A323605
Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).
Original entry on oeis.org
2, 3, 7, 43, 13, 3263443, 547, 29881, 5295435634831, 181, 2287, 73
Offset: 0
-
with(numtheory):
u:=1: P:=NULL: to 9 do P:=P,sort([op(divisors(u+1))])[2]: u:=u*(u+1) od:
P;
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f(n)=if(n<1, n>=0, f(n-1)+f(n-1)^2); \\ A007018
a(n)=divisors(f(n)+1)[2]; \\ Michel Marcus, Jan 20 2019
A375543
Sylvester primes. Yet another proof of the infinity of primes.
Original entry on oeis.org
2, 3, 7, 43, 13, 139, 547, 607, 1033, 181, 1987, 73, 2287, 29881, 13999, 17881, 31051, 52387, 67003, 74203, 128551, 352867, 635263, 74587, 1286773, 2271427, 27061, 164299, 20929, 1171, 298483, 1679143, 3229081, 3263443, 120823, 447841, 2408563, 333457, 30241, 4219
Offset: 1
The generation of the sequence starts:
n selected factors of S(i), i<n a(n)
[1] {} {2} -> 2,
[2] {2} {2, 3} -> 3,
[3] {2, 3} {2, 3, 7} -> 7,
[4] {2, 3, 7} {2, 3, 7, 43} -> 43,
[5] {2, 3, 7, 43} {2, 3, 7, 13, 43, 139} -> 13.
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fact := n -> NumberTheory:-PrimeFactors(n):
SylvesterPrimes := proc(len) local p, d, w, n;
p := 1; d := {}; w := {};
for n from 1 to len do
p := p*(p + 1);
d := fact(p) minus w;
w := w union {min(d)};
od end:
SylvesterPrimes(8);
isSylvesterPrime := proc(p) local s, M;
M := NULL: s := 2:
while not member(s, [M]) do
M := M, s;
s := (s^2 + s) mod p;
if s = 0 then return true fi;
od: false end:
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Module[{nmax = 20, a = {}, p = 1, f}, Do[p *= p + 1; f = 2; While[MemberQ[a,f] || !Divisible[p, f], f = NextPrime[f]]; AppendTo[a, f], nmax]; a] (* Paolo Xausa, Sep 03 2024 *)
-
from sympy import sieve
from itertools import count, islice
def smallest_new_primefactor(n, pf):
return next(pi for i in count(1) if (pi:=sieve[i]) not in pf and n%pi==0)
def agen(): # generator of terms
p, d, w, pf = 1, set(), set(), set()
while True:
p = p*(p + 1)
m = smallest_new_primefactor(p, w)
w |= {m}
yield m
print(list(islice(agen(), 20)))
# Michael S. Branicky, Sep 02 2024 after Peter Luschny
-
# Returns the first 24 terms in less than 60 seconds.
def SylvesterPrimes(len: int) -> list[int]:
M: list[int] = []
p = q = 1
for n in range(1, len + 1):
p = p * (p + 1)
pq = p // q
for s in Primes():
if pq % s == 0 and not s in M:
M.append(s)
q = q * s
print(n, s)
break
return M
SylvesterPrimes(24) # Peter Luschny, Sep 05 2024
A256147
First repeated number in Sylvester's sequence modulo prime(n).
Original entry on oeis.org
1, 1, 2, 1, 3, 1, 4, 2, 7, 3, 2, 6, 2, 1, 7, 7, 7, 17, 7, 3, 1, 43, 66, 2, 72, 51, 7, 50, 32, 3, 111, 85, 26, 1, 44, 31, 7, 7, 96, 157, 23, 1, 88, 3, 97, 7
Offset: 1
a(4) = 1, because the fourth prime is 7 and Sylvester's sequence modulo 7 is 2, 3, 0, 1, 1, 1, ...
a(5) = 3, because the fifth prime is 11 and Sylvester's sequence modulo 11 is 2, 3, 7, 10, 3, 7, 10, 3, 7, 10, ... (3 is the first number repeated).
- J. J. Sylvester, Postscript to Note on a Point in Vulgar Fractions. American Journal of Mathematics Vol. 3, No. 4 (Dec., 1880): 388 - 389.
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