A134406 -id:A134406 - cp's OEIS frontend

A245463 Smallest k such that A002522(k) and A002522(k+2n) are successive primes of the form m^2+1.

2, 6, 84, 66, 26, 134, 40, 94, 986, 184, 1524, 716, 864, 1246, 2986, 784, 350, 2174, 4796, 496, 7674, 13136, 3390, 12636, 5880, 9904, 16446, 37410, 6646, 10430, 56774, 31870, 9054, 24606, 12986, 54284, 35000, 124320, 114216, 58576, 88854, 85416, 18854, 3536

Offset: 1

Michel Lagneau, Jul 22 2014

nonn

A002522(n) = n^2+1.

Subsequence of A005574.

			a(3) = 84 because A002522(84)=7057 and A002522(84+2*3)= 8101 are two consecutive primes of the form m^2+1.

Jens Kruse Andersen, Table of n, a(n) for n = 1..163

Cf. A002496, A002522, A005574, A134406, A206400.

T:=array(1..44):
for n from 1 by 2 to 88 do:
   z:=0:ii:=0:
    for k from 2 to 10^7 while(z=0) do:
    p:=k^2+1:
    if type(p,prime)=false
     then
     ii:=ii+1:
     else
      if ii=n
      then
      printf ( "%d %d \n",(n+1)/2,k-n-1):T[(n+1)/2]:=k-n-1:z:=1:
      else
     fi:
     ii:=0:
    fi:
   od:
od:
print(T):

for(n=1, 44, m=2; until(m==k+2*n, k=m; until(isprime(m^2+1), m++)); print1(k", ")) \\ Jens Kruse Andersen, Jul 22 2014

A248511 Difference between k and the least prime factor of k^2+1 where k is the n-th number with k^2+1 composite.

Original entry on oeis.org

1, 3, 5, 3, 7, 9, 7, 11, 13, 15, 13, 17, 19, 17, 21, 23, 25, 23, 27, 13, 29, 27, 31, 21, 33, 35, 33, 37, 39, 37, 41, 31, 43, 17, 45, 43, 47, 9, 49, 47, 51, 53, 55, 53, 57, 47, 59, 57, 61, 47, 63, 65, 63, 67, 57, 69, 67, 71, 73, 23, 75, 73, 77, 43, 79, 77, 81

Offset: 1

Michel Lagneau, Oct 07 2014

nonn

a(n) = A134407(n) - least prime divisor of A134406(n).

			a(1) = 1 because the first composite is 3^2+1 = 2*5 and 3-2 = 1.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A002496, A020639, A089120, A134406, A134407.

with(numtheory):
   for n from 1 to 200 do:
    p:=n^2+1:x:=factorset(p):d:=n-x[1]:
    if type(p,prime)=false
    then
    printf(`%d, `,d):
    else
    fi:
   od:

A249122 a(n) = floor(n / lpf(n^2 + 1)) where lpf(n^2 + 1) is the smallest prime divisor of n^2 + 1.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 2, 6, 0, 7, 0, 8, 3, 9, 0, 10, 4, 11, 0, 12, 0, 13, 5, 14, 1, 15, 6, 16, 2, 17, 0, 18, 7, 19, 0, 20, 8, 21, 3, 22, 1, 23, 9, 24, 1, 25, 10, 26, 0, 27, 0, 28, 11, 29, 4, 30, 12, 31, 3, 32, 0, 33, 13, 34, 5, 35, 14, 36, 0, 37, 1

Offset: 1

Michel Lagneau, Oct 21 2014

nonn

a(n) = floor(n / A089120(n)).

a(A002496(n)) = 0 and a(A247340(n)) = 1 where A002496 are the primes of form m^2 + 1 and A247340(n) = {3, 8, 30, 46, 50, 76, ...} are the numbers m such that m^2 + 1 = p*q, p and q primes => p | a^2+1 and q | b^2+1 for some a,b < m.

			a(8) = 1 because 30^2 + 1 = 17*53 and floor(30/17) = 1.
Or a(8) = a(A247340(2)) = 1.

Michel Lagneau, Table of n, a(n) for n = 1..20000

Cf. A002496, A089120, A134406, A247340.

with(numtheory):
   for n from 1 to 200 do:
    p:=n^2+1:x:=factorset(p):d:=floor(n/x[1]):
    printf(`%d, `, d):
   od:

Table[Floor[n/ FactorInteger[n^2+1][[ 1, 1]]], {n, 100}]

a(n) = n\factor(n^2+1)[1, 1]; \\ Michel Marcus, Oct 25 2014

A276460 Numbers k such that for any positive integers a < b, if a * b = k then b - a is a square.

