A243780
Primes p for which p^i + 4 is prime for i = 1, 3 and 5.
Original entry on oeis.org
7, 5503, 8779, 14629, 15877, 21013, 23599, 51199, 61483, 70237, 78163, 79333, 80149, 96667, 113089, 113359, 133153, 140053, 149377, 150889, 184039, 198967, 228199, 251287, 255637, 295843, 301123, 303613, 356929, 382843, 385393, 393709, 420037, 457363, 458119
Offset: 1
p=7 is in this sequence as p + 4, p^3 + 4, p^5 + 4 (11, 347, 16811) are all prime.
p=5503 is in this sequence as p + 4 = 5507 (prime), p^3 + 4 = 166647398531 (prime) and p^5 + 4 = 5046584669419727747 (prime).
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Select[Range[500000], PrimeQ[#] && AllTrue[#^{1, 3, 5} + 4, PrimeQ] &] (* Amiram Eldar, Apr 04 2020 *)
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s=[]; forprime(p=2, 200000, if(isprime(p+4) && isprime(p^3+4) && isprime(p^5+4), s=concat(s, p))); s \\ Colin Barker, Jun 11 2014
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import sympy.ntheory as snt
n=2
while n>1:
n1=n+4
n2=((n**3)+4)
n3=((n**5)+4)
##Check if n1 , n2 and n3 are also primes.
if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True:
print(n,n1,n2,n3)
n=snt.nextprime(n)
A243734
Primes p for which p + 4, p^2 + 4 and p^3 + 4 are primes.
Original entry on oeis.org
3, 7, 103, 277, 487, 967, 4783, 5503, 5923, 8233, 21013, 26317, 27943, 41593, 55213, 78307, 78853, 86197, 89653, 94723, 99013, 123727, 148153, 157177, 166627, 172867, 177883, 179107, 185893, 192883, 194713, 203767, 204517, 223633, 225217, 227593, 236893
Offset: 1
p = 3 is in this sequence because p + 4 = 7, p^2 + 4 = 13 and p^3 + 4 = 31 are all primes.
p : p+4, p^2+4, p^3+4
7 : 11, 53, 347
103: 107, 10613, 1092731
277: 281, 76733, 21253937
487: 491, 237173, 115501307
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s=[]; forprime(p=2, 200000, if(isprime(p+4) && isprime(p^2+4) && isprime(p^3+4), s=concat(s, p))); s \\ Colin Barker, Jun 11 2014
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import sympy.ntheory as snt
n=2
while n > 1 and n < 10**6:
n1=n+4
n2=((n**2)+4)
n3=((n**3)+4)
##Check if n1, n2 and n3 are also primes.
if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True:
print(n, end=', ')
n=snt.nextprime(n)
A243861
Primes p for which p^i - 4 is prime for i = 1, 3, 5 and 7.
Original entry on oeis.org
971, 12641, 205607, 228341, 276557, 412343, 1012217, 1101323, 1902881, 2171021, 2477411, 2692121, 4116377, 4311677, 6060953, 6182993, 6388913, 6444863, 8341121, 8551451, 9507527, 10523141, 10997411, 11444093, 14101361, 14656307, 14813147, 15435587, 17337521
Offset: 1
Prime p=971 is in this sequence because p-4 = 967 (prime), p^3-4 = 915498607 (prime), p^5-4 = 863169625893847 (prime), and p^7-4 = 813831713247384370687 (prime).
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import sympy.ntheory as snt
n=2
while n>1:
n1=n-4
n2=((n**3)-4)
n3=((n**5)-4)
n4=((n**7)-4)
##Check if n1 , n2, n3 and n4 are also primes.
if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True and snt.isprime(n4)== True:
print(n, n1, n2, n3, n4)
n=snt.nextprime(n)
A243859
Primes p for which p^i + 4 is prime for i = 1, 3, 5 and 7.
Original entry on oeis.org
7, 133153, 184039, 356929, 469363, 982843, 2154487, 2552713, 2686573, 3378103, 3847867, 4270069, 4341373, 4564363, 4584847, 4964899, 5366017, 5600989, 6185173, 6592609, 6595597, 6629683, 6768409, 8232277, 9028429, 9964177, 10009339, 12107089, 13266553, 13600189
Offset: 1
p=7 is in this sequence as p + 4 = 11 (prime), p^3 + 4 = 347 (prime), p^5 + 4 = 16811 (prime), and p^7 + 4 = 823547 (prime).
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p := 2:
for n from 1 do
if isprime(p+4) and isprime(p^3+4) and isprime(p^5+4) and isprime(p^7+4) then
print(p) ;
end if;
p := nextprime(p) ;
end do: # R. J. Mathar, Jun 13 2014
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Select[Prime[Range[900000]],AllTrue[#^{1,3,5,7}+4,PrimeQ]&] (* Harvey P. Dale, Apr 12 2022 *)
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import sympy.ntheory as snt
n=2
while n>1:
n1=n+4
n2=((n**3)+4)
n3=((n**5)+4)
n4=((n**7)+4)
##Check if n1 , n2, n3 and n4 are also primes.
if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True and snt.isprime(n4)== True:
print(n, n1, n2, n3, n4)
n=snt.nextprime(n)
Showing 1-4 of 4 results.
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