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.

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A242330 Numbers k such that k^2 + 2 is a semiprime.

Original entry on oeis.org

2, 6, 7, 11, 12, 17, 18, 27, 29, 35, 37, 42, 43, 48, 51, 53, 54, 55, 60, 65, 66, 69, 72, 73, 75, 79, 83, 84, 87, 90, 93, 97, 115, 119, 125, 132, 133, 135, 137, 141, 144, 150, 153, 155, 159, 161, 165, 169, 174, 183, 186, 187, 189, 191, 192, 195, 198
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 6, 38, 51, 123, 146, 291, 326, 731, 843, 1227, 1371, 1766, 1851, 2306, 2603, 2811, 2918, 3027, 3602, ....
There are no four consecutive terms in this sequence, that is, a(n) > a(n-3) + 3 (check mod 6). Probably sieve theory can show that this sequence has density 0. - Charles R Greathouse IV, Feb 24 2023

Links

Crossrefs

Cf. A056899, A067201, A085722, A242331, A242332, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..200] | IsSemiprime(s) where s is n^2+2];
  • Mathematica
    Select[Range[300], PrimeOmega[#^2 + 2] == 2 &]
  • PARI
    issemi(n)=forprime(p=2,997,if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
is(n)=issemi(n^2+2) \\ Charles R Greathouse IV, Feb 24 2023

Formula

a(n) > 2n for n > 1. - Charles R Greathouse IV, Feb 24 2023

A240587 Primes p of the form n^2 + 123456789 where 123456789 is the first zeroless pandigital number.

Original entry on oeis.org

123457189, 123459289, 123465253, 123466789, 123470713, 123481753, 123482389, 123486373, 123489913, 123501733, 123505189, 123510613, 123535189, 123545593, 123564373, 123571033, 123584953, 123587833, 123592213, 123610453, 123631513, 123641689, 123657493
Offset: 1

Views

Author

K. D. Bajpai, Apr 08 2014

Keywords

Examples

			123457189 is a prime and appears in the sequence because 123457189 = 20^2 + 123456789.
123459289 is a prime and appears in the sequence because 123459289 = 50^2 + 123456789.

Links

Crossrefs

Cf. A000040, A002496, A028871, A050289, A056899.

Programs

  • Maple
    KD := proc() local a; a:=n^2+123456789; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..1000);
  • Mathematica
    Select[Table[k^2+123456789,{k,1,3000}],PrimeQ]

A176969 Numbers n such that n^2 + 13^2 is prime.

Original entry on oeis.org

2, 8, 10, 12, 20, 22, 28, 30, 32, 38, 42, 48, 58, 60, 62, 68, 80, 90, 100, 108, 110, 112, 122, 128, 138, 142, 148, 150, 168, 172, 180, 190, 198, 200, 202, 210, 228, 230, 232, 238, 242, 248, 258, 262, 268, 280, 282, 302, 310, 318, 340, 342, 360, 362, 368, 378
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 29 2010

Keywords

Comments

The n^2 + d conjecture is a famous and still unsolved problem.
It states that there exist an infinite number of primes whose values are of the form n^2 + d for some integer n.
This is case d = 13^2.

Examples

			2^2 + 13^2 = 173 = prime(40), 2 is first term.
12^2 + 13^2 = 313 = prime(65) = palprime(11), 12 is 4th term.
310^2 + 13^2 = 96269 = prime(9274) = palprime(106), 310 the 49th term.

References

  • J. Matousek, J. Nesetril: Diskrete Mathematik: eine Entdeckungsreise, Springer-Lehrbuch, 2. Aufl., Berlin, 2007
  • M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006

Crossrefs

Cf. A000040, A000290, A056899, A113536, A176371.

Programs

  • PARI
    isok(n) = isprime(n^2 + 13^2) \\ Michel Marcus, Jun 28 2013

A176978 Numbers n such that n^2 + 13 and n^2 + 13^2 are primes.

Original entry on oeis.org

2, 10, 12, 28, 42, 60, 68, 108, 110, 122, 142, 180, 198, 200, 268, 282, 340, 378, 380, 418, 430, 488, 500, 502, 530, 612, 742, 788, 802, 838, 840, 912, 942, 948, 952, 1010, 1080, 1102, 1148, 1232, 1270, 1428, 1452, 1472, 1502, 1522, 1538, 1720, 1778, 1830
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 30 2010

Keywords

Comments

Numbers are subsequence of A113536 and A176969
See comments and references of A176969

Examples

			2^2 + 13 = 17 = prime(7), 2^2 + 13^2 = 173 = prime(40), 2 is first term.

Links

Crossrefs

Cf. A000040, A000290, A056899, A113536, A176371, A176969.

