A014442 -id:A014442 - cp's OEIS frontend

A217393 Smallest k > 0 such that 1 + n^2 and 1 + (n+k)^2 have the same largest factor, or 0 if no such k exists.

0, 1, 4, 9, 3, 25, 0, 10, 23, 81, 39, 5, 8, 169, 83, 225, 24, 39, 143, 361, 17, 53, 7, 529, 263, 625, 19, 101, 363, 53, 12, 41, 43, 21, 543, 1225, 63, 9, 683, 1521, 29, 269, 25, 61, 923, 127, 221, 365, 1103, 22, 1199, 437, 175, 2809, 68, 3025, 182, 557, 1623, 157

Offset: 1

Michel Lagneau, Oct 02 2012

nonn

Numbers k such that A014442(n) = A014442(n+k), otherwise 0.
A014442(n) is the largest prime factor of n^2 + 1.
a(n) = 0 when A014442(n) is the last possible largest prime, for instance a(1) = 0, a(7) = 0 whose corresponding largest primes are respectively 2 and 5. The general case for the numbers n such that a(n) = 0 is difficult.

			a(1) = 0 because A014442(1) = 2 is the unique largest prime of A014442(n);
a(2) = 1 because A014442(2) = 5 and A014442(2+1) = 5;
a(3) = 4 because A014442(3) = 5 and A014442(3+4) = 5;
a(4) = 9 because A014442(4) = 17 and A014442(4+17) = 17.
a(57) = 182 because A014442(57) = 13 and A014442(182+57) = 13.

Cf. A014442.

with(numtheory):T:=array(1..300): for n from 1 to 300 do:x:=factorset(n^2+1):n1:=nops(x): T[n] := x[n1]:od:for a from 1 to 60 do:p:=T[a]:ii:=0:for b from a to 10000 do: z:=factorset(b^2+1): n2:=nops(z):if z[n2]=p and ii=0 then b0:=b:ii:=1:else if z[n2]=p and ii=1 then b1:=b:printf(`%d, `,b1-b0):ii:=2:else fi:fi:od:if ii=1 then printf(`%d, `,0):else fi:od:

A217448 Least k > 0 such that 1 + n^2 and 1 + (n+k)^2 have the same smallest prime factor.

Original entry on oeis.org

2, 6, 2, 26, 2, 74, 2, 4, 2, 404, 2, 6, 2, 366, 2, 514, 2, 4, 2, 1564, 2, 6, 2, 1106, 2, 4010, 2, 4, 2, 34, 2, 6, 2, 10, 2, 2594, 2, 4, 2, 22334, 2, 6, 2, 16, 2, 58, 2, 4, 2, 64, 2, 6, 2, 29062, 2, 18710, 2, 4, 2, 10, 2, 6, 2, 42, 2, 17428, 2, 4, 2, 16, 2, 6

Offset: 1

Michel Lagneau, Oct 03 2012

Alternate title: Least k > 0 such that A089120(n) = A089120(n+k).
A089120(n): smallest prime factor of n^2 + 1.
Conjecture: a(n) exists for all n.

			a(10) = 404 because 10^2 + 1 = 101, (10+404)^2+1 = 101*1697 so A089120(10) = A089120(414) = 101;
a(170) = 404274 because 170^2 + 1 = 28901, (170+404274)^2+1 = 163574949137 = 28901* 5659837 so A089120(170) = A089120(40444) = 28901.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A089120, A217393, A014442.

with(numtheory):T:=array(1..100): for n from 1 to 100 do:x:=factorset(n^2+1):n1:=nops(x): T[n] := x[1]:od:for a from 1 to 80 do:p:=T[a]:ii:=0:for k from 1 to 50000 while(ii=0) do: z:=factorset((a+k)^2+1): n2:=nops(z):if z[1]=p then printf(`%d, `,k):ii:=1:else fi:od:od:

sspf[n_]:=Module[{c=FactorInteger[1+n^2][[1,1]],k=1},While[ FactorInteger[ 1+ (n+k)^2][[1,1]]!=c,k++];k]; Array[sspf,80] (* Harvey P. Dale, Oct 12 2012 *)

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A217448(n) = { my(spf=A020639(1+(n^2)), x); for(k=1,oo,x=1+((n+k)^2); if(!(x%spf) && A020639(x)==spf,return(k))); }; \\ Antti Karttunen, May 24 2021

A240554 Square array of the greatest prime factor of n^k + 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 2, 5, 2, 1, 5, 5, 3, 2, 1, 3, 17, 7, 17, 2, 1, 7, 13, 13, 41, 11, 2, 1, 2, 37, 7, 257, 61, 13, 2, 1, 3, 5, 31, 313, 41, 73, 43, 2, 1, 5, 13, 43, 1297, 521, 241, 547, 257, 2, 1, 11, 41, 19, 1201, 101, 601, 113, 193, 19, 2, 1, 3, 101, 73, 241

Offset: 1

T. D. Noe, Apr 07 2014

T. D. Noe, Rows n = 1..60 of triangle, flattened

Cf. A003992 (n^k), A014442 (k=2), A081256 (k=3), A096172 (k=4).