Original entry on oeis.org

0, 1, 2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 901, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 10001, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 20737, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177, 52901

Offset: 1

Michel Lagneau, Sep 03 2016

nonn

A majority of numbers are primes of form m^2+1 (A002496), and it appears that the composite numbers of the form m^2+1: 901, 10001, 20737, 75077, 234257, 266257, 276677, 571537,... are semiprimes.

For n >1, a(n)==1,5 mod 12 and a(n)==1,5 mod 16.

			901 is in the sequence because 901 = 1*901 = 17*53 => 901-1 = 30^2 and 53-17 = 6^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002496, A134406.

t={};Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=IntegerQ[Sqrt[Abs[ds[[k]]-ds[[-k]]]]]),k++];If[ok,AppendTo[t,n]]],{n,2,10^5}];t

from _future_ import division
from sympy import divisors
from gmpy2 import is_square
A276460_list = [0]
for m in range(10**3):
    k = m**2+1
    for d in divisors(k):
        if d > m:
            A276460_list.append(k)
            break
        if not is_square(k//d - d):
            break # Chai Wah Wu, Sep 04 2016

Terms 0, 1 added by Chai Wah Wu, Sep 04 2016

A359185 Numbers k such that for any positive integers x,y, if x*y=k then (x+y)^2+1 is a prime number.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 23, 25, 39, 53, 55, 73, 83, 89, 109, 119, 133, 149, 155, 159, 169, 179, 203, 223, 229, 239, 263, 269, 283, 299, 305, 313, 339, 349, 383, 395, 419, 439, 443, 463, 469, 473, 543, 569, 593, 643, 653, 673, 689, 699, 703, 713, 739, 763, 859, 863, 889, 909

Offset: 1

Michel Lagneau, Dec 19 2022

nonn

Conjecture: if a term k is a perfect square > 1, then sqrt(k) is in the sequence A236068 (Primes p such that f(f(p)) is prime, where f(z) = z^2 + 1).
The conjecture is false.  A counterexample is 296147^2 = 87703045609 where 296147 = 47 * 6301. - Robert Israel, Mar 05 2024
The primes of the sequence are in A157468.
All terms except 1 are congruent to 3, 5 or 9 (mod 10). - Robert Israel, Mar 05 2024

			909 is in the sequence because 909 = 3^2*101 with 3 decompositions:
909 = 1*909 and (1+909)^2+1 = 910^2+1 = 828101 is prime;
909 = 3*303 and (3+303)^2+1 = 306^2+1 = 93637 is prime;
909 = 9*101 and (9+101)^2+1 = 110^2+1 = 12101 is prime.

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

Cf. A001358, A002496, A134406, A157468, A236068.

filter:= proc(n) local F;
  F:= select(t -> t^2 <= n, numtheory:-divisors(n));
  andmap(t -> isprime((t + n/t)^2+1), F)
end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Mar 05 2024

t={};Do[ds=Divisors[n];k=1;While[k<=(Length[ds]+1)/2&&(ok=PrimeQ[(ds[[k]]+ds[[-k]])^2+1]),k++];If[ok,AppendTo[t,n]],{n,1,2000}];t

isok(k) = fordiv(k, d, if ((d<=k/d) && !isprime((d+k/d)^2+1), return(0));); return(1); \\ Michel Marcus, Dec 19 2022

A180508 Numbers k such that k^2 + 1 = p*q, p and q primes and | p-q | <= k.

Original entry on oeis.org

3, 8, 46, 50, 76, 100, 144, 266, 274, 334, 504, 516, 526, 566, 670, 726, 756, 810, 836, 844, 1064, 1086, 1164, 1250, 1300, 1714, 1740, 1800, 1826, 1834, 1946, 1950, 2014, 2194, 2220, 2440, 2450, 2466, 2494, 2560, 2610

Offset: 1

Michel Lagneau, Jan 20 2011

nonn

|p - q| = k for k = 3, 8, 144.

			46 is in the sequence because 46^2 + 1 = 29*73, and 73-29 = 44 < 46.

Cf. A134406 A027862 A134407 A002522 A005574.

with(numtheory):for n from 1 to 4000 do: x:=n^2+1:y:=factorset(x):yy:=bigomega(x):if
  yy=2 and (y[2]-y[1] < n or y[2]-y[1] = n) then printf(`%d, `,n):else fi:od:

A206246 Numbers n such that the greatest prime divisor p of n^2+1 has the property that (p - n)^2 + 1 = p.

