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.

A352837 Primes p such that, if q is the next prime, p + q^2 is a prime times a power of 10.

Original entry on oeis.org

2, 409, 709, 1249, 3229, 4621, 6709, 7069, 7129, 7489, 8209, 9241, 9661, 10789, 11149, 13381, 13789, 15349, 17509, 20641, 21169, 21961, 23509, 25819, 26449, 28309, 28729, 32029, 32089, 34849, 35089, 35809, 37321, 37369, 39769, 40129, 42169, 42349, 42709, 42901, 42979, 44179, 46489, 47269, 47809
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Apr 05 2022

Keywords

Comments

Primes prime(k) such that when any trailing zeros are removed from A349660(k), the result is prime.

Examples

			a(3) = 709 is a term because 709 and 719 are consecutive primes, 709 + 719^2 = 517670, and 51767 is prime.
		

Crossrefs

Programs

  • Maple
    R:= NULL: count:= 0:
    q:= 2:
    while count < 100 do
      p:= q; q:= nextprime(p);
      v:= p+q^2;
      if v mod 10 = 0 then v:= v/10^min(padic:-ordp(v,2),padic:-ordp(v,5)) fi;
      if isprime(v) then count:= count+1; R:= R,p; fi;
    od:
    R;
  • Mathematica
    f[n_] := n/10^IntegerExponent[n, 10]; Select[Range[50000], PrimeQ[#] && PrimeQ[f[# + NextPrime[#]^2]] &] (* Amiram Eldar, Apr 07 2022 *)

A352851 a(n) = prime(n)^2 + prime(n+1).

Original entry on oeis.org

7, 14, 32, 60, 134, 186, 308, 384, 558, 872, 998, 1410, 1724, 1896, 2262, 2868, 3542, 3788, 4560, 5114, 5408, 6324, 6978, 8018, 9510, 10304, 10716, 11558, 11994, 12896, 16260, 17298, 18908, 19470, 22352, 22958, 24812, 26736, 28062, 30108, 32222, 32952, 36674, 37446, 39008, 39812, 44744, 49956
Offset: 1

Views

Author

Robert Israel, Apr 05 2022

Keywords

Examples

			a(3) = 5^2 + 7 = 32.
		

Crossrefs

Programs

  • Maple
    seq(ithprime(i)^2+ithprime(i+1), i=1..100);
  • Mathematica
    a[n_] := Prime[n]^2 + Prime[n + 1]; Array[a, 50] (* Amiram Eldar, Apr 07 2022 *)

Formula

a(n) = A001248(n) + A000040(n+1).
a(n) = 2*A140511(n), for n>1. - Michel Marcus, Mar 25 2023

A352126 Primes p such that, if q is the next prime, both p+q^2 and p^2+q are primes times powers of 10.

Original entry on oeis.org

2, 4806589, 8369989, 11168569, 20666869, 25068349, 25465249, 29046469, 37597849, 40593349, 44242669, 45405889, 47975869, 49637149, 50057569, 51468349, 57060469, 59570449, 64602589, 64707889, 65940769, 70752049, 75879169, 81799789, 87845869, 90277249, 92415649, 93315889, 95458249, 97225069
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Apr 05 2022

Keywords

Comments

Primes prime(k) such that when any trailing zeros are removed from A349660(k) and A352851(k), the results are prime.
Except for 2, each term and the next prime == 19 (mod 30).

Examples

			a(3) = 8369989 is a term because it is prime, the next prime is 8370049,
8369989+8370049^2 = 70057728632390, 8369989^2+8370049 = 70056724230170, and 7005772863239 and 7005672423017 are prime.
		

Crossrefs

Intersection of A352837 and A352852.

Programs

  • Maple
    R:= NULL: count:= 0:
    q:= 2:
    while count < 30 do
      p:= q; q:= nextprime(p);
      w:= p+q^2;
      m:= padic:-ordp(w,2);
      if padic:-ordp(w,5) <> m then next fi;
      if m > 0 then w:= w/10^m fi;
      if not isprime(w) then next fi;
      v:= p^2+q;
      m:= padic:-ordp(v,2);
      if padic:-ordp(v,5) <> m then next fi;
      if m > 0 then v:= v/10^m fi;
      if isprime(v) then count:= count+1; R:= R, p; fi
    od:
    R;
Showing 1-3 of 3 results.