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 1-7 of 7 results.

A081256 Greatest prime factor of n^3 + 1.

Original entry on oeis.org

2, 3, 7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 13, 307, 7, 127, 421, 463, 13, 79, 601, 31, 37, 757, 271, 67, 19, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 43, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163
Offset: 1

Views

Author

Jan Fricke, Mar 14 2003

Keywords

Comments

Record values appear to match the terms of A002383 for n>1. - Bill McEachen, Oct 18 2023

Links

Crossrefs

Cf. A081257, A081258, A096173, A096174.
Cf. A006530, A001093, A002383, A014442, A096172.

Programs

  • Maple
    A081256 := proc(n)
    A006530(n^3+1) ;
end proc:
seq(A081256(n),n=1..20) ; # R. J. Mathar, Feb 13 2014
  • Mathematica
    Table[Max[Transpose[FactorInteger[n^3 + 1]][[1]]], {n, 25}]
  • PARI
    a(n)=my(f=factor(n^3+1)); f[#f~,1] \\ Charles R Greathouse IV, Mar 08 2017
  • PARI
    A081256(n)=vecmax(factor(n^3+1)[,1]) \\ It seems slightly slower to get the last element using ...[-1..-1][1]. - M. F. Hasler, Jun 15 2018

Formula

a(n) = A006530(A001093(n)). - M. F. Hasler, Jun 13 2018
a(n) >= 31 for n >= 70 (Buchmann et al., 1991). - Amiram Eldar, Oct 25 2024

Extensions

More terms from Harvey P. Dale, Mar 22 2003
More terms from Hugo Pfoertner, Jun 20 2004

A096173 Numbers k such that k^3+1 is an odd semiprime.

Original entry on oeis.org

2, 4, 6, 16, 18, 22, 28, 42, 58, 60, 70, 72, 78, 100, 102, 106, 112, 148, 156, 162, 190, 210, 232, 280, 310, 330, 352, 358, 382, 396, 448, 456, 490, 568, 606, 672, 756, 786, 820, 826, 828, 856, 858, 876, 928, 970, 982, 1008, 1012, 1030, 1068, 1092, 1108, 1150
Offset: 1

Views

Author

Hugo Pfoertner, Jun 20 2004

Keywords

Comments

Numbers n such that n^3 + 1 is a semiprime, because then n^3 + 1 must be odd, since n^3 + 1 = (n+1)*(n^2 - n + 1) is a semiprime only if n+1 is odd. - Jonathan Sondow, Feb 02 2014
Obviously, n + 1 is always a prime number. Sequence is intersection of A006093 and A055494. - Altug Alkan, Dec 20 2015

Examples

			a(1)=2 because 2^3+1=9=3*3, a(13)=100: 100^3+1=1000001=101*9901.

Links

Crossrefs

Cf. A001358; A081256: largest prime factor of k^3+1; A096174: (k^3+1)/(k+1) is prime; A046315, A237037, A237038, A237039, A237040.
Cf. A006093, A055494.

Programs

  • Magma
    [n: n in [1..2*10^3] | IsPrime(n+1) and IsPrime(n^2-n+1)]; // Vincenzo Librandi, Dec 21 2015
  • Maple
    select(n -> isprime(n+1) and isprime(n^2-n+1), [seq(2*i,i=1..1000)]); # Robert Israel, Dec 20 2015
  • Mathematica
    Select[Range[1200], PrimeQ[#^2 - # + 1] && PrimeQ[# + 1] &] (* Jonathan Sondow, Feb 02 2014 *)
  • PARI
    for(n=1, 1e5, if(bigomega(n^3+1)==2, print1(n, ", "))); \\ Altug Alkan, Dec 20 2015

Formula

a(n) = 2*A237037(n) = (A237040(n)-1)^(1/3). - Jonathan Sondow, Feb 02 2014

Extensions

Corrected by Zak Seidov, Mar 08 2006

A237040 Semiprimes of the form k^3 + 1.

Original entry on oeis.org

9, 65, 217, 4097, 5833, 10649, 21953, 74089, 195113, 216001, 343001, 373249, 474553, 1000001, 1061209, 1191017, 1404929, 3241793, 3796417, 4251529, 6859001, 9261001, 12487169, 21952001, 29791001, 35937001, 43614209, 45882713, 55742969, 62099137, 89915393, 94818817, 117649001
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Comments

k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.
Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?
In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - Charles R Greathouse IV, Jul 02 2017

Examples

			9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.

Links

Crossrefs

Cf. A001358, A002383, A002496, A046315, A081256, A096173, A096174, A237037, A237038, A237039.
Cf. A242262 (semiprimes of the form k^3 - 1).

