and solved by Watson. - Peter Munn, Aug 03 2023

			a(4) = 5 because the sum of the first 4 positive squares is 1 + 4 + 9 + 16 = 30, the nearest square is 25 and 30 - 25 = 5. - _Paolo Xausa_, Jul 05 2022

W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tidsskr. 34 (1952), pp 65-72.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, entry 24, p 101.

Paolo Xausa, Table of n, a(n) for n = 0..9999
Michael A. Bennett, Lucas' Square Pyramid Problem Revisited.
Richard E. Borcherds, How to construct the Leech lattice, YouTube video, 2022.
J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice, Bulletin (New Series) of the American Mathematical Society, Volume 6, Number 2, March 1982.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer, New York, NY, 1999, pp. 524-528.
Anji Dong, Katerina Saettone, Kendra Song, and Alexandru Zaharescu, An Equidistribution Result for Differences Associated to Square Pyramidal Numbers II, arXiv:2505.04166 [math.NT], 2025.
E. Lucas, Problem 1180, Nouvelles Ann. Math. (2) 14 (1875), p 336.
G. N. Watson, The problem of the square pyramid, Messenger of Mathematics 48 (1918), pp. 1-22.
Wikipedia, Leech lattice.

Cf. A000290, A000330, A053188, A351831, A353295, A354330, A363284.

nterms=66;Array[Abs[(s=#(#+1)(2#+1)/6)-Round[Sqrt[s]]^2]&,nterms,0]

from math import isqrt
def a(n):
    t = n*(n+1)*(2*n+1)//6
    r = isqrt(t)
    return min(t - r**2, (r+1)**2 - t)
print([a(n) for n in range(66)]) # Michael S. Branicky, Feb 21 2022

From Paolo Xausa, Jul 05 2022: (Start)
a(n) = A053188(A000330(n)).
a(n) = abs(A000330(n) - A353295(n)). (End)

Name edited by Peter Munn, Aug 04 2023

A353295 Square nearest to the sum of the first n positive squares.

Original entry on oeis.org

0, 1, 4, 16, 25, 49, 100, 144, 196, 289, 400, 484, 625, 841, 1024, 1225, 1521, 1764, 2116, 2500, 2916, 3364, 3844, 4356, 4900, 5476, 6241, 6889, 7744, 8464, 9409, 10404, 11449, 12544, 13689, 14884, 16129, 17689, 19044, 20449, 22201, 23716, 25600, 27556, 29241

Offset: 0

Paolo Xausa, Jun 04 2022

			a(4) = 25 because the sum of the first 4 positive squares is 1 + 4 + 9 + 16 = 30, and the nearest square is 25.

Paolo Xausa, Table of n, a(n) for n = 0..9999

Cf. A000290, A000330, A053187, A351830, A354330.

nterms=100;Array[Round[Sqrt[#(#+1)(2#+1)/6]]^2&,nterms,0]

from math import isqrt
def a(n):
    s = n*(n+1)*(2*n+1)//6
    r = isqrt(s)
    d1, d2 = s-r**2, (r+1)**2-s
    return r**2 if d1 <= d2 else (r+1)**2
print([a(n) for n in range(45)]) # Michael S. Branicky, Jun 05 2022

a(n) = A053187(A000330(n)).

A354329 Triangular number nearest to the sum of the first n positive triangular numbers.

Original entry on oeis.org

0, 1, 3, 10, 21, 36, 55, 78, 120, 171, 210, 276, 351, 465, 561, 666, 820, 990, 1128, 1326, 1540, 1770, 2016, 2278, 2628, 2926, 3240, 3655, 4095, 4465, 4950, 5460, 5995, 6555, 7140, 7750, 8385, 9180, 9870, 10731, 11476, 12403, 13203, 14196, 15225, 16290, 17205

Offset: 0

Paolo Xausa, Jun 04 2022

			a(4) = 21 because the sum of the first 4 positive triangular numbers is 1 + 3 + 6 + 10 = 20, and the nearest triangular number is 21.

Wikipedia, Triangular number.

Cf. A000217, A000292, A053616, A229118, A354330.

nterms=100;Table[t=Floor[Sqrt[n(n+1)(n+2)/3]];(t^2+t)/2,{n,0,nterms-1}]

a(n)=my(t=sqrtint(n*(n+1)*(n+2)/3));(t^2+t)/2;
vector(100,n,a(n-1))

from math import isqrt
def A354329(n): return (m:=isqrt(n*(n*(n + 3) + 2)//3))*(m+1)>>1 # Chai Wah Wu, Jul 15 2022

a(n) = (t^2+t)/2, where t = floor(sqrt(n*(n+1)*(n+2)/3)).

A366092 Distance from the sum of the first n primes to the nearest prime.

Original entry on oeis.org

2, 0, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 0, 3, 2, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 5, 2, 1, 4, 1, 4, 1, 2, 3, 4, 5, 2, 3, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 10, 1, 0, 11, 2, 1, 0, 3, 2, 3, 2, 7, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 5, 4, 3, 10, 3

Offset: 0

Paolo Xausa, Sep 29 2023

Positions of zeros are given by A013916.

Positions of records are given by A366093.

			a(3) = 1 because the sum of the first 3 primes is 2 + 3 + 5 = 10, the nearest prime is 11 and 11 - 10 = 1.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000040, A007504, A013916, A051699, A366093, A366094.

Cf. also A351830, A354330.

pDist[n_]:=If[PrimeQ[n],0,Min[NextPrime[n]-n,n-NextPrime[n,-1]]];
A366092list[nmax_]:=Map[pDist,Prepend[Accumulate[Prime[Range[nmax]]],0]];
A366092list[100]

from sympy import prime, nextprime, prevprime
def A366092(n): return min((m:=sum(prime(i) for i in range(1,n+1)))-prevprime(m+1),nextprime(m)-m) if n else 2 # Chai Wah Wu, Oct 03 2023

a(n) = A051699(A007504(n)).

a(n) = abs(A007504(n) - A366094(n)).

cp's OEIS Frontend

A351830 Distance from the n-th square pyramidal number (sum of the first n positive squares) to the nearest square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A353295 Square nearest to the sum of the first n positive squares.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A354329 Triangular number nearest to the sum of the first n positive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A366092 Distance from the sum of the first n primes to the nearest prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula