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A367187 Numbers which are the sum of a prime number and a Fibonacci number of index >1 in at least two ways.

4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 68, 69, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 91, 92, 94, 96, 98, 100

Offset: 1

Yuda Chen, Nov 08 2023

nonn

			4 is a term since 4 = 1+3 = 2+2.
5 is a term since 5 = 2+3 = 3+2.
57 is a term since 57 = 2+55 = 23+34.

Cf. A000040, A000045, A367186.

Subsequence of A132147.

isfib(n) = if (n>0, my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8));
isok(k) = sum(i=1, primepi(k), isfib(k-prime(i))) > 1; \\ Michel Marcus, Nov 09 2023

A367186 Numbers that can be written as 2^k + prime in more than one way.

Original entry on oeis.org

4, 6, 7, 9, 11, 13, 15, 18, 19, 21, 23, 25, 27, 31, 33, 35, 37, 39, 43, 45, 47, 49, 51, 55, 57, 61, 63, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 95, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 121, 123, 125, 129, 131, 133, 135, 139, 141, 143, 145, 147, 151, 153, 155

Offset: 1

Yuda Chen, Nov 08 2023

nonn

Numbers m such that A109925(m) > 1.

			4 is a term since 4 = 2^0 + 3 = 2^1 + 2 which is 2 ways.
6 is a term since 6 = 2^0 + 5 = 2^2 + 2.

Subsequence of A118955.

Cf. A000040, A000079, A109925.

isok(m) = sum(k=0, logint(m,2), isprime(m-2^k)) > 1; \\ Michel Marcus, Nov 10 2023

from itertools import count, islice
from sympy import isprime
def A367186_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = 0
        for i in range(n.bit_length()-1,-1,-1):
            if isprime(n-(1<1:
                yield n
                break
A367186_list = list(islice(A367186_gen(),30)) # Chai Wah Wu, Nov 29 2023

A352903 a(n) is the minimum number of steps required to construct a segment of length sqrt(n) in compass-and-straightedge construction.

Original entry on oeis.org

1, 5, 3, 2, 6, 5, 5, 5, 3, 5, 5, 4, 5, 5, 5, 3, 6, 5, 6, 6, 5, 6, 6, 5, 4, 6, 5, 5, 6, 6, 6, 6, 5, 6, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 4, 6, 6, 5, 7, 6, 6, 6, 6, 6, 7, 5, 6, 7, 6, 4, 6, 6, 6, 7, 6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 6, 5, 7, 7, 5, 7, 7, 7

Offset: 1

Yuda Chen, Apr 07 2022

Compass-and-straightedge construction allows the use of only a straightedge (without scale) and a collapsing compass. A "step" consists of constructing a line or constructing a circle with a point as its center. Constructing an intersection uses 0 steps.
Given two points in the plane, separated by a unit distance, a segment of length sqrt(n) cannot be constructed in fewer than a(n) steps.
Proving that "a(2021) is not more than 10" was the 2021 Chinese Mathematical Olympiad's Problem 5. According to Y. Ai et al., it is known that a(2021) is not more than 8 and not less than 7 because the maximum k such that a(k)=6 is 1024.
The sequence greatest k such that a(k) = n begins at 1, 4, 16, 64, 256, 1024, 170569, ... - Jinyuan Wang, Jul 18 2023

Y. Ai et al., Analysis of Questions and answers about the National Senior High School's Mathematical Olympiad in 2021 (Final) (in Chinese).
L. Jin, The generalized version of 2021 CMO Problem 5 (in Chinese).
Jinyuan Wang, Illustration of initial terms
Jinyuan Wang, PARI program
Index to sequences related to Olympiads.

a(6) corrected by and more terms from Jinyuan Wang, Jul 17 2023

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User: Yuda Chen

A367187 Numbers which are the sum of a prime number and a Fibonacci number of index >1 in at least two ways.

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A367186 Numbers that can be written as 2^k + prime in more than one way.

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Author

Keywords

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Examples

Crossrefs

Programs

PARI

Python

A352903 a(n) is the minimum number of steps required to construct a segment of length sqrt(n) in compass-and-straightedge construction.

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