A094479 -id:A094479 - cp's OEIS frontend

A045637 Primes of the form p^2 + 4, where p is prime.

13, 29, 53, 173, 293, 1373, 2213, 4493, 5333, 9413, 10613, 18773, 26573, 27893, 37253, 54293, 76733, 85853, 94253, 97973, 100493, 120413, 139133, 214373, 237173, 253013, 299213, 332933, 351653, 368453, 375773, 458333, 552053, 619373

Offset: 1

Felice Russo

These are the only primes that are the sum of two primes squared. 11 = 3^2 + 2 is the only prime of the form p^2 + 2 because all primes greater than 3 can be written as p=6n-1 or p=6n+1, which allows p^2+2 to be factored. - T. D. Noe, May 18 2007
Infinite under the Bunyakovsky conjecture. - Charles R Greathouse IV, Jul 04 2011
All terms > 29 are congruent to 53 mod 120. - Zak Seidov, Nov 06 2013

			29 belongs to the sequence because it equals 5^2 + 4.

T. D. Noe, Table of n, a(n) for n = 1..1000
Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.

The corresponding primes p are in A062324.
Subsequence of A005473 (and thus A185086).
Cf. A094473-A094479.

Select[Prime[ # ]^2+4&/@Range[140], PrimeQ]

forprime(p=2,1e4,if(isprime(t=p^2+4),print1(t","))) \\ Charles R Greathouse IV, Jul 04 2011

a(n) = A062324(n)^2 + 4. - Zak Seidov, Nov 06 2013

Edited by Dean Hickerson, Dec 10 2002

A158361 Primes p with property that Q = p^4 + 2^4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 37, 41, 59, 61, 71, 79, 97, 131, 139, 179, 223, 227, 229, 241, 283, 313, 317, 359, 367, 379, 383, 389, 439, 449, 461, 487, 503, 521, 569, 593, 617, 619, 631, 661, 683, 709, 733, 811, 821, 853, 911, 977, 1049, 1061, 1063, 1069, 1091, 1093, 1117

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2009

Q is always congruent to 1 (mod 4).
Q is divisible by 17 if p is congruent to 1, 4, 13, or 16 (mod 17).
It is conjectured that sequence a(n) is infinite.
Q is in A094479. - Zak Seidov, Jul 08 2020

			3 is in the sequence since for p=3: p^4+2^4 = 3^4+16 = 97 is prime.
29 is not in the sequence since 29^4+2^4 = 707297 = 73 x 9689 is not prime.

Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
Richard Guy, "Unsolved Problems in Number Theory"

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A062324, A094479, A157950, A157764.

[p: p in PrimesUpTo(2000) | IsPrime(p^4+16)]; // Vincenzo Librandi, Jun 18 2014

Select[Range[10^3], PrimeQ[#] && PrimeQ[#^4 + 16] &] (* Vincenzo Librandi, Jun 18 2014 *)
Select[Prime[Range[200]],PrimeQ[#^4+16]&] (* Harvey P. Dale, Jun 23 2014 *)

isA158361(n) = isprime(n) && isprime(n^4+16)

Corrected and edited by Michael B. Porter, Dec 17 2009

A193411 Primes which are sums of two or more distinct 4th powers of primes.

Original entry on oeis.org

97, 641, 2417, 14657, 17123, 17683, 43283, 46309, 83537, 112163, 126739, 129221, 129749, 130337, 145043, 145603, 173539, 176021, 176549, 214483, 216259, 229189, 242419, 243109, 244901, 257141, 279857, 280547, 294563, 295123, 297589, 310819, 325541, 365779

Offset: 1

Jonathan Vos Post, Jul 25 2011

nonn

Primes in A130833. Primes which are sums of exactly two distinct 4th powers of primes must be in A094479 primes of the form p^4 + 16 where p is also a prime.

The first term that arises in more than one way is 6625607 = 2^4+5^4+7^4+11^4+17^4+23^4+41^4+43^4 = 2^4+5^4+7^4+13^4+17^4+29^4+31^4+47^4. - Robert Israel, Apr 27 2020

			a(5) = 17123 = 3^4 + 7^4 + 11^4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000583, A030514, A094479, A192926.