Cf. A240548-A240553 (k=5 to 10).

Table[FactorInteger[(n-k)^k + 1][[-1,1]], {n, 12}, {k, n}]

A247339 a(n) is the least number k such that the greatest prime divisor of k^2+1 is the smallest prime divisor of n^2+1.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 14, 1, 16, 1, 2, 1, 20, 1, 2, 1, 24, 1, 26, 1, 2, 1, 4, 1, 2, 1, 5, 1, 36, 1, 2, 1, 40, 1, 2, 1, 5, 1, 12, 1, 2, 1, 9, 1, 2, 1, 54, 1, 56, 1, 2, 1, 5, 1, 2, 1, 4, 1, 66, 1, 2, 1, 5, 1, 2, 1, 74, 1, 23, 1, 2, 1, 6, 1, 2

Offset: 1

Michel Lagneau, Sep 14 2014

nonn

a(n)=n if n^2+1 is prime and a(n)=1 if n is odd.

Conjecture: for all integer n, there exists at least an integer m <= n such that the smallest prime factor of n^2+1 is also the greatest prime factor of m^2+1. - Michel Lagneau, Sep 27 2015

			a(34)=5 because the greatest prime divisor of 5^2+1 = 2*13 is the smallest prime divisor of 34^2+1 =13*89.

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

Cf. A002522, A089120, A014442.

with(numtheory):nn:=2000:T:=array(1..nn):U:=array(1..nn):
  for i from 1 to nn do:
    x:=factorset(i^2+1):T[i]:=x[1]:U[i]:=i:
  od:
    for n from 1 to 100 do:
     ii:=0:
      for k from 1 to 50000 while(ii=0) do:
       y:=factorset(k^2+1):n0:=nops(y):q:=y[n0]:
        if q=T[n]
         then
         ii:=1: printf(`%d, `,k):
         else
        fi:
     od:
   od:

Table[k = 1; While[FactorInteger[k^2 + 1][[-1, 1]] != FactorInteger[n^2 + 1][[1, 1]], k++]; k, {n, 82}] (* Michael De Vlieger, Sep 27 2015 *)

a(n) = {f = factor(n^2+1)[1,1]; k = 1; while (! ((g=factor(k^2+1)) && (g[#g~,1] == f)), k++); k;} \\ Michel Marcus, Sep 14 2014

A248746 a(n) is the index k of the greatest prime divisor A002313(k) of n^2 + 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 3, 7, 13, 9, 5, 4, 22, 15, 26, 5, 3, 20, 39, 4, 12, 8, 51, 31, 60, 10, 18, 41, 8, 6, 7, 14, 11, 54, 105, 16, 4, 65, 121, 5, 35, 6, 17, 83, 10, 4, 45, 97, 9, 106, 48, 29, 209, 11, 221, 3, 59, 133, 28, 138, 66, 38, 25, 155, 294, 43, 6, 174, 5

Offset: 1

Michel Lagneau, Oct 13 2014

nonn

a(n) is the number k such that A002313(k) = A014442(n).

			a(5)=3 because A002313(3)=13 and 5^2+1 = 2*13 with A002313(3)= A014442(5).

Michel Lagneau, Table of n, a(n) for n = 1..5000

Cf. A014442 (greatest prime divisor of n^2+1), A002313 (primes congruent to 1 or 2 modulo 4).

Cf. also A002522.

with(numtheory):T:=array(1..50000):T[1]:=2:kk:=1:nn:=10^5:
for i from 1 to nn do:
  p:=4*i+1:
  if type(p,prime)=true
  then
    kk:=kk+1:T[kk]:=p:
    else
    fi:
   od:
     for k from 1 to 5000 do:ii:=0:
      y:=factorset(k^2+1):n2:=nops(y):t:=y[n2]:
        for l from 1 to kk while(ii=0)do :
        if t=T[l]
         then
         printf(`%d, `,l):
         else
        fi:
     od:
    od:

A250069 a(n) = n^2 mod gpf(n^2 + 1) where gpf(k) is the greatest prime dividing k.

