A384797 -id:A384797 - cp's OEIS frontend

A384798 Records in A384797.

2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 31, 34, 37, 38, 39, 40, 42, 44, 49, 51, 56, 58, 64, 69, 71, 73, 77, 79, 82, 92, 94, 101, 104, 108, 109, 111, 116, 118, 120, 129, 136, 140, 141, 155, 160, 172, 174, 181, 190, 193, 194, 208, 209

Offset: 1

Hugo Pfoertner, Jun 17 2025

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A047800, A384797, A384799.

from itertools import count, islice
def A384798_gen(): # generator of terms
    s, m = {0}, 1
    for n in count(1):
        c = n+1
        for i in range(n+1):
            if (k:=n**2+i**2) in s:
                c -= 1
            else:
                s.add(k)
        if c>m:
            yield c
            m = c
A384798_list = list(islice(A384798_gen(),30)) # Chai Wah Wu, Jul 07 2025

A384799 Positions of records in A384797.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 22, 24, 27, 28, 30, 33, 35, 36, 40, 45, 48, 52, 54, 55, 56, 60, 65, 70, 78, 80, 90, 95, 100, 105, 108, 110, 120, 130, 135, 140, 150, 155, 156, 160, 165, 168, 180, 190, 200, 205, 210, 225, 240, 255, 260, 270, 280, 285, 300

Offset: 1

Hugo Pfoertner, Jun 17 2025

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A047800, A384797, A384798.

from itertools import count, islice
def A384799_gen(): # generator of terms
    s, m = {0}, 1
    for n in count(1):
        c = n+1
        for i in range(n+1):
            if (k:=n**2+i**2) in s:
                c -= 1
            else:
                s.add(k)
        if c>m:
            yield n
            m = c
A384799_list = list(islice(A384799_gen(),30)) # Chai Wah Wu, Jul 07 2025

A385754 Positive numbers not occurring in A384797.

Original entry on oeis.org

1, 6, 16, 20, 25, 30, 33, 41, 48, 53, 57, 59, 62, 67, 74, 75, 78, 86, 90, 93, 98, 100, 107, 110, 113, 114, 123, 128, 130, 135, 138, 142, 145, 151, 153, 157, 159, 162, 165, 168, 178, 183, 191, 202, 204, 211, 212, 220, 223, 229, 232, 245, 254, 255, 283, 286, 291, 301

Offset: 1

Hugo Pfoertner, Jul 08 2025

nonn

This sequence is to A384797 what A363762 is to A077773.

Cf. A047800, A077773, A363762, A384797.

A047800 Number of different values of i^2 + j^2 for i,j in [0, n].

Original entry on oeis.org

1, 3, 6, 10, 15, 20, 27, 34, 42, 51, 61, 71, 83, 94, 106, 120, 135, 148, 165, 180, 198, 216, 235, 252, 273, 294, 315, 337, 360, 382, 408, 431, 457, 484, 508, 536, 567, 595, 624, 653, 687, 715, 749, 781, 813, 850, 884, 919, 957, 993, 1031, 1069, 1108, 1142

Offset: 0

Wouter Meeussen

a(n-1) is the number of distinct distances on an n X n pegboard. What is its asymptotic growth? Can it be efficiently computed for large n? - Charles R Greathouse IV, Jun 13 2013

Conjecture (after Landau and Erdős): a(n) ~ c * n^2 / sqrt(log(n)), where c = 0.79... . - Vaclav Kotesovec, Mar 10 2016

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from T. D. Noe)
Paul Erdős, On sets of distances of n points, American Mathematical Monthly 53, pp. 248-250 (1946).
Vaclav Kotesovec, Graph - The asymptotic ratio
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig B. G. Teubner, 1909, p. 643.
Hugo Pfoertner, Plot of asymptotic ratio, 1.44*10^6 terms, logarithmic scale for n, indicating c > 0.8.

Cf. A034966, A047801, A160663, A384797.

import Data.List (nub)
a047800 n = length $ nub [i^2 + j^2 | i <- [0..n], j <- [i..n]]
-- Reinhard Zumkeller, Oct 03 2012

Table[ Length@Union[ Flatten[ Table[ i^2+j^2, {i, 0, n}, {j, 0, n} ] ] ], {n, 0, 49} ]
nmax = 100; sq = Table[i^2 + j^2, {i, 0, nmax}, {j, 0, nmax}]; Table[Length@Union[Flatten[Table[Take[sq[[j]], n + 1], {j, 1, n + 1}]]], {n, 0, nmax}] (* Vaclav Kotesovec, Mar 09 2016 *)

a(n) = n++; #vecsort(vector(n^2, i, ((i-1)\n)^2+((i-1)%n)^2), , 8) \\ Charles R Greathouse IV, Jun 13 2013; edited by Michel Marcus, Jul 06 2025

a(n) = #setbinop((i,j)->i^2+j^2, [0..n]); \\ Michel Marcus, Jul 07 2025

def A047800(n): return len(set(i**2+j**2 for i in range(n+1) for j in range(i+1))) # Chai Wah Wu, Jul 07 2025

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A384798 Records in A384797.

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A384799 Positions of records in A384797.

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A385754 Positive numbers not occurring in A384797.

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A047800 Number of different values of i^2 + j^2 for i,j in [0, n].

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