N:= 5*10^5: # for all terms <= N
S1:= {}:
S2:= {}:
p:= 1:
R:= {}:
do
  p:= nextprime(p);
  if p^4 > N then break fi;
  s:= p^4;
  nS2:= select(`<=`,map(`+`,S1 union S2, s), N);
  S2:= S2 union nS2;
  S1:= S1 union {s};
  R:= R union select(isprime, nS2);
od:
sort(convert(R,list)); # Robert Israel, Apr 27 2020

nn = 9; Select[Sort[Table[Dot[IntegerDigits[i, 2, nn], Prime[Range[nn]]^4], {i, 2^nn-1}]], # < Prime[nn-1]^4 + Prime[nn]^4 && PrimeQ[#] &] (* T. D. Noe, Jul 27 2011 *)

list(lim)=my(v=List(), t1, t2, t3, t4, t5, t6, t7); forprime(p=2, (lim-16)^(1/4), forprime(q=2, min(p-1, (lim-p^4)^(1/4)), t1=p^4+q^4; if(isprime(t1), listput(v, t1)); forprime(r=2, min(q-1, (lim-t1)^(1/4)), t2=t1+r^4; if(isprime(t2), listput(v, t2)); forprime(s=2, min(r-1, (lim-t2)^(1/4)), t3=t2+s^4; if(isprime(t3), listput(v, t3)); forprime(t=2, min(s-1, (lim-t3)^(1/4)), t4=t3+t^4; if(isprime(t4), listput(v, t4)); forprime(u=2, min(t-1, (lim-t4)^(1/4)), t5=t4+u^4; if(isprime(t5), listput(v, t5)); forprime(w=2, min(u-1, (lim-t5)^(1/4)), t6=t5+w^4; if(isprime(t6), listput(v, t6)); forprime(x=2, min(w-1, (lim-t6)^(1/4)), t7=t6+x^4; if(isprime(t7), listput(v, t7)); if(x>2&&t7+16<=lim&&isprime(t7+16), listput(v, t7+16)))))))))); vecsort(Vec(v), , 8);
list(4044955) \\ Charles R Greathouse IV, Jul 27 2011

a(7)-a(33) from Charles R Greathouse IV, Jul 25 2011

A244344 Numbers such that the largest prime factor equals the sum of the 4th power of the other prime factors.

Original entry on oeis.org

582, 1164, 1746, 2328, 3492, 4656, 5238, 6410, 6984, 9312, 10476, 12820, 13968, 15714, 18624, 20952, 25640, 27936, 31428, 32050, 33838, 37248, 41904, 47142, 51280, 55872, 56454, 62856, 64100, 67676, 74496, 83808, 94284, 102560, 111744, 112908, 125712, 128200

Offset: 1

Michel Lagneau, Jun 26 2014

nonn

Observation: it seems that the prime divisors of a majority of numbers n are of the form {2, p, q} with q = 2^4 + p^4, but there exists more rarely odd numbers with more prime divisors (example from Michel Marcus: 3955413 = 3*7*11*17123).

			582 is in the sequence because the prime divisors of 582 are 2, 3 and 97 => 2^4 + 3^4 = 97.

Amiram Eldar, Table of n, a(n) for n = 1..1500

Cf. A094479, A193411, A185077.

fpdQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},Max[f]-Total[Most[f]^4]==0];Union[Select[Range[2,5*10^5],fpdQ]]

A287927 Numbers k such that A287925(k) is a prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 12, 13, 17, 18, 20, 22, 25, 32, 34, 41, 48, 49, 50, 53, 61, 65, 66, 72, 73, 75, 76, 77, 85, 87, 89, 93, 96, 98, 104, 108, 113, 114, 115, 121, 124, 127, 130, 141, 142, 147, 156, 165, 176, 178, 179, 180, 182, 183, 187, 196, 197, 208, 214

Offset: 1

XU Pingya, Jun 03 2017

Corresponding primes are in A094479.

Cf. A287925, A094479, A287924.

Select[Table[n, {n, 215}], PrimeQ[Prime[#]^4 + 2^4] &]

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A045637 Primes of the form p^2 + 4, where p is prime.

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A158361 Primes p with property that Q = p^4 + 2^4 is prime.

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A193411 Primes which are sums of two or more distinct 4th powers of primes.

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A244344 Numbers such that the largest prime factor equals the sum of the 4th power of the other prime factors.

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A287927 Numbers k such that A287925(k) is a prime.

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