A353295 -id:A353295 - 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

A366094 Least prime nearest to the sum of the first n primes.

Original entry on oeis.org

2, 2, 5, 11, 17, 29, 41, 59, 79, 101, 127, 157, 197, 239, 281, 331, 379, 439, 499, 569, 641, 709, 787, 877, 967, 1061, 1163, 1259, 1373, 1481, 1597, 1721, 1847, 1987, 2129, 2273, 2423, 2579, 2749, 2917, 3089, 3271, 3449, 3637, 3833, 4027, 4229, 4441, 4663, 4889

Offset: 0

Paolo Xausa, Sep 29 2023

			a(3) = 11 because the sum of the first 3 primes is 2 + 3 + 5 = 10 and the nearest prime is 11.
a(10) = 127 because the sum of the first 10 primes is 129, which is equidistant from the nearest primes (127 and 131), and 127 is the smaller one.

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

Cf. A000040, A007504, A366092.

Cf. A353295, A354329.

pNearest[n_]:=If[PrimeQ[n],n,With[{np=NextPrime[n],pp=NextPrime[n,-1]},If[np-nA366094list[nmax_]:=Prepend[Map[pNearest,Accumulate[Prime[Range[nmax]]]],2];
A366094list[100]

from sympy import prime, nextprime, prevprime
def A366094(n): return (p if ((m:=sum(prime(i) for i in range(1,n+1)))<<1)-(p:=prevprime(m+1))<=(k:=nextprime(m)) else k) if n else 2 # Chai Wah Wu, Oct 03 2023

cp's OEIS Frontend

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

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