A042999 -id:A042999 - cp's OEIS frontend

A319041 Numbers k > 1 such that Pell(k) == -1 (mod k).

3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 251, 269, 277, 283, 293, 307, 317, 331, 347, 349, 373, 379, 389, 397, 419, 421, 443, 461, 467, 491, 499, 509, 523, 541, 547, 557

Offset: 1

Jon E. Schoenfield, Sep 08 2018

nonn

It appears that most of the terms of this sequence are primes. The composite terms are 741, 3827, 11395, 13067, 27971, ... (A319043).
The primes in the sequence give A003629 (primes == +-3 (mod 8)), since for primes p we have Pell(p) == (2/p) (mod p) where (2/p) is the Legendre symbol. - Jianing Song, Sep 10 2018
It appears that this sequence is (A042999 \ {2}) UNION A319043. - Georg Fischer, Oct 17 2018

			k = 3 is in the sequence since Pell(3) = 5 = 3*2 - 1 == -1 (mod 3).
k = 7 is not in the sequence: Pell(7) = 169 = 7*24 + 1 !== -1 (mod 7).

Seiichi Manyama, Table of n, a(n) for n = 1..9999 (offset adapted by _Georg Fischer_, Jan 31 2019)

Cf. A000129 (Pell numbers), A003629, A042999, A319040, A319042, A319043.

Select[Range[800], Mod[Fibonacci[-#, 2], -#]== -1 &] (* Vincenzo Librandi, Sep 09 2018 after Alonso del Arte; {1} removed by Georg Fischer, Jan 31 2019 *)

A289398 Least integer m > n such that (n^2 + m^2)/2 is a square.

Original entry on oeis.org

7, 14, 21, 28, 35, 42, 17, 56, 63, 70, 77, 84, 91, 34, 105, 112, 31, 126, 133, 140, 51, 154, 47, 168, 175, 182, 189, 68, 203, 210, 49, 224, 231, 62, 85, 252, 259, 266, 273, 280, 113, 102, 301, 308, 315, 94, 79, 336, 71, 350, 93, 364, 371, 378, 385, 136, 399, 406, 413, 420

Offset: 1

Zak Seidov, Jul 05 2017

From first 100 terms, in 65 cases a(n) = 7*n. In general, a(n) <= 7*n.
From Robert Israel, Jul 07 2017: (Start)
For any p in A042999, a(n) == 0 (mod p) if and only if n == 0 (mod p), with a(p*k) = p*a(k).
Thus if n = m*r where all prime factors of m are in A042999, a(n) = m*a(r).
In particular, if all prime factors of n are in A042999, then a(n) = 7*n.
Conjecture: this is "if and only if".
(End)
Alternatively: A306236(n) is the smallest integer m > n with integer j > m that makes n^2, m^2 and j^2 an arithmetic progression. This is the sequence of j. - Jinyuan Wang, Feb 09 2019.

			a(1)=7: (1^2 + 7^2)/2 = 5^2;
a(7)=17: (7^2 + 17^2)/2 = 5^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A042999, A075892, A306236.

f:= proc(n) local m; for m from n+2 by 2 do if issqr((n^2+m^2)/2) then return m fi od end proc:
map(f, [$1..100]); # Robert Israel, Jul 07 2017

n=0;Table[n++;m=n+1;While[!IntegerQ[Sqrt[(n^2+m^2)/2]],m++];m,{100}]

a(n) = my(m=n+1); while(!issquare((n^2+m^2)/2), m++); m; \\ Michel Marcus, Jul 07 2017

from itertools import count
from sympy.ntheory.primetest import is_square
def A289398(n): return next(m for m in count(n+2,2) if is_square(n**2+m**2>>1)) # Chai Wah Wu, Mar 02 2025

A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 17, 22, 23, 24, 25, 26, 28, 31, 33, 36, 38, 39, 40, 41, 42, 47, 49, 54, 55, 57, 58, 60, 63, 64, 65, 68, 70, 71, 73, 74, 79, 81, 86, 87, 88, 89, 90, 92, 95, 96, 97, 100, 102, 103, 104, 105, 106, 111, 112, 113, 118, 119, 121, 122, 124, 127, 129

Offset: 1

Peter Munn, Feb 13 2024

Construction by subgroup generation: (Start)
The set of numbers congruent to 1 modulo 8 (A017077) contains all the odd squares and generates a subgroup of the positive rational numbers (under multiplication) that contains no additional integers. The subgroup has an infinite number of cosets. The rest of the construction process extends the subgroup, reducing the number of cosets to 2, by choosing additional generators that are semiprime.
First we extend the subgroup to include all nonzero integer squares. As we already have the odd squares, we need only add 4, the square of the smallest prime, as a generator. The extended subgroup has only 8 cosets and its integer members are listed in A234000. To achieve a subgroup with 2 cosets we now add squarefree semiprime generators. The 2 smallest, 6 and 10, suffice.
The resulting subgroup has this sequence's terms as its integer members.
(End)
The equivalent process starting with numbers congruent to 1 modulo 3 (or 1 modulo 6) produces A189715. If we take its intersection with this sequence we get A370268, which starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). Similarly, if we start with numbers congruent to 1 modulo 5 (or 1 modulo 10) and take the resulting set's intersection with this sequence we get a set starting with the first 32 nonzero numbers of the form x^2 - 10y^2 (see A242664).
The construction process leads to a number of properties:
- The sequence is closed under multiplication and all integer ratios between terms are in the sequence.
- The sequence and its complement have the property that the terms of one can be generated by halving the even terms of the other. Each has asymptotic density 1/2.
Numbers whose squarefree part is congruent to {1,7} mod 8 or {6,10} mod 16.