Programs

  • Mathematica
    Select[Range[2000],AllTrue[#^2+{13,13^2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 28 2015 *)
  • PARI
    isok(n) = isprime(n^2 + 13) && isprime(n^2 + 13^2); \\ Michel Marcus, Aug 27 2013

A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 5, 4, 7, 4, 8, 5, 8, 5, 10, 4, 10, 6, 9, 6, 11, 4, 12, 6, 10, 6, 12, 5, 12, 7, 10, 6, 14, 4, 14, 6, 10, 8, 13, 6, 13, 6, 12, 6, 16, 4, 14, 8, 10, 8, 14, 5, 16, 7, 12, 6, 14, 6, 16, 8, 12, 6, 16, 4, 16, 8, 11, 10, 15, 6, 14, 6, 14, 6, 20, 4, 16, 8, 10, 10
Offset: 2

Views

Author

T. D. Noe, Apr 28 2011

Keywords

Comments

d(n-1) + d(n+1) is a measure of the compositeness of the numbers next to n. Sequence A189825 lists the first occurrence of each number.
It is conjectured that every number greater than 3 occurs an infinite number of times. Note that an infinite number of 4's is equivalent to there being an infinite number of twin primes (A001097). An infinite number of 5's is equivalent to there being an infinite number of primes of the form p^2-2 (A028871) or p^2+2 (A056899) for prime p. An infinite number of 6's is equivalent to there being an infinite number of primes of the form p^3-2 (A066878), p^3+2 (A048636), p*q-2 (A063637), or p*q+2 (A063638), where p and q are distinct primes.

Examples

			a(5) = d(4) + d(6) = 3 + 4 = 7.

Links

Crossrefs

Cf. A175144, A189825.

Programs

  • Mathematica
    Table[DivisorSigma[0,n-1] + DivisorSigma[0,n+1], {n, 2, 100}]
First[#]+Last[#]&/@Partition[DivisorSigma[0,Range[80]],3,1] (* Harvey P. Dale, May 27 2013 *)

A234812 Primes p of the form n + 987654321 where 987654321 is the largest zeroless pandigital number.

Original entry on oeis.org

987654323, 987654337, 987654347, 987654359, 987654361, 987654377, 987654379, 987654383, 987654419, 987654439, 987654443, 987654461, 987654463, 987654467, 987654511, 987654539, 987654581, 987654583, 987654601, 987654607, 987654611, 987654673, 987654677, 987654791
Offset: 1

Views

Author

K. D. Bajpai, Apr 19 2014

Keywords

Examples

			987654323 is a prime and appears in the sequence because 987654323 = 2 + 987654321.
987654337 is a prime and appears in the sequence because 987654337 = 16 + 987654321.

Links

Crossrefs

Cf. A000040, A002496, A028871, A050289, A056899, A240587.

Programs

  • Maple
    KD := proc() local a; a:=n+987654321; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..1000);
  • Mathematica
    Select[Table[k + 987654321, {k,1,1000}], PrimeQ]
c=0; a=n+987654321; Do[If[PrimeQ[a], c=c+1; Print[c," ",a]], {n,0,200000}] (* b-file *)

A091199 Numbers k such that (6k-3)^2 + 2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 14, 17, 18, 19, 20, 21, 25, 29, 37, 38, 40, 41, 42, 43, 46, 50, 51, 52, 54, 58, 59, 61, 62, 63, 65, 70, 71, 72, 74, 75, 76, 80, 84, 86, 92, 93, 94, 96, 101, 102, 109, 113, 114, 117, 119, 126, 130, 135, 137, 140, 145, 148, 150, 151, 152, 156, 160
Offset: 1

Views

Author

Zak Seidov, Feb 22 2004

Keywords

Comments

Arises from A056899: primes of the form m^2+2; m should be of the form 6n-3, hence this sequence.

Examples

			10 is a member because (60-3)^2 + 2 = 3251 is prime.

Links

Crossrefs

Cf. A056899, A056900.

Programs

Formula

a(n) = A056900(n-1) + 1. - Jeppe Stig Nielsen, May 14 2017

Extensions

More terms from Ray Chandler and Robert G. Wilson v, Feb 25 2004

A129389 Numbers k such that the mean of 5 consecutive squares starting with k^2 is prime.

Original entry on oeis.org

1, 7, 13, 19, 31, 37, 43, 55, 79, 97, 103, 109, 115, 121, 145, 169, 217, 223, 235, 241, 247, 253, 271, 295, 301, 307, 319, 343, 349, 361, 367, 373, 385, 415, 421, 427, 439, 445, 451, 475, 499, 511, 547, 553, 559, 571, 601, 607, 649, 673, 679, 697, 709, 751
Offset: 1

Views

Author

Zak Seidov, Apr 12 2007

Keywords

Comments

Sum of 5 consecutive squares starting with k^2 is equal to 5*(6 + 4*k + k^2) and mean is (6 + 4*k + k^2) = (k+2)^2 + 2. Hence a(n) = A067201(n+2).
Also, numbers k such that A000217(k) + A000217(k+3) is prime. - Bruno Berselli, Apr 17 2013

Examples

			(1^2 + ... + 5^2)/5 = 11, which is prime;
(7^2 + ... + 11^2)/5 = 83, which is prime;
(13^2 + ... + 17^2)/5 = 227, which is prime.

