A219108 -id:A219108

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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

A356872

a(n) = k is the smallest number such that 3*k+1 contains n distinct prime factors.

+10
2

1, 3, 23, 303, 4363, 56723, 1077743, 33410043, 718854803, 22284498903, 824526459423, 35454637755203, 1588862487308763, 68321086954276823, 4167586304210886223, 213640038906023626563, 13032042373267441220363, 873146839008918561764343, 63739719247651055008797063

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

OFFSET

1,2

LINKS

David A. Corneth, Table of n, a(n) for n = 1..349

FORMULA

From Michael S. Branicky, Sep 02 2022: (Start)

a(n) >= ceiling((A002110(n)-1)/3).

a(n) <= (c*A002110(n+1)/3-1)/3 for n > 1, and c = 1 or 2 chosen so the expression is an integer, with equality holding for c = 1 for n = 2, 3, 6, 7, ... . (End)

PROG

(Python)

from sympy import factorint, isprime

from itertools import count, islice

def f(n): return 1 if isprime(n) else len(factorint(n))

def agen():

n = 1

for k in count(0):

v = f(3*k+1)

while v >= n: yield k; n += 1

print(list(islice(agen(), 7))) # Michael S. Branicky, Sep 02 2022

CROSSREFS

Cf. A002110, A180278, A219108.

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, Sep 02 2022

EXTENSIONS

a(8) from Michael S. Branicky, Sep 02 2022

a(9)-a(19) from Jon E. Schoenfield, Sep 02 2022

STATUS

approved

A356873

a(n) is the smallest number k such that 2^k+1 has at least n distinct prime factors.

+10
0

0, 5, 14, 18, 30, 42, 78, 78, 78, 90, 150, 150, 210, 210, 234, 234, 270, 390, 390, 390, 390, 450, 510, 630, 630, 630, 810, 810, 810, 966, 966, 1170, 1170, 1170, 1170, 1170, 1170, 1170

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

OFFSET

1,2

COMMENTS

From Jon E. Schoenfield, Sep 04 2022: (Start)

a(39) <= a(40) <= a(41) <= 1530.

a(42) <= a(43) <= a(44) <= 1890.

a(45) <= a(46) <= 2070.

a(47) <= a(48) <= ... <= a(54) = 2730. (End)

LINKS

Table of n, a(n) for n=1..38.

MATHEMATICA

a[n_] := Block[{k=0}, While[ Length@ FactorInteger[2^k + 1] < n, k++]; k]; Array[a, 12] (* Giovanni Resta, Oct 13 2022 *)

PROG

(Python)

from sympy import factorint, isprime

from itertools import count, islice

def f(n): return 1 if isprime(n) else len(factorint(n))

def agen():

n = 1

for k in count(0):

v = f(2**k+1)

while v >= n: yield k; n += 1

print(list(islice(agen(), 10))) # Michael S. Branicky, Sep 02 2022

(PARI) a(n) = my(k=1); while (omega(2^k+1) < n, k++); k; \\ Michel Marcus, Sep 05 2022

CROSSREFS

Cf. A046799, A071852.

Cf. A180278, A219108, A356872.

KEYWORD

nonn,hard,more

AUTHOR

Alex Ratushnyak, Sep 02 2022

EXTENSIONS

a(11)-a(38) from Michael S. Branicky, Sep 02 2022 using A071852

STATUS

approved

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