cp's OEIS Frontend

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.

A369056 Numbers k of the form 4m+3 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 31, 35, 43, 47, 59, 63, 67, 79, 83, 99, 107, 115, 127, 139, 143, 159, 163, 171, 175, 179, 207, 219, 223, 227, 235, 243, 259, 279, 283, 295, 303, 307, 319, 323, 339, 347, 367, 379, 387, 399, 403, 415, 427, 443, 463, 499, 515, 523, 531, 547, 559, 571, 579, 595, 603, 619, 639, 643, 655, 659, 675
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Comments

Numbers k in A004767 for which A369054(k) = 0.
Numbers k of the form 4m-1 such that they are not arithmetic derivative (A003415) of any term of A046316.
Question: Is it possible that this sequence might be finite (although very long)? See comments in A369055.

Links

Crossrefs

Setwise difference A004767 \ A369251.
Cf. A003415, A046316, A369054, A369055.
Subsequences: A369248 (terms that are multiples of 3), A369249 (primes in this sequence).
Cf. also A369250 (4m+3 primes missing from this sequence).

Programs

  • Maple
    N:= 1000: # for terms <= N
S:= {seq(i,i=3..N,4)}:
P:= select(isprime, [seq(i,i=3..N/3,2)]):
for i from 1 to nops(P) do
  p:= P[i];
  for j from i to nops(P) do
    q:= P[j];
    if 2*p*q + q^2 > N then break fi;
    for k from j to nops(P) do
      r:= P[k];
      v:= p*q + p*r + q*r;
      if v > N then break fi;
      S:= S minus {v};
od od od:
sort(convert(S,list)); # Robert Israel, Apr 17 2024
  • PARI
    isA369056(n) = ((3==(n%4)) && !A369054(n)); \\ Needs also program from A369054.

A369464 Numbers for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88
Offset: 1

Views

Author

Antti Karttunen, Jan 24 2024

Keywords

Crossrefs

Complement of A369251. Numbers not in A369252.
Union of A004773 and A369056.
Positions of 0's in A369054.
Cf. A098700, A369248, A369249, A369463 (subsequences).

Programs

  • PARI
    isA369251(n) = if(3!=(n%4),0, my(v = [3,3], ip = #v, r); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r), return(1))); if(!ip, return(0)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
isA369464(n) = !isA369251(n);

A369250 Primes for which there is at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

71, 103, 131, 151, 167, 191, 199, 211, 239, 251, 263, 271, 311, 331, 359, 383, 419, 431, 439, 467, 479, 487, 491, 503, 563, 587, 599, 607, 631, 647, 691, 719, 727, 739, 743, 751, 811, 823, 839, 859, 863, 887, 911, 919, 971, 983, 991, 1019, 1031, 1051, 1063, 1091, 1103, 1151, 1163, 1187, 1223, 1231, 1279, 1283, 1291
Offset: 1

Views

Author

Antti Karttunen, Jan 22 2024

Keywords

Comments

All such primes are by necessity of the form 4m+3 (in A002145). See A369249 for those 4m+3 primes that do not have such a representation.
Also by necessity, in these cases the primes in the sum (p*q + p*r + q*r) must all be distinct, that is, we actually need p < q < r, otherwise the sum would not be a prime.

Examples

			71 is present as 71 = (3*5) + (3*7) + (5*7) = A003415(105).

Links

Crossrefs

Primes in A369251.
Setwise difference A002145 \ A369249.
Subsequence of A189441.
Cf. A003415, A369054, A369056.

Programs

Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.