A257366 -id:A257366

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A180278

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

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..18.

FORMULA

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

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

EXAMPLE

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}.

MATHEMATICA

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 *)

PROG

(Python)

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

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

CROSSREFS

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

KEYWORD

nonn,hard,more

AUTHOR

Michel Lagneau, Jan 17 2011

EXTENSIONS

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

STATUS

approved

A258929

a(n) is the unique even-valued residue modulo 5^n of a number m such that m^2+1 is divisible by 5^n.

+10
1

2, 18, 68, 182, 1068, 1068, 32318, 280182, 280182, 3626068, 23157318, 120813568, 1097376068, 1097376068, 11109655182, 49925501068, 355101282318, 355101282318, 15613890344818, 15613890344818, 365855836217682, 2273204469030182, 2273204469030182, 49956920289342682

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For any positive integer n, if a number of the form m^2+1 is divisible by 5^n, then m mod 5^n must take one of two values--one even, the other odd. This sequence gives the even residue. (The odd residues are in A259266.)

LINKS

Table of n, a(n) for n=1..24.

EXAMPLE

If m^2+1 is divisible by 5, then m mod 5 is either 2 or 3; the even value is 2, so a(1)=2.

If m^2+1 is divisible by 5^2, then m mod 5^2 is either 7 or 18; the even value is 18, so a(2)=18.

If m^2+1 is divisible by 5^3, then m mod 5^3 is either 57 or 68; the even value is 68, so a(3)=68.

CROSSREFS

Cf. A048898, A048899, A257366, A259266.

KEYWORD

nonn

AUTHOR

Jon E. Schoenfield, Jun 15 2015

EXTENSIONS

More terms and additional comments from Jon E. Schoenfield, Jun 23 2015

STATUS

approved

A259266

a(n) is the unique odd-valued residue modulo 5^n of a number m such that m^2+1 is divisible by 5^n.

+10
1

3, 7, 57, 443, 2057, 14557, 45807, 110443, 1672943, 6139557, 25670807, 123327057, 123327057, 5006139557, 19407922943, 102662389557, 407838170807, 3459595983307, 3459595983307, 79753541295807, 110981321985443, 110981321985443, 9647724486047943, 9647724486047943

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For any positive integer n, if a number of the form m^2+1 is divisible by 5^n, then m mod 5^n must take one of two values--one even, the other odd. This sequence gives the odd residue. (The even residues are in A258929.)

LINKS

Table of n, a(n) for n=1..24.

EXAMPLE

If m^2+1 is divisible by 5, then m mod 5 is either 2 or 3; the odd value is 3, so a(1)=3.

If m^2+1 is divisible by 5^2, then m mod 5^2 is either 7 or 18; the odd value is 7, so a(2)=7.

If m^2+1 is divisible by 5^3, then m mod 5^3 is either 57 or 68; the odd value is 57, so a(3)=57.

CROSSREFS

Cf. A048898, A048899, A257366, A258929.

KEYWORD

nonn

AUTHOR

Jon E. Schoenfield, Jun 23 2015

STATUS

approved

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