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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A186665 Total number of positive integers below 10^n requiring 10 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 72, 812, 7876, 70630, 673358, 6676098, 66678723, 666681488, 6666684311
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + a(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003344, A186666.

Extensions

a(5)-a(7) from Lars Blomberg, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186667 Total number of positive integers below 10^n requiring 11 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 70, 760, 7434, 68348, 669289, 6670284, 66671226, 666672168, 6666673077
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + a(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003345, A186668.

Extensions

a(5)-a(7) from Lars Blomberg, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186671 Total number of positive integers below 10^n requiring 13 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 66, 706, 6945, 67173, 667369, 6667582, 66667813, 666668052, 6666668292
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + a(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A046044, A186672.

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186673 Total number of positive integers below 10^n requiring 14 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 66, 698, 6833, 66965, 667080, 6667198, 66667327, 666667454, 6666667590
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + a(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Links

Crossrefs

Cf. A046045, A186674.

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186675 Total number of positive integers below 10^n requiring 15 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 66, 690, 6761, 66834, 666903, 6666972, 66667041, 666667102, 6666667173
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + a(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Links

Crossrefs

Cf. A046046, A186676.

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186677 Total number of positive integers below 10^n requiring 16 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 4, 47, 288, 587, 874, 1178, 1487, 1803, 2089, 2388
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + a(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A046047, A186678.

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186680 Total number of positive integers below 10^n requiring 17 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 3, 33, 63, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + a(n) + A186682(n) + A186684(n) = A002283(n)
a(n) = 65 for n >= 5. - Nathaniel Johnston, May 09 2011
Continued fraction expansion of (826055+sqrt(4229))/2503382. - Bruno Berselli, May 10 2011

Links

Crossrefs

Cf. A046048, A099591.

Programs

  • Mathematica
    PadRight[{0, 3, 33, 63}, 100, 65] (* Paolo Xausa, Jul 31 2024 *)

Formula

G.f.: x^2*(3+30*x+30*x^2+2*x^3)/(1-x). - Bruno Berselli, May 10 2011

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
Terms after a(7) from Nathaniel Johnston, May 09 2011

A186682 Total number of positive integers below 10^n requiring 18 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 2, 19, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + a(n) + A186684(n) = A002283(n)
a(n) = 24 for n >= 4. - Nathaniel Johnston, May 09 2011
Continued fraction expansion of (185-sqrt(145))/355. - Bruno Berselli, May 10 2011

Links

Crossrefs

Cf. A046049.

Programs

  • Mathematica
    PadRight[{0, 2, 19}, 100, 24] (* Paolo Xausa, Jul 30 2024 *)

Formula

G.f.: x^2*(2+17*x+5*x^2)/(1-x). - Bruno Berselli, May 10 2011

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
Terms after a(7) from Nathaniel Johnston, May 09 2011

A186670 Total number of n-digit numbers requiring 12 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 62, 656, 6402, 60411, 600390, 6000430, 60000450, 600000451, 6000000424
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + a(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

Links

Crossrefs

Cf. A003346, A186669.

Formula

a(n) = A186669(n) - A186669(n-1).

Extensions

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A105041 Positive integers k such that k^7 + 1 is semiprime.

Original entry on oeis.org

2, 10, 16, 18, 46, 52, 66, 72, 78, 106, 136, 148, 226, 228, 240, 262, 282, 330, 442, 508, 616, 630, 732, 750, 756, 768, 810, 828, 910, 936, 982, 1032, 1060, 1128, 1216, 1302, 1366, 1558, 1626, 1696, 1698, 1758, 1800, 1810, 1830, 1932, 1996, 2002, 2026, 2080
Offset: 1

Views

Author

Jonathan Vos Post, Apr 03 2005

Keywords

Comments

We have the polynomial factorization n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1). Hence after the initial n=1 prime, the binomial can at best be semiprime and that only when both (n+1) and (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) are primes.

Examples

			n n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
2 129 = 3 * 43
10 10000001 = 11 * 909091
16 268435457 = 17 * 15790321
18 612220033 = 19 * 32222107
46 435817657217 = 47 * 9272716111

Links

Crossrefs

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

Programs

  • Magma
    IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..2100] | IsSemiprime(n^7+1)]; // Vincenzo Librandi, Mar 12 2015
  • Mathematica
    Select[Range[0,200000], PrimeQ[# + 1] && PrimeQ[(#^7 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)
Select[Range[2500], Plus@@Last/@FactorInteger[#^7 + 1]==2 &] (* Vincenzo Librandi, Mar 12 2015 *)
Select[Range[2100],PrimeOmega[#^7+1]==2&] (* Harvey P. Dale, Jun 18 2019 *)
  • PARI
    is(n)=isprime(n+1) && isprime((n^7+1)/(n+1)) \\ Charles R Greathouse IV, Aug 31 2021

Formula

a(n)^7 + 1 is semiprime. a(n)+1 is prime and a(n)^6 - a(n)^5 + a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.

Extensions

More terms from R. J. Mathar, Dec 14 2009
Previous Showing 11-20 of 27 results. Next

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