A356110
Numbers k such that k^2 + {1,3,7,13,31} are prime.
Original entry on oeis.org
4, 10, 14290, 43054, 109456, 315410, 352600, 483494, 566296, 685114, 927070, 1106116, 1248796, 1501174, 1997986, 2399204, 2501404, 2553100, 2726840, 2874680, 3291760, 4129394, 4473766, 4794520, 4901144, 6350306, 7444070, 7753456, 7892504, 8009536, 8069540
Offset: 1
4^2 + {1,3,7,13,31} = {17,19,23,29,47} are all prime.
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q:= k-> andmap(j-> isprime(k^2+j), [1,3,7,13,31]):
select(q, [$0..1000000])[]; # Alois P. Heinz, Jul 27 2022
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Select[Range[10^6], AllTrue[#^2 + {1,3,7,13,31}, PrimeQ] &] (* Amiram Eldar, Jul 27 2022 *)
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from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,31})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 27 2022
A356175
Numbers k such that k^2 + {1,3,7,13,163} are prime.
Original entry on oeis.org
2, 4, 10, 14290, 64390, 74554, 83464, 93460, 132304, 238850, 262630, 277630, 300206, 352600, 376190, 404954, 415180, 610340, 806180, 984686, 1025650, 1047050, 1106116, 1382860, 2014624, 2440714, 2525870, 2538344, 2760026, 2826380, 3145600, 3508586, 3715156
Offset: 1
2 is a term since 2^2 + {1,3,7,13,163} = {5,7,11,17,167} are all primes.
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q:= k-> andmap(j-> isprime(k^2+j), [1, 3, 7, 13, 163]):
select(q, [$0..1000000])[]; # Alois P. Heinz, Jul 28 2022
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Select[Range[4*10^6], AllTrue[#^2 + {1, 3, 7, 13, 163}, PrimeQ] &] (* Amiram Eldar, Jul 28 2022 *)
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is(k)=my(v=[1,3,7,13,163],ok=1);for(i=1,#v,if(!isprime(k^2+v[i]),ok=0;break));ok
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from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,163})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 28 2022
A178639
Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.
Original entry on oeis.org
10, 12, 200, 268, 340, 418, 488, 530, 838, 840, 1102, 1720, 1830, 2240, 2410, 2768, 3148, 3202, 3318, 3322, 3958, 4162, 4610, 5080, 5672, 5700, 5722, 5870, 6178, 6302, 6480, 7490, 8130, 8262, 8888, 9132, 9602, 9618, 10638
Offset: 1
Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 31 2010
m=10 is in the sequence because 10^2 + 13 = 113 = prime(30), 10^2 + 13^2 = 269 = prime(57), 10^2 + 13^3 = 2297 = prime(342).
m=8888 is in the sequence because 8888^2 + 13 = 78996557 = prime(4614261), 8888^2 + 13^2 = 78996713 = prime(4614269), 8888^2 + 13^3 = 78998741 = prime(4614379).
m=6480 yields a prime 6480^2 + 13^k even for k=0.
m=7490 yields a prime 7490^2 + 13^k even for k=0 and k=4.
- B. Bunch: The Kingdom of Infinite Number: A Field Guide, W. H. Freeman, 2001.
- R. Courant, H. Robbins: What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1996.
- G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (5th edition), Oxford University Press, 1980.
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