Original entry on oeis.org

1, 4, 4, 16, 12, 36, 4, 12, 40, 100, 60, 28, 16, 196, 112, 256, 28, 12, 180, 400, 16, 96, 52, 576, 312, 676, 72, 156, 420, 52, 36, 40, 108, 88, 612, 1296, 136, 16, 760, 1600, 28, 352, 36, 148, 1012, 72, 16, 460, 1200, 60, 1300, 540, 280, 2916, 88, 3136, 12

Offset: 1

Michel Lagneau, Nov 11 2014

For n > 1, a(n) == 0 (mod 4).

			a(5)=12 because 5^2 mod A014442(5) = 25 mod 13 = 12.

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

Cf. A006530, A014442.

with(numtheory):
for n from 1 to 500 do:
   p:=n^2+1:x:=factorset(p):n0:=nops(x):r:=irem(n^2,x[n0]):
   printf(`%d, `, r):
  od:

Table[Mod[n^2,FactorInteger[n^2+1,FactorComplete->True][[-1,1]]],{n,100}]

a(n) = lift(Mod(n, vecmax(factor(n^2+1)[,1]))^2); \\ Michel Marcus, Sep 13 2017

a(n) = n^2 mod A014442(n) where A014442(n) is the greatest prime factor of n^2 + 1.

Edited: exchanged name with an old comment. Old name as an alternative formula. Keyword easy added. - Wolfdieter Lang, Nov 29 2014

Redundancy in Name and in Formula section removed (at the suggestion of Michel Marcus) by Jon E. Schoenfield, Sep 13 2017

A252096 Largest prime divisor of n^2+1 - smallest prime divisor of n^2+1.

Original entry on oeis.org

0, 0, 3, 0, 11, 0, 3, 8, 39, 0, 59, 24, 15, 0, 111, 0, 27, 8, 179, 0, 15, 92, 51, 0, 311, 0, 71, 152, 419, 36, 35, 36, 107, 76, 611, 0, 135, 12, 759, 0, 27, 348, 35, 136, 1011, 44, 15, 456, 1199, 20, 1299, 536, 279, 0, 87, 0, 11, 668, 1739, 264, 1859, 764, 395

Offset: 1

Michel Lagneau, Dec 14 2014

			a(5)= 11 because 5^2+1 = 2*13 and 13-2 = 11.

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

Cf. A002522 (n^2+1), A005574 (n^2+1 is prime).

Cf. A014442 (largest prime factor of n2+1), A089120 (smallest prime factor).

with(numtheory):
a:= n-> (f-> max(f[])-min(f[]))(factorset(n^2+1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 07 2015

f[n_]:=Transpose[FactorInteger[n^2+1]][[1]]; Table[Last[f[n]-First[f[n]]], {n, 200}]

a(n) = {my(f = factor(n^2+1)); f[#f~,1] - f[1,1];} \\ Michel Marcus, Dec 15 2014

a(n) = A014442(n) - A089120(n).

a(A005574(n)) = 0. - Michel Marcus, Dec 15 2014

A252890 Number of times the greatest prime factor of n^2 + 1 is a factor in all numbers <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 6, 1, 1, 4, 1, 3, 1, 1, 2, 7, 1, 1, 2, 1, 1, 1, 1, 2, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 6, 1, 3, 4, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1

Offset: 1

Michel Lagneau, Dec 24 2014

nonn

The greatest prime factor is counted with multiplicity (see the example).

a(n)=1 iff n^2 + 1 is prime.

			a(7)=4 because 7^2 + 1 = 50 and 5 is 4 times a factor:
2^2+1 = 5;
3^2+1 = 10 = 2*5;
7^2+1 = 50 = 2*5*5 (two times).

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

Cf. A089120, A014442, A078897.

with(numtheory): with(padic,ordp):
f:= proc(n) local p ,q, n0;
  q:=factorset(n^2+1);n0:=nops(q);p:= q[n0];
  add(ordp(k^2+1, p), k=1..n);
end proc:
seq(f(n), n=1.. 100);
# Using code from Robert Israel adapted for this sequence. See A078897.

A321069 Greatest prime factor of n^3+2.

Original entry on oeis.org

3, 5, 29, 11, 127, 109, 23, 257, 43, 167, 43, 173, 733, 1373, 307, 683, 983, 2917, 2287, 4001, 157, 71, 283, 223, 5209, 47, 127, 3659, 24391, 587, 9931, 113, 433, 6551, 809, 569, 307, 27437, 433, 10667, 439, 239, 1559, 223, 91127, 16223, 4153, 457, 39217, 62501

Offset: 1

Charles R Greathouse IV, Oct 27 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, The largest prime factor of x^3+2, Proceedings of the London Mathematical Society, 82:3 (2000), pp. 554-596.
Christopher Hooley, On the greatest prime factor of a cubic polynomial, Journal für die reine und angewandte Mathematik, 303 (1978), pp. 21-50.
A. J. Irving, The largest prime factor of x^3+2, arXiv:1412.0024 [math.NT], 2014.

