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

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

Author

Jonathan Vos Post, Jul 25 2011

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Extensions

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