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.

A001539 a(n) = (4*n+1)*(4*n+3).

Original entry on oeis.org

3, 35, 99, 195, 323, 483, 675, 899, 1155, 1443, 1763, 2115, 2499, 2915, 3363, 3843, 4355, 4899, 5475, 6083, 6723, 7395, 8099, 8835, 9603, 10403, 11235, 12099, 12995, 13923, 14883, 15875, 16899, 17955, 19043, 20163, 21315, 22499, 23715, 24963, 26243, 27555, 28899
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Sequence arises from reading the line from 3, in the direction 3, 35, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
log(sqrt(2)+1)/sqrt(2) = 0.62322524... = 2/3 - 2/35 + 2/99 - 2/195 + 2/323, ... = (1 - 1/3) + (1/7 - 1/5) + (1/9 - 1/11) + (1/15 - 1/13) + (1/17 - 1/19) + (1/23 - 1/21) + ... - Gary W. Adamson, Mar 01 2009
Numbers k such that k+1 is a square and k+5 is divisible by 8. - Bruno Berselli, Sep 27 2017
The concatenation of 8*A000217(n) and 99 is a term of the sequence. Example: for A000217(5) = 15, 8*15 = 120 and 12099 = a(27). In general, a(5*n+2) = 800*A000217(n) + 99. - Bruno Berselli, Sep 29 2017

Links

Crossrefs

Bisection of A000466.
Cf. A000217, A000290, A001533, A001538, A004767, A016286, A016813, A016826, A157142, A133766, A154633.

Programs

  • Mathematica
    CoefficientList[Series[(3 + 26 x + 3 x^2)/(1 - x)^3, {x, 0, 41}], x] (* or *) Table[(4 n + 1) (4 n + 3), {n, 0, 41}] (* Michael De Vlieger, Sep 29 2017 *)
  • Maxima
    makelist((4*n+1)*(4*n+3), n, 0, 30); /* Martin Ettl, Nov 12 2012 */
  • PARI
    a(n)=(4*n+1)*(4*n+3) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = A016826(n) - 1 = (A001533(n)+5)/4 = (A001538(n)+16)/9.
Sum_{k>=0} 1/a(k) = Pi/8. - Benoit Cloitre, Aug 20 2002
G.f.: (3 + 26*x + 3*x^2)/(1 - x)^3. - Jaume Oliver Lafont, Mar 07 2009
a(n) = 32*n + a(n-1) for n > 0, a(0)=3. - Vincenzo Librandi, Nov 12 2010
a(n) = a(m) + 16*(n-m)*(n+m+1). The previous formula is obtained for m = n-1. - Bruno Berselli, Sep 29 2017
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A016813(n)*A004767(n).
Product_{n>=0} (1 - 1/a(n)) = sqrt(2)*cos(Pi/(2*sqrt(2))).
Product_{n>=0} (1 + 1/a(n)) = sqrt(2). (End)
From Elmo R. Oliveira, Oct 23 2024: (Start)
E.g.f.: exp(x)*(3 + 16*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A103854 Positive integers n such that n^6 + 1 is semiprime.

Original entry on oeis.org

2, 4, 10, 36, 56, 94, 126, 224, 260, 270, 300, 350, 686, 716, 780, 1036, 1070, 1080, 1156, 1174, 1210, 1394, 1416, 1434, 1440, 1460, 1524, 1550, 1576, 1616, 1654, 1660, 1700, 1756, 1860, 1980, 2054, 2084, 2096, 2116, 2224, 2454, 2600, 2664, 2770, 2864
Offset: 1

Views

Author

Jonathan Vos Post, Mar 31 2005

Keywords

Comments

n^6+1 can only be prime when n = 1, n^6+1 = 2. This is because the sum of cubes formula gives the polynomial factorization n^6+1 = (n^2+1) * (n^4 - n^2 + 1). Hence n^6+1 can only be semiprime when both (n^2+1) and (n^4 - n^2 + 1) are primes.

Examples

			n n^6+1 = (n^2+1) * (n^4 - n^2 + 1)
2 65 = 5 * 13
4 4097 = 17 * 241
10 1000001 = 101 * 9901
36 2176782337 = 1297 * 1678321
56 30840979457 = 3137 * 9831361
94 689869781057 = 8837 * 78066061
126 4001504141377 = 15877 * 252031501
224 126324651851777 = 50177 * 2517580801

Links

Crossrefs

Subsequence of A005574.
Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Programs

  • Mathematica
    semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ 2Range@1526, semiprimeQ[ #^6 + 1] &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[200000], PrimeQ[#^2 + 1] && PrimeQ[(#^6 + 1)/(#^2 + 1)] &] (* Robert Price, Mar 11 2015 *)
  • PARI
    is(n)=my(s=n^2); isprime(s+1) && isprime(s^2-s+1) \\ Charles R Greathouse IV, Aug 31 2021

Formula

a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.

Extensions

More terms from Robert G. Wilson v, May 26 2006

A105066 Positive integers n such that n^8 + 1 is semiprime.

Original entry on oeis.org

6, 9, 10, 13, 16, 18, 20, 22, 26, 28, 32, 33, 34, 38, 42, 43, 47, 50, 51, 52, 53, 56, 58, 60, 66, 68, 69, 70, 72, 81, 84, 92, 94, 98, 102, 104, 110, 116, 120, 134, 136, 138, 144, 145, 160, 162, 164, 166, 170, 172, 174, 178, 185, 188, 192, 196, 198, 200, 204, 205, 210
Offset: 1

Views

Author

Jonathan Vos Post, Apr 05 2005

Keywords

Comments

n^8 + 1 is an irreducible polynomial over the integers and thus can be prime (1^8+1=2, 2^8+1=257, 4^8+1=65537) as well as semiprime.

Examples

			6^8+1 = 1679617 = 17 * 98801,
16^8+1 = 4294967297 = 641 * 6700417,
72^8+1 = 722204136308737 = 12110113 * 59636449 where the two factors have the same number of digits.

Links

Crossrefs

Cf. A000040, A001538, A103854, A104238, A105041.

Programs

  • Mathematica
    fQ[n_] := Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == {1, 1}; Select[ Range[220], fQ[ #^8 + 1] &] (* Robert G. Wilson v, Apr 06 2005 *)
Select[Range[300],PrimeOmega[#^8+1]==2&] (* Harvey P. Dale, Nov 19 2018 *)

Formula

a(n)^8+1 is an element of A001538.

Extensions

More terms from Robert G. Wilson v, Apr 06 2005
Showing 1-3 of 3 results.

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

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