A219017 -id:A219017 - cp's OEIS frontend

A180278 Smallest nonnegative integer k such that k^2 + 1 has exactly n distinct prime factors.

0, 1, 3, 13, 47, 447, 2163, 24263, 241727, 2923783, 16485763, 169053487, 4535472963, 36316463227, 879728844873, 4476534430363, 119919330795347, 1374445897718223, 106298577886531087

Offset: 0

Michel Lagneau, Jan 17 2011

			a(2) = 3 because the 2 distinct prime factors of 3^2 + 1 are {2, 5};
a(10) = 16485763 because the 10 distinct prime factors of 16485763^2 + 1 are {2, 5, 13, 17, 29, 37, 41, 73, 149, 257}.

Cf. A001221, A002144, A002522, A128428, A164511, A185952, A219017, A219108, A257366.

a[n_] := a[n] = Module[{k = 1}, If[n == 0, Return[0]]; Monitor[While[PrimeNu[k^2 + 1] != n, k++]; k, {n, k}]]; Table[a[n], {n, 0, 8}] (* Robert P. P. McKone, Sep 13 2023 *)

a(n)=for(k=0, oo, if(omega(k^2+1) == n, return(k))) \\ Andrew Howroyd, Sep 12 2023

from itertools import count
from sympy import factorint
def A180278(n):
    return next(k for k in count() if len(factorint(k**2+1)) == n) # Pontus von Brömssen, Sep 12 2023

a(n) >= sqrt(A185952(n)-1). - Charles R Greathouse IV, Feb 17 2015

a(n) <= A164511(n). - Daniel Suteu, Feb 20 2023

a(9), a(10) and example corrected; a(11) added by Donovan Johnson, Aug 27 2012
a(12) from Giovanni Resta, May 10 2017
a(13)-a(17) from Daniel Suteu, Feb 20 2023
Name clarified and incorrect programs removed by Pontus von Brömssen, Sep 12 2023
a(18) from Max Alekseyev, Feb 24 2024

A082863 Number of distinct prime factors of n^2-1.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 2, 4, 3, 2, 4, 3, 3, 4, 3, 4, 3, 4, 2, 3, 3, 3, 4, 4, 3, 4, 2, 3, 2, 4, 3, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 3, 3, 3, 3, 3

Offset: 2

Jon Perry, May 24 2003

nonn

This is a very slowly growing sequence - by n=100000 the maximum value is 8.

If n is in A014574 then a(n) = 2. - Robert Israel, Aug 05 2014

			a(11)=3 because (11-1)*(11+1)=10*12=2^3*3*5, which has 3 distinct prime factors, namely 2,3 and 5.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A001221, A014574, A219017 (greedy inverse).

A082863 := proc(n)
        A001221(n^2-1) ;
end proc: # R. J. Mathar, Aug 05 2014
# alternative:
A082863:= n -> nops(numtheory:-factorset(n^2-1)):
seq(A082863(n), n=1..100); # Robert Israel, Aug 05 2014

Table[PrimeNu[n^2-1],{n,2,100}] (* Harvey P. Dale, Jul 05 2011 *)

for (n=2,100,print1(omega((n-1)*(n+1))","))

a(n) = A001221((n-1)*(n+1)).

a(n) = A001221(n-1) + A001221(n+1) + ((-1)^n - 1)/2. - Robert Israel, Aug 05 2014

A365326 a(n) is the smallest positive number k such that k^2 - 1 and k^2 + 1 each have exactly n distinct prime divisors.

Original entry on oeis.org

2, 5, 13, 83, 463, 4217, 169333, 2273237, 23239523, 512974197, 5572561567

Offset: 1

Juri-Stepan Gerasimov, Sep 01 2023

Cf. A001221, A180278, A219017.

Cf. A088075 (with k instead of k^2).

isok(k, n) = (omega(k^2-1)==n) && (omega(k^2+1)==n);
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Sep 03 2023

a(n) >= max(A219017(n), A180278(n)). - Daniel Suteu, Sep 03 2023

a(9)-a(11) from Amiram Eldar, Sep 03 2023

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A180278 Smallest nonnegative integer k such that k^2 + 1 has exactly n distinct prime factors.

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A082863 Number of distinct prime factors of n^2-1.

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A365326 a(n) is the smallest positive number k such that k^2 - 1 and k^2 + 1 each have exactly n distinct prime divisors.

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