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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Jonathan Vos Post, Jul 25 2011

Keywords

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

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

A130833 Sums of two or more distinct 4th powers of primes.

Original entry on oeis.org

97, 641, 706, 722, 2417, 2482, 2498, 3026, 3042, 3107, 3123, 14657, 14722, 14738, 15266, 15282, 15347, 15363, 17042, 17058, 17123, 17139, 17667, 17683, 17748, 17764, 28577, 28642, 28658, 29186, 29202, 29267, 29283, 30962, 30978, 31043, 31059, 31587, 31603
Offset: 1

Views

Author

Jonathan Vos Post, Jul 21 2011

Keywords

Comments

This is to cubes and A030078 as A192926 is to 4th powers and A030514. The subsequence of primes which are sums of two or more distinct 4th powers of primes begins 97, 641, 2417 (A193411).
The first term that arises in more than one way is 6539044 = 11^4 + 23^4 + 41^4 + 43^4 = 13^4 + 29^4 + 31^4 + 47^4. - Robert Israel, Apr 27 2020

Examples

			a(1) = 97 = 2^4 + 3^4.
a(2) = 641 = 2^4 + 5^4.
a(3) = 706 = 3^4 + 5^4.
a(4) = 722 = 2^4 + 3^4 + 5^4.

Links

Crossrefs

Cf. A000040, A000583, A030514, A192926, A193411.

Programs

  • Maple
    N:= 40000: # for all terms <= N
S1:= {}:
S2:= {}:
p:= 1:
do
  p:= nextprime(p);
  if p^4 > N then break fi;
  s:= p^4;
  S2:= S2 union select(`<=`,map(`+`,S1 union S2, s), N);
  S1:= S1 union {s};
od:
sort(convert(S2,list)); # Robert Israel, Apr 27 2020
  • Mathematica
    nn=6; t = Sort@ Flatten@ Table[ n^4, {n, Prime@ Range@ nn}]; Select[Sort[
Plus @@@ Subsets[t, {2, nn}]], # < Prime[nn-1]^4 + Prime[nn]^4 &] (* Robert G. Wilson v, Jul 22 2011 *)

Formula

{A030078(i) + A030078(j) for i not equal to j} UNION {A030078(i) + A030078(j) + A030078(k) for i not equal to j not equal to k} UNION {A030078(i) + A030078(j) + A030078(k) + A030078(L) for i not equal to j not equal to k not equal to L}...

A192231 Numbers k without prime numbers in the range (k-3*sqrt(sqrt(k)), k].

Original entry on oeis.org

1, 123, 124, 125, 126, 306, 330, 538, 539, 540, 904, 905, 906, 1147, 1148, 1149, 1150, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1689, 1690, 1691, 1692, 1971, 1972, 2200, 2201
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 26 2011

Keywords

Comments

a(934) = 20831532 is probably the last term. Any further terms must be greater than 1.5 * 10^18. - Charles R Greathouse IV, Jun 27 2011

Examples

			a(2)=123 because (123-3*sqrt(sqrt(123)), 123]=(123-9,9907.., 123]=(113,0092.., 123].

Links

Crossrefs

Subsequence of A192226.
Cf. A001223, A188817, A189025, A192926.

Programs

Extensions

a(2) added by Alonso del Arte, Jun 27 2011
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