A219108 -id:A219108 - 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

A356872 a(n) = k is the smallest number such that 3*k+1 contains n distinct prime factors.

Original entry on oeis.org

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

Offset: 1

Alex Ratushnyak, Sep 02 2022

nonn

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

Cf. A002110, A180278, A219108.

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

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)

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

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

A356873 a(n) is the smallest number k such that 2^k+1 has at least n distinct prime factors.

Original entry on oeis.org

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

Offset: 1

Alex Ratushnyak, Sep 02 2022

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)

Cf. A046799, A071852.

Cf. A180278, A219108, A356872.

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

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

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

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

cp's OEIS Frontend

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

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A356872 a(n) = k is the smallest number such that 3*k+1 contains n distinct prime factors.

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A356873 a(n) is the smallest number k such that 2^k+1 has at least n distinct prime factors.

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Mathematica

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Python

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