Greatest prime factors of polynomials: A006530 (n), A076565 (2n+1), A076566 (3n+3), A076567 (4n+6), A164314 (n^2-2), A076605 (n^2-1), A014442 (n^2+1), A069902 (n^2+n), A074399 (n^2+n), A199423 (2n^2+n), A089619 (2n^2+2n+1), A037464 (4n^2-1), A253254 (9n^2-7n), A093074 (n^3-n), A081257 (n^3-1), A081256 (n^3+1), A321069(n^3+2), A281793 (n^3+n^2+n+1), A281793 (n^4-1), A096172 (n^4+1), A190136 (n^4 + 6n^3 + 11n^2 + 6n), A140538 (2n^4+1), A240548 (n^5+1), A281794 (n^5+n^3+n^2+1), A240549 (n^6+1), A240550 (n^7+1), A240551 (n^8+1), A240552 (n^9+1), A240553 (n^10+1).

[Maximum(PrimeDivisors(n^3 + 2)): n in [1..60]]; // Vincenzo Librandi, Oct 27 2018

Table[FactorInteger[n^3 + 2] [[-1, 1]], {n, 80}] (* Vincenzo Librandi, Oct 27 2018 *)

a(n) = vecmax(factor(n^3+2)[,1]); \\ Michel Marcus, Oct 27 2018

A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 18, 34, 55, 57, 89, 123, 144, 233, 239, 322, 377, 411, 500, 568, 610, 746, 788, 843, 987, 1487, 1542, 1568, 1636, 2207, 2584, 2707, 3173, 3639, 3793, 3804, 3817, 4050, 4181, 4217, 4594, 4662, 5270, 5778, 6107, 6613, 8595, 8972, 10341, 10569

Offset: 1

Michel Lagneau, Mar 11 2022

nonn

A281618 is a subsequence.
The corresponding greatest prime Fibonacci factors of the sequence are 2, 5, 5, 13, 5, 13, 13, 89, 89, 13, 233, 89, 233, ...
The Fibonacci numbers of the sequence are 1, 2, 3, 5, 8, 34, 55, 89, 144, 233, 377, 610, 987, 2584, 4181, 10946, 17711, ... (subsequence of A000045).
The Lucas numbers of the sequence are 1, 2, 3, 7, 18, 123, 322, 843, 2207, 5778, 39603, 103682, ... (subsequence of A000032).
The prime numbers of the sequence are 2, 3, 5, 7, 89, 233, 239, 1487, 2207, 2707, 3793, 4217, 11789, 11981, 13763, ... including the prime Fibonacci numbers 2, 3, 5, 89, 233, 1066340417491710595814572169, ... (subsequence of A005478).

			18 is in the sequence because 18^2 + 1 = 5^2*13 and 13 is a Fibonacci number.

Cf. A000032, A000045, A002522, A005478, A014442, A248516, A338762, A339461.

q:= n-> (t-> ormap(issqr, [t+4, t-4]))(5*max(numtheory[factorset](n^2+1))^2):
select(q, [$1..12000])[];  # Alois P. Heinz, Mar 11 2022

With[{f = Fibonacci[Range[21]], m = f[[-1]]}, Select[Range[m], MemberQ[f, FactorInteger[#^2 + 1][[-1, 1]]] &]] (* Amiram Eldar, Mar 11 2022 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(m) = isfib(vecmax(factor(m^2+1)[,1])); \\ Michel Marcus, Mar 11 2022

cp's OEIS Frontend

A217393 Smallest k > 0 such that 1 + n^2 and 1 + (n+k)^2 have the same largest factor, or 0 if no such k exists.

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A217448 Least k > 0 such that 1 + n^2 and 1 + (n+k)^2 have the same smallest prime factor.

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A240554 Square array of the greatest prime factor of n^k + 1, read by antidiagonals.

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A247339 a(n) is the least number k such that the greatest prime divisor of k^2+1 is the smallest prime divisor of n^2+1.

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A248746 a(n) is the index k of the greatest prime divisor A002313(k) of n^2 + 1.

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A250069 a(n) = n^2 mod gpf(n^2 + 1) where gpf(k) is the greatest prime dividing k.

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A252096 Largest prime divisor of n^2+1 - smallest prime divisor of n^2+1.

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A252890 Number of times the greatest prime factor of n^2 + 1 is a factor in all numbers <= n.

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