Brennan G. Benfield - cp's OEIS frontend

A375519 Number of positive integers with Pisano period equal to n.

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 10, 0, 1, 0, 2, 0, 8, 0, 4, 0, 1, 0, 9, 0, 1, 0, 11, 0, 8, 0, 2, 0, 3, 0, 55, 0, 6, 0, 2, 0, 6, 0, 11, 0, 3, 0, 49, 0, 1, 0, 8, 0, 8, 0, 2, 0, 13, 0, 133, 0, 1, 0, 6, 0, 20, 0, 46, 0, 1, 0, 49, 0, 3, 0, 27, 0, 81, 0, 4, 0, 10, 260, 0, 2, 0, 38

Offset: 1

Oliver Lippard and Brennan G. Benfield, Jul 29 2024

nonn

See also A375089, which is the main entry.

[This was the version of A375089 that was originally submitted. I think both belong in the OEIS. - N. J. A. Sloane, Aug 28 2024]

Cf. A001175, A375089.

from functools import lru_cache
from math import gcd, lcm
from sympy import factorint, divisors, fibonacci
def A375519(n):
    @lru_cache(maxsize=None)
    def A001175(n):
        if n == 1:
            return 1
        f = factorint(n).items()
        if len(f) > 1:
            return lcm(*(A001175(a**b) for a,b in f))
        else:
            k,x = 1, (1,1)
            while x != (0,1):
                k += 1
                x = (x[1], (x[0]+x[1]) % n)
            return k
    a, b = fibonacci(n+1), fibonacci(n)
    return sum(1 for d in divisors(gcd(a-1,b),generator=True) if A001175(d)==n) # Chai Wah Wu, Aug 28 2024

a(2n) = A375089(n). - Chai Wah Wu, Aug 28 2024

A375089 Number of positive integers with Pisano period equal to 2n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 3, 2, 1, 10, 1, 2, 8, 4, 1, 9, 1, 11, 8, 2, 3, 55, 6, 2, 6, 11, 3, 49, 1, 8, 8, 2, 13, 133, 1, 6, 20, 46, 1, 49, 3, 27, 81, 4, 1, 260, 2, 38, 20, 11, 1, 106, 21, 78, 20, 4, 7, 874, 1, 6, 81, 48, 29, 49, 3, 27, 42, 108, 1, 1319, 3, 14, 174, 23, 13, 101, 1, 444

Offset: 1

Oliver Lippard and Brennan G. Benfield, Jul 29 2024

nonn

a(n) = 0 for all odd values n > 3 since the Pisano period of m is always even except when m=1 or 2.
The Pisano period of m divides n if and only if F_m = 0 (mod n) and F_{m+1} = 1 (mod n), hence n | gcd(F_m,F_{m+1}-1).
Conjecture: If n is of the form 12*5^j (i.e. 2n is of the form 24*5^j which has the unique property that pi(2n)=2n), then a(n) > a(m) for all m < n.
Conjecture: Every natural number appears on this list.

			a(9) = 3 because the Pisano periods of 76, 38, and 19, but no others, are 2*9=18.
There are no numbers with Pisano period 2 or 4, so a(1) = a(2) = 0.

Chai Wah Wu, Table of n, a(n) for n = 1..314 (terms 1..239 from Oliver Lippard)
B. Benfield and O. Lippard, Fixed points of K-Fibonacci Pisano periods, arXiv:2404.08194 [math.NT], 2024.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica 16 (1969), 105-110.

Cf. A000045, A001175.

See A375519 for the "equal to n" version.

from functools import lru_cache
from math import gcd, lcm
from sympy import factorint, divisors, fibonacci
def A375089(n):
    @lru_cache(maxsize=None)
    def A001175(n):
        if n == 1:
            return 1
        f = factorint(n).items()
        if len(f) > 1:
            return lcm(*(A001175(a**b) for a,b in f))
        else:
            k,x = 1, (1,1)
            while x != (0,1):
                k += 1
                x = (x[1], (x[0]+x[1]) % n)
            return k
    a, b = fibonacci((m:=n<<1)+1), fibonacci(m)
    return sum(1 for d in divisors(gcd(a-1,b),generator=True) if A001175(d)==m) # Chai Wah Wu, Aug 28 2024

def a(n):
    num=0
    if n<3:
        return 0
    x=gcd(fibonacci(2*n),fibonacci(2*n+1)-1)
    for d in divisors(x):
        if BinaryRecurrenceSequence(1,1,0,1).period(d)==2*n:
            num+=1
    return num
for i in range(1,101):
    print(a(i))

A374486 Numbers k such that Taxicab(2,j,k) exists for large j.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 37, 39, 40, 42, 44, 48, 50, 51, 52, 53, 56, 59, 62, 66, 68, 70, 72, 74, 77, 79, 87, 91, 92, 96, 97, 103, 108, 112, 115, 117, 120, 121, 124, 130, 131, 138, 148, 149, 161, 164, 176, 184, 185, 194, 200

Offset: 1

Oliver Lippard and Brennan G. Benfield, Aug 04 2024

nonn

Here Taxicab(2,j,k) denotes the smallest number (if it exists) that is the sum of j perfect squares in exactly k ways. For sufficiently large N, Taxicab(2,j,k) either always exists for j > N or always does not exist for j > N.
Conjecture: Infinitely many positive integers are in this sequence, and infinitely many positive integers are not in this sequence.
Conjecture: This sequence grows exponentially. Computationally it appears to have asymptotic a(n) = 1.03691*exp(0.594473*n^(1/2)).

			For k = 3, Taxicab(2,j,3) does not exist for all j > 9, hence 3 is not a member of the sequence.