Links

Crossrefs

Cf. A056899, A067201, A129388.
Cf. A000217, A128815 (numbers n such that A000217(n)+A000217(n+2) is prime). [Bruno Berselli, Apr 17 2013]

Programs

  • Magma
    [n: n in [1..800] | IsPrime(n^2+4*n+6)]; /* or, from the second comment: */ A000217:=func; [n: n in [1..800] | IsPrime(A000217(n)+A000217(n+3))]; // Bruno Berselli, Apr 17 2013
  • Mathematica
    Select[Range[800], PrimeQ[#^2 + 4 # + 6] &] (* Bruno Berselli, Apr 17 2012 *)
  • SageMath
    [n for n in (1..1000) if is_prime(n^2+4*n+6)] # G. C. Greubel, Feb 04 2024

A132282 Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.

Original entry on oeis.org

2, 3, 29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793
Offset: 1

Views

Author

Jonathan Vos Post, Aug 16 2007

Keywords

Comments

The corresponding near-cube prime indices q are A132281. Analog of near-square primes. After a(1) = 2, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.

Examples

			a(1) = 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1^3 + 2 = 3 is prime and 1 is noncomposite.
a(3) = 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29^3 + 2 = 24391 is prime and 29 is prime.
45^3 + 2 = 91127 is prime, but not in this sequence because 45 is not prime.
63^3 + 2 = 250049 is prime, but not in this sequence because 63 is not prime.
a(6) = 71^3 + 2 = 357913 is prime.
a(7) = 83^3 + 2 = 571789 is prime.
a(8) = 113^3 + 2 = 1442899 is prime.

Links

Crossrefs

Cf. A000040, A048636, A056899, A059100, A067200, A067201, A084380, A132281.

Programs

  • Mathematica
    Join[{2, 5}, Select[Prime[Range[200]]^3 + 2, PrimeQ[ # ] &]] (* Stefan Steinerberger, Aug 17 2007 *)
  • PARI
    v=[2,3]; forprime(p=3, 1e4, if(isprime(t=p^3+2), v=concat(v, t))); t \\ Charles R Greathouse IV, Feb 14 2011

Formula

a(n) = A132281(n)^3 + 2. {p in A000040 such that for some q = 0, 1, or q in A000040, we have p = A067200(q) = A084380(q) = q^3 + 2 is in A000040}.
a(n) = A048636(n-2) for n >= 3. - Georg Fischer, Nov 03 2018

Extensions

More terms from Stefan Steinerberger, Aug 17 2007
a(2) corrected by Charles R Greathouse IV, Feb 14 2011

A164518 Primes of the form A162143(k) + 2.

Original entry on oeis.org

11027, 65027, 74531, 119027, 184043, 308027, 314723, 370883, 423803, 603731, 783227, 804611, 815411, 915851, 938963, 1238771, 1279163, 1461683, 1490843, 1535123, 1550027, 1718723, 2556803, 2673227, 2812331, 3059003, 3493163
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 14 2009

Keywords

Comments

Primes of the form 2 + q^2*r^2*s^2 where q, r, and s are three distinct primes.

Examples

			a(1) = 11027 = A000040(1337) = A162143(7) + 2.

Links

Crossrefs

Cf. A056899, A162143, A164517.

Programs

  • Maple
    N:= 10^7: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N-2)/15))]):
R:= NULL:
for i from 1 to nops(P) do
  for j from 1 to i-1 while (3*P[i]*P[j])^2<=N-2 do
    for k from 1 to j-1 do
      p:= (P[i]*P[j]*P[k])^2+2;
      if p > N then break fi;
      if isprime(p) then R:= R, p fi
od od od:
sort([R]); # Robert Israel, Jun 05 2018
  • Mathematica
    f[n_]:=FactorInteger[n][[1,2]]==2&&Length[FactorInteger[n]]==3&&FactorInteger[n][[2, 2]]==2&&FactorInteger[n][[3,2]]==2; lst={};Do[p=Prime[n];If[f[p-2], AppendTo[lst,p]],{n,4,9!}];lst
With[{nn=30},Take[Union[Select[Times@@(#^2)+2&/@Subsets[Prime[ Range[ nn]], {3}],PrimeQ]],nn]] (* Harvey P. Dale, Mar 14 2016 *)

Extensions

Edited by R. J. Mathar, Aug 21 2009
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