Original entry on oeis.org

1, 3, 7, 13, 21, 31, 43, 91, 111, 183, 211, 241, 273, 381, 421, 553, 601, 651, 703, 1261, 1333, 1561, 1641, 2863, 2971, 3081, 3193, 4291, 4423, 5403, 5551, 6973, 7141, 8011, 8191, 8743, 8931, 11991, 12211, 13341, 13573, 14281, 14521, 15253, 15501, 15751, 16003

Offset: 1

Michel Lagneau, Feb 05 2012

nonn

For the n > 1 in this sequence, n^2+1 is composite. The corresponding primes p are A002496(n) repeated two times for n > 1 : {2, 5, 5, 17, 17, 37, 37, 101, 101, 197,...}.

Because this sequence is connected with A002496, it is conjectured that the set of this numbers is infinite.

			31 is in the sequence because 31^2 + 1 = 2*13*37 and (37 - 31)^2 + 1 = 37.
43 is in the sequence because 43^2 + 1 = 2*5*5*37 and (37 - 43)^2 + 1 = 37.

Cf. A002496, A134406.

with(numtheory):for n from 1 to 20000 do:x:=n^2+1:y:=factorset(x):n1:=nops(y):p:=y[n1]:q:=(p-n)^2+1:if q=p then printf(`%d, `,n): else fi:od:

pn2pQ[n_]:=Module[{p=FactorInteger[n^2+1][[-1,1]]},(p-n)^2+1==p]; Select[ Range[20000],pn2pQ] (* Harvey P. Dale, Nov 20 2019 *)

A351177 Number of distinct residues of k^(n^2) (mod n^2+1), k=0..n^2.

Original entry on oeis.org

2, 2, 10, 2, 26, 2, 42, 8, 82, 2, 122, 16, 170, 2, 226, 2, 290, 12, 362, 2, 170, 50, 530, 2, 626, 2, 90, 80, 842, 70, 962, 36, 130, 92, 1226, 2, 1370, 138, 1522, 2, 1626, 178, 1554, 152, 2026, 152, 2210, 232, 2402, 12, 2602, 272, 2810, 2, 306, 2, 1010, 338, 3482

Offset: 1

Michel Lagneau, Mar 18 2022

nonn

a(A005574(n)) = 2.
a(n) = n for n = 2, 8, 128, ...
a(n) = n^2+1 (subsequence of A134406) for n = 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, ...
a(n) > 2 and a(n) <= n for n = 8, 18, 50, 60, 64, 72, 98, 112, 128, 132, 162, ... .
For n odd, gcd(a(n),n) = 1 except for n = 7, 27, 63, 75, 93, 105, 111, 125, 135, 153, 177, 207, 213, ...
For n even, gcd(a(n),n) = 2 for n in {A005574} union {22, 34, 38, 42, 46, 50, 58, 62, 78, 82, 86, 98, 102, 106, 114, 118, 122, 138, ...}
gcd(a(n),n) > 2 for n = 7, 8, 12, 18, 27, 28, 30, 32, 44, 48, 52, 60, 63, 64, 68, ...

			a(2) = 2 because k^(2^2) == 0, 1 (mod 5) implies 2 distinct residues.
The table of k^(n^2) (mod n^2+1) of residues starts in row n=1 with columns k>=2 as:
0,1;
0,1,1,1,1;
0,1,2,3,4,5,6,7,8,9;
0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25;
Its row sums are 1, 4, 45, 16, 325, ...

Cf. A005574, A134406, A195637.

a:= n-> nops ({seq (k&^(n^2) mod (n^2+1), k=0..n^2)}):
seq (a(n), n=1..100);

Table[Length[Union[PowerMod[Range[0,n^2],n^2,n^2+1]]],{n,100}]

a(n) = #Set(vector(n^2+1, k, k--; Mod(k, n^2+1)^n^2)); \\ Michel Marcus, Mar 18 2022

cp's OEIS Frontend

A245463 Smallest k such that A002522(k) and A002522(k+2n) are successive primes of the form m^2+1.

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A248511 Difference between k and the least prime factor of k^2+1 where k is the n-th number with k^2+1 composite.

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A249122 a(n) = floor(n / lpf(n^2 + 1)) where lpf(n^2 + 1) is the smallest prime divisor of n^2 + 1.

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A276460 Numbers k such that for any positive integers a < b, if a * b = k then b - a is a square.

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A359185 Numbers k such that for any positive integers x,y, if x*y=k then (x+y)^2+1 is a prime number.

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A180508 Numbers k such that k^2 + 1 = p*q, p and q primes and | p-q | <= k.

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A206246 Numbers n such that the greatest prime divisor p of n^2+1 has the property that (p - n)^2 + 1 = p.

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A351177 Number of distinct residues of k^(n^2) (mod n^2+1), k=0..n^2.

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