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 1-5 of 5 results.

A045542 Sub-perfect powers: perfect powers (squares, cubes etc.) minus 1.

Original entry on oeis.org

3, 7, 8, 15, 24, 26, 31, 35, 48, 63, 80, 99, 120, 124, 127, 143, 168, 195, 215, 224, 242, 255, 288, 323, 342, 360, 399, 440, 483, 511, 528, 575, 624, 675, 728, 783, 840, 899, 960, 999, 1023, 1088, 1155, 1224, 1295, 1330, 1368, 1443, 1520, 1599, 1680, 1727
Offset: 1

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Author

William M. Glasgow (billg(AT)wakely.com)

Keywords

Comments

Goldbach showed that Sum 1/a(n) = 1, see A214390, A214391.
The only primes in the sequence are 3,7,31,127,... the Mersenne primes (A000668). - Zak Seidov, Dec 08 2011
Repdigits of two or more digits, interpreted in the smallest possible base. E.g., the smallest base for 222 is 3, 222 in base 3 is 26, and 26 is in the sequence. - Franklin T. Adams-Watters, Aug 11 2014

References

  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd edition, p. 66.

Links

Crossrefs

Cf. A000668, A001597.

Programs

  • Haskell
    a045542 n = a045542_list !! (n-1)
a045542_list = map (subtract 1) $ tail a001597_list
-- Reinhard Zumkeller, Jul 15 2012
  • Mathematica
    f[upto_] := Union[Flatten[Table[n^pwr - 1, {pwr, 2, Log[2,upto+1]}, {n, 2, (upto+1)^(1/pwr)}]]]; f[1763] (* Zak Seidov, Dec 08 2011 *)
Select[Range[2000],GCD@@FactorInteger[#][[All,2]]>1&]-1 (* Harvey P. Dale, Jan 31 2023 *)
  • PARI
    list(lim)=my(v=List()); for(e=2,logint(lim\=1,2), for(k=2,sqrtnint(lim,e), listput(v,k^e-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Formula

a(n) = A001597(n + 1) - 1.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2002

A216765 Perfect powers (squares, cubes, etc.) plus 1.

Original entry on oeis.org

5, 9, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729, 1765
Offset: 1

Views

Author

Jonathan Vos Post, Sep 15 2012

Keywords

Comments

Integers of the form m^k + 1 for integers m, k >= 2.

Examples

			a(1) = 2^2 + 1; a(2) = 2^3 + 1; a(3) = 3^2 + 1; a(4) = 2^4 + 1.

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6, p. 113.

Links

Crossrefs

Cf. A001597, A045542, A175064, A195055, A214390, A214391.

Formula

a(n) = A001597(n+1) + 1 = A045542(n) + 2. [corrected by Georg Fischer, Jun 21 2020]
Sum_{n>=1} 1/a(n) = Pi^2/3 - 5/2 (Lev, 2002). - Amiram Eldar, Oct 15 2020

A214390 Numerators of Sum_{n>=1} 1/A045542(n).

Original entry on oeis.org

1, 10, 101, 187, 149, 1021, 33016, 6845, 37447, 343471, 217843, 269387, 652735, 658741, 84404851, 594869, 99801, 19571431, 845870737, 6799977641, 826494610001, 4703352797483, 5311077568679, 5955647363593, 5974287995603, 2396798638405, 2403189712237
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 15 2012

Keywords

Comments

Goldbach-Euler: sum a(n)/A214391(n) -> 1, n -> oo.

Examples

			.      n     a(n) / A214391(n)
. ----------------------------
.      1    3.0
.     10    1.1827024697863866
.    100    1.0142002275801738
.   1000    1.0011756130379812 .

Links

Crossrefs

Cf. A214391 (denominators).

Programs

  • Haskell
    import Data.Ratio ((%), numerator)
a214390 n = a214390_list !! (n-1)
a214390_list = map numerator $ scanl1 (+) $ map (1 %) a045542_list

A216766 Numerators of partial sums of 1/A216765(n).

Original entry on oeis.org

1, 14, 37, 719, 5056, 151513, 1759463, 68163191, 352149277, 360867217, 15078888947, 1546201093747, 95491548878617, 10736398220663, 1400899861968427, 41036431877859203, 41386424246755373, 8212624279323157381, 256265816149636840711, 29139716513641120366493
Offset: 1

Views

Author

Jonathan Vos Post, Sep 15 2012

Keywords

Comments

Partial sums of the reciprocals of (perfect powers -- squares, cubes, etc. -- plus 1).

Examples

			The partial sums are of the sequence of fractions: 1/5 + 1/9 + 1/10 + 1/17 + 1/26 + 1/28 + 1/33 + 1/37 + 1/50, ... and thus the partial sums are 1/5, 14/45, 37/90, 719/1530, 5056/9995, 151513/278460, 1759463/3063060, 68163191/113333220, 352149277/566666100, 360867217/566666100, 15078888947/23233310100, ...

Links

Crossrefs

Cf. A001597, A045542, A175064, A214390, A214391, A216765, A216767 (denominators).

Programs

  • Mathematica
    Numerator[FoldList[Plus, 1/(1 + Select[Range[250], GCD @@ FactorInteger[#][[;; , 2]] > 1 &])]] (* Amiram Eldar, May 05 2022 *)

Formula

a(n) = numerator(Sum_{k=1..n} 1/A216765(k)).
Limit_{n->oo} a(n)/A216767(n) = Pi^2/3 - 5/2. - Amiram Eldar, May 05 2022

A216767 Denominators of partial sums of 1/A216765(n).

Original entry on oeis.org

5, 45, 90, 1530, 9945, 278460, 3063060, 113333220, 566666100, 566666100, 23233310100, 2346564320100, 143140423526100, 15904491502900, 2051679403874100, 59498702712348900, 59498702712348900, 11721244434332733300, 363358577464314732300
Offset: 1

Views

Author

Jonathan Vos Post, Sep 16 2012

Keywords

Comments

A216766 is the numerators of the partial sums of 1/A216765(n).

Examples

			 The partial sums are of the sequence of fractions: 1/5 + 1/9 + 1/10 + 1/17 + 1/26 + 1/28 + 1/33 + 1/37 + 1/50, ... and thus the partial sums are 1/5, 14/45, 37/90, 719/1530, 5056/9995, 151513/278460, 1759463/3063060, 68163191/113333220, 352149277/566666100, 360867217/566666100, 15078888947/23233310100, ...

Links

Crossrefs

Cf. A001597, A045542, A175064, A214390, A214391, A216765, A216766 (numerators).

Programs

  • Mathematica
    Denominator[FoldList[Plus, 1/(1 + Select[Range[250], GCD @@ FactorInteger[#][[;; , 2]] > 1 &])]] (* Amiram Eldar, May 05 2022 *)

Formula

a(n) = denominator(Sum_{k=1..n} 1/A216765(k)).
Showing 1-5 of 5 results.

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

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