A059591 -id:A059591 - cp's OEIS frontend

A059321 Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).

7, 70, 38, 41, 117, 378, 500, 682, 776, 3861, 4052, 515, 5744, 1710, 6613, 1744, 11018, 13241, 3458, 5099, 1393, 16610, 26884, 15006, 2072, 13637, 31361, 4443, 26508, 7850, 37520, 31152, 39922, 37107, 6072, 4005, 32491, 4030, 43211, 12238

Offset: 1

Marc LeBrun, Jan 26 2001

a(2) = 70 since A002144(2)=13, 70^2+1 = 4091 = 13^2 * 29 and for no k<70 does 13^2 divide k^2+1. Related to period-1 continued fractions.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Klaus Brockhaus)

Cf. A002144, A059591, A059592.

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A059321_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**2))
A059321_list = list(islice(A059321_gen(),20)) # Chai Wah Wu, May 04 2024

A059592 Square-full part of n^2 + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 17, 1, 1, 29, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 13, 1, 1, 1

Offset: 0

Marc LeBrun, Jan 25 2001

Related to period-1 continued fractions [z,z,z,...].

A124808 gives number of numbers m <= n with a(m)=1. - Reinhard Zumkeller, Nov 08

			a(7)=5 since 7^2 + 1 = 50 = 25*2 = (5^2)*2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 (corrected by _Sean A. Irvine_)

Cf. A000188, A059591, A007913, A002522.

a(n) = core(n^2+1, 1)[2]; \\ Michel Marcus, Nov 12 2023

a(n) = A000188(A002522(n)).

a(n)^2 * A059591(n) = n^2 + 1.

A069987 Squarefree numbers of form k^2 + 1.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1090, 1157, 1226, 1297, 1370, 1522, 1601, 1765, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 2917, 3026

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 01 2002

nonn

Heath-Brown (following Estermann) shows that, for any e > 0, there are k sqrt(x) + O(x^{7/24 + e}) members of this sequence up to x, for k = Product(1 - 2/p^2) = 0.8948412245... (A335963) where the product is over primes p = 1 mod 4. - Charles R Greathouse IV, Nov 19 2012, corrected by Amiram Eldar, Jul 08 2020

Integers k for which the period of the continued fraction of sqrt(k) is 1. - Michel Marcus, Apr 12 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), pp. 653-662.
D. R. Heath-Brown, Square-free values of n^2 + 1, arXiv:1010.6217 [math.NT], 2010-2012.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155 (2012), pp. 1-13.

Cf. A059591, A002496, A124809, A005117, A002522, A335963.

select(numtheory:-issqrfree, [seq(n^2+1, n=1..100)]); # Robert Israel, Feb 09 2016

Select[ Range[10^4], IntegerQ[ Sqrt[ # - 1]] && Union[ Transpose[ FactorInteger[ # ]] [[2]]] [[ -1]] == 1 &]
Select[Range[60]^2+1,SquareFreeQ] (* Harvey P. Dale, Mar 21 2013 *)

for(n=1,100,if(issquarefree(n^2+1),print1(n^2+1,",")))

a(n) = A049533(n)^2 + 1.

Edited and extended by Robert G. Wilson v, Benoit Cloitre and Vladeta Jovovic, May 04 2002

A282092 Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.

Original entry on oeis.org

7, 18, 117, 239, 378, 843, 2207, 2943, 4443, 4662, 6072, 8307, 8708, 9872, 31561, 103682, 271443, 853932, 1021693, 3539232, 3699356, 6349657, 6907607, 7042807, 7249325, 9335094, 12623932, 12752043, 12813848, 22211431, 33385282, 42483057, 52374157, 105026693

Offset: 1

Michel Lagneau, Feb 06 2017

nonn

For the pairs (m, k), is k always unique?

The pairs (m, k) are (7, 3), (18, 8), (117, 43), (239, 5), (378, 132), (843, 377), (2207, 987), (2943, 73), (4443, 53), (4662, 1568), (6072, 5118), (8307, 743), (8708, 2112), (9872, 2738), ...

			7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5.

Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).

Cf. A002522, A059591, A059592, A124809.

Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* Michael De Vlieger, Feb 07 2017 *)

isok(n)=ok = 0; vn = factor(n^2+1)[,1]; for (k=1, n-1, if (factor(k^2+1)[,1] == vn, ok = 1; break);); ok; \\ Michel Marcus, Feb 09 2017

squeeze(f)=factorback(f)\2
list(lim)=my(v=List(),m=Map(),t); for(n=1,lim, t=squeeze(factor(n^2+1)[,1]); if(mapisdefined(m,t), listput(v,n), mapput(m,t,0))); Vec(v) \\ Charles R Greathouse IV, Feb 12 2017

use ntheory qw(:all);
for (my ($m, %t) = 1 ; ; ++$m) {
my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
push @{$t{$k}}, $m;
if (@{$t{$k}} >= 2) {
print'('.join(', ',reverse(@{$t{$k}})).")\n";
}
} # Daniel Suteu, Feb 08 2017

a(15)-a(29) from Daniel Suteu, Feb 08 2017
a(30) from Daniel Suteu, Feb 10 2017
a(31)-a(34) from Joerg Arndt, Feb 11 2017

cp's OEIS Frontend

A059321 Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).

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A059592 Square-full part of n^2 + 1.

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A069987 Squarefree numbers of form k^2 + 1.

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A282092 Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.

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