E. Grosswald. Representations of Integers as Sums of Squares. Springer New York, NY, 1985.

Oliver Lippard, Table of n, a(n) for n = 1..372
B. Benfield, O. Lippard, and A. Roy, End Behavior of Ramanujan's Taxicab Numbers, arXiv:2404.08190 [math.NT], 2024.

Cf. A025416, A080673, A295702, A295795

A375018 Numbers k such that repeated application of the Pisano period eventually gives 24.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 32, 34, 36, 37, 38, 39, 42, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 63, 64, 67, 68, 69, 72, 73, 74, 76, 78, 79, 81, 83, 84, 87, 91, 92, 94, 96, 97, 98

Offset: 1

Oliver Lippard and Brennan G. Benfield, Aug 04 2024

nonn

This sequence is infinite. A number n is a fixed point if the Pisano period of n is equal to n. The trajectory of k is the sequence of values the Pisano period takes on under repeated iteration, starting at k and leading to a fixed point; this sequence is the sequence of integers such that the trajectory leads to 24.

			a(1)=2 because 2 is the smallest number with Pisano period trajectory terminating at 24: pi(2)=3, pi(3)=8, pi(8)=12, pi(12)=24.

B. Benfield and O. Lippard, Fixed points of K-Fibonacci Pisano periods, arXiv:2404.08194 [math.NT], 2024.
J. Fulton and W. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arith., 2(16):105-110, 1969.
E. Trojovska, On periodic points of the order of appearance in the Fibonacci sequence, Mathematics, 2020.

Cf. A001175.

L=[]
for i in range(2,101):
    a=i
    y=BinaryRecurrenceSequence(1,1,0,1).period(Integer(i))
    while a!=y:
        a=y
        y=BinaryRecurrenceSequence(1,1,0,1).period(Integer(a))
    if a==24:
        L.append(i)
print(L)

A352083 Numbers k such that (3^k - k^3)/2 is prime.

Original entry on oeis.org

5, 13, 205, 409, 413, 545, 88649

Offset: 1

Brennan G. Benfield, Mar 03 2022

All terms must be odd. - Michael S. Branicky, Mar 03 2022

No further terms less than 25000. - Michael S. Branicky, Mar 04 2022

			5 is a term since (3^5 - 5^3)/2 = 59 is prime.

Select[Range[1, 600, 2], PrimeQ[(3^# - #^3)/2] &] (* Amiram Eldar, Mar 03 2022 *)

isok(k) = if (denominator(x=(3^k-k^3)/2) == 1, ispseudoprime(x)); \\ Michel Marcus, Mar 03 2022

from sympy import isprime
def afind(limit):
    for k in range(1, limit+1, 2):
        if isprime((3**k - k**3)//2):
            print(k, end=", ")
afind(1000) # Michael S. Branicky, Mar 03 2022

for n in srange(1,10000):
             if ((3^n-n^3)//2).is_prime():
                  print(n)

a(7) from Michael S. Branicky, Oct 03 2024

A346160 a(n) is the smallest K such that the power partition function P_n(k) is log-concave for all k > K.

Original entry on oeis.org

0, 25, 1041, 15655, 637854, 2507860, 35577568

Offset: 0

Brennan G. Benfield, Sep 28 2021

The power partition function P_n(k) is a restriction on the partition function. P_n(k) equals the number of ways a positive integer k can be written as the sum of perfect n powers. DeSalvo and Pak showed that the partition function (P_1(k)) is log-concave for all k > 24.

			For n=0, P_0(k)^2 >= P_0(k-1)*P_0(k+1) for all k > 0.
For n=1, P_1(k)^2 >= P_1(k-1)*P_1(k+1) for all k > 24.
For n=2, P_2(k)^2 >= P_2(k-1)*P_2(k+1) for all k > 1042.
For n=3, P_3(k)^2 >= P_3(k-1)*P_3(k+1) for all k > 15656.
No further terms are known.

Stephen DeSalvo and Igor Pak, Log-concavity of the partition function, The Ramanujan Journal, 38 (2015), 61-73.

Brennan Benfield and Arindam Roy, Log-concavity And The Multiplicative Properties of Restricted Partition Functions, arXiv:2404.03153 [math.NT], 2024.
Stephen DeSalvo and Igor Pak, Log-Concavity of the Partition Function, arXiv:1310.7982 [math.CO], 2013-2014.

Cf. A000041, A001156, A003108, A046042.

def power_partition_generating_series(s, n=20):
    R = ZZ['q']
    ans = R.one()
    m = 1
    while m**s < n:
        l = [0] * n
        l[0] = 1
        for i in range(1, (n + m**s - 1) // m**s):
            l[i*m**s] = 1
        ans = ans.mul_trunc(R(l), n)
        m += 1
    return ans
%time
seq = power_partition_generating_series(2, 15000).coefficients()
last_neg = None
for n in range(2, len(seq) - 1):
    d = seq[n]**2 / seq[n-1] / seq[n+1]
    if d < 1:
        last_neg = n
print(last_neg)
# Vincent Delecroix, Dec 28 2022

Data corrected by Brennan G. Benfield, Dec 28 2022

cp's OEIS Frontend

User: Brennan G. Benfield

A375519 Number of positive integers with Pisano period equal to n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

A375089 Number of positive integers with Pisano period equal to 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Sage

A374486 Numbers k such that Taxicab(2,j,k) exists for large j.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A375018 Numbers k such that repeated application of the Pisano period eventually gives 24.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

A352083 Numbers k such that (3^k - k^3)/2 is prime.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

PARI

Python

Sage

Extensions

A346160 a(n) is the smallest K such that the power partition function P_n(k) is log-concave for all k > K.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

SageMath

Extensions