Programs

  • Magma
    IsSemiprime:= func; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017
  • Mathematica
    L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)
  • PARI
    lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ Michel Marcus, Jun 27 2017
  • PARI
    list(lim)=my(v=List(),n,t); forprime(p=3,sqrtnint(lim\1-1,3)+1, if(isprime(t=p^2-3*p+3), listput(v,t*p))); Vec(v) \\ Charles R Greathouse IV, Jul 02 2017

Formula

a(n) = A096173(n)^3 + 1 = 8*A237037(n)^3 + 1.

A237037 Numbers k such that (2*k)^3 + 1 is a semiprime.

Original entry on oeis.org

1, 2, 3, 8, 9, 11, 14, 21, 29, 30, 35, 36, 39, 50, 51, 53, 56, 74, 78, 81, 95, 105, 116, 140, 155, 165, 176, 179, 191, 198, 224, 228, 245, 284, 303, 336, 378, 393, 410, 413, 414, 428, 429, 438, 464, 485, 491, 504, 506, 515, 534, 546, 554, 575, 596, 611, 638, 641, 648, 659, 680, 683, 711, 714, 725, 744, 765, 774, 791
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Comments

Numbers k such that 2*k+1 and 4*k^2 - 2*k + 1 are both prime.
Same as k/2 such that k^3 + 1 is a semiprime, because then k must be even.

Examples

			(2*1)^3 + 1 = 9 = 3*3 is a semiprime, so a(1) = 1.

Links

Crossrefs

Cf. A001358, A046315, A081256, A096173, A096174, A237038, A237039, A237040.

Programs

  • Mathematica
    Select[Range[800], PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Range[800],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Nov 28 2024 *)

Formula

a(n) = A096173(n)/2 = (1/2)*(A237040(n)-1)^(1/3).

A237038 Primes p such that (2*p)^3 + 1 is a semiprime.

Original entry on oeis.org

2, 3, 11, 29, 53, 179, 191, 491, 641, 659, 683, 1103, 1499, 1901, 2129, 2543, 2549, 3803, 3851, 4271, 4733, 4943, 5303, 5441, 6101, 6329, 6449, 7193, 7211, 8093, 8513, 9059, 9419, 10091, 10271, 10733, 10781, 11321, 12203, 12821, 13451, 14561, 15233, 15803, 17159, 17333, 18131, 19373, 19919
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Comments

Same as Sophie Germain primes p such that 4*p^2 - 2*p + 1 is also prime (because (2*p)^3 + 1 = (2*p + 1)(4*p^2 - 2*p + 1)).
Primes in A237037.
For n>1, 8*a(n)^3 is a solution for the equation phi(x+1) - phi(x) = x/2. - Farideh Firoozbakht, Dec 17 2014

Examples

			11 is prime and (2*11)^3 + 1 = 10649 = 23*463 is a semiprime, so 11 is a member.

Links

Crossrefs

Cf. A000010, A001358, A005384, A046315, A081256, A096173, A096174, A237037, A237039, A237040.

Programs

  • Mathematica
    Select[Range[20000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Prime[Range[2500]],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Jun 28 2021 *)

Formula

a(n) = (1/2)*(A237039(n)-1)^(1/3).

A237039 Semiprimes of the form (2*p)^3 + 1, where p is prime.

Original entry on oeis.org

65, 217, 10649, 195113, 1191017, 45882713, 55742969, 946966169, 2106997769, 2289529433, 2548895897, 10735357817, 26946035993, 54958685609, 77199941513, 131561576057, 132495001193, 440016501017, 456888832409, 623273556089, 848202406697, 966188398457
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Examples

			(2*2)^3 + 1 = 65 = 5*13 is a semiprime, so a(1) = 2.

Links

Crossrefs

Cf. A001358, A046315, A081256, A096173, A096174, A237037, A237038, A237040.

Programs

  • Mathematica
    L = Select[Range[5000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]; (2 L)^3 + 1
Select[Table[(2p)^3+1,{p,Prime[Range[1000]]}],PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 21 2021 *)

Formula

a(n) = (2*A237038(n))^3 + 1.

A096176 Numbers k such that (k^3-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218
Offset: 1

Views

Author

Hugo Pfoertner, Jun 22 2004

Keywords

Comments

Numbers k > 1 such that k^2 + k + 1 is a prime. - Vincenzo Librandi, Nov 16 2010
Therefore essentially the same as A002384. - Georg Fischer, Oct 06 2018

Examples

			a(5) = 8 because (8^3-1)/(8-1) = 511/7 = 73 is prime.

Links

Crossrefs

Cf. A096174 (n^3+1)/(n+1) is prime, A081257 largest prime factor of n^3-1, A096175 n^3-1 is an odd semiprime.
Cf. A028491, A004061. - Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009
Cf. A002384.

Programs

Extensions

3 and 5 added by Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009
Corrected terms, including many previously omitted terms, from Harvey P. Dale, Sep 10 2009
Showing 1-7 of 7 results.

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