			7 is prime, so 7 is its only prime factor, which has the form 8m-1. So 7 has an even number (zero) of prime factors not of the form 8m+-1, and therefore is in the sequence. In terms of the subgroup generators described at the start of the comments, (13*8+1) * 4 / (6*10) = 105 * 4/60 = 7.
110 = 2 * 5 * 11, so it has 3 prime factors and all 3 do not have the form 8m+-1. 3 is odd, so 110 is not in the sequence.

Disjoint union of A004215, A055042, A055043 and A234000.
See the comments for the relationships with A002481, A017077, A189715, A242664, A370268.
Cf. A042999 (primes), A059897.

isok(k) = {c = core(k); c%8 == 1 || c%8 == 7 || c%16 == 6 || c%16 == 10}

def A370267(n):
    def f(x): return n+x-sum(((y:=x>>(i<<1))-7>>3)+(y-1>>3)+2 for i in range((x.bit_length()>>1)+1))-sum(((z:=x>>(i<<1)+1)-5>>3)+(z-3>>3)+2 for i in range(x.bit_length()-1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Mar 19 2025

{a(n) : n >= 1} = {A059897(i,j) : i in A234000, j in {1, 6, 10, 15}}.

A370268 Intersection of A189715 and A370267.

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49, 54, 55, 58, 60, 63, 64, 70, 73, 79, 81, 87, 88, 90, 96, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 144, 145, 150, 151, 154, 159, 160, 166, 168, 169, 175, 177, 186, 193, 196, 198, 199, 202, 214, 216, 217, 220, 223, 225, 231, 232, 240, 241, 247

Offset: 1

Peter Munn, Feb 13 2024

A189715 and A370267 are closely related in that they may be generated by the same process, but starting from numbers of the form 6m+1 and 8m+1 respectively - see A370267 for details.
Independent definition: numbers with an even number of prime factors not of the form 3m+1 and an even number of prime factors not of the form 8m+-1 (counting repetitions).
The sequence starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). After the absence of 0, this sequence next differs from A002481 by including 247, 391, 442, ... . From these early intermittent differences, the densities of the two sequences diverge progressively, driven by the absence from A002481 of many of the squarefree composite numbers that are present here though their prime factors are not. (Both sequences are closed under multiplication.) Asymptotic densities are 1/4 and 0 respectively.
Likewise, if we list the even terms halved, we find a similar relationship to the nonzero terms of A002480. The first 66 terms match, then we find we have generated intermittent extra terms: 221, 299, 323, ... .
Numbers whose squarefree part is congruent to {1,7} mod 24, {10,22} mod 48, {15,33} mod 72, or {6,42} mod 144. (Each congruence describes a coset of A334832 under A059897(.,.) as described in A334832. This sequence corresponds to the subgroup of the quotient group generated by {6,7,10}.)

Intersection of A189715 and A370267.
A002481\{0}, A334832 are subsequences.
Cf. A002480, A042999, A045309, A059897.

isok(k) = {c = core(k); c%24 == 1 || c%24 == 7 || c%48 == 10 || c%48 == 22 || c%72 == 15 || c%72 == 33 || c%144 == 6 || c%144 == 42}

{a(n) : n >= 1} = {A059897(i,j*k) : i in A334832, j in {1,7}, k in {1,6,10,15}}.

A373560 a(n) is the smallest multiple of prime(n)^2 that starts a run of 5 consecutive integers with 6 divisors, or -1 if no such multiple exists.

Original entry on oeis.org

-1, -1, -1, 10093613546512321, -1, -1, 7700031346933907521, -1, 5344962129269790721, -1, 20453982425165652721, -1, 8163195338222675521, -1, 2467958104789157112721, -1, -1, -1, -1, 14666767069023896053921, 212170739123852995921, 287954235303137500060321, -1, 84769922583214545304321

Offset: 1

Jon E. Schoenfield, Jun 09 2024

sign

Terms were obtained using the b-file at A141621.
a(n) = -1 if prime(n) is not in A001132.
Conjecture: the converse is also true.

			a(1) = a(2) = a(3) = -1 because the first of five consecutive integers having six divisors is never a multiple of 2^2, 3^2, or 5^2.
a(4) = 10093613546512321 because it is the smallest term in A141621 that is a multiple of prime(4)^2 = 49.
a(9) = 5344962129269790721 because it is the smallest term in A141621 that is a multiple of prime(9)^2 = 23^2.

Cf. A001132, A030515, A042999, A054753, A141621.

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A319041 Numbers k > 1 such that Pell(k) == -1 (mod k).

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A289398 Least integer m > n such that (n^2 + m^2)/2 is a square.

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A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

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A370268 Intersection of A189715 and A370267.

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A373560 a(n) is the smallest multiple of prime(n)^2 that starts a run of 5 consecutive integers with 6 divisors, or -1 if no such multiple exists.

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