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.

A108866 Numerator of Sum_{k=1..n} 2^k/k.

Original entry on oeis.org

0, 2, 4, 20, 32, 256, 416, 4832, 8192, 42496, 74752, 1467392, 2650112, 62836736, 115552256, 42790912, 79691776, 2535587840, 4766040064, 170851041280, 1617069867008, 3070050172928, 5843921666048, 256460544016384, 490390373269504, 4697678227177472, 9016382767235072
Offset: 0

Views

Author

N. J. A. Sloane, Jul 12 2005

Keywords

Comments

Conjecture: for n > 3, numerator(-2/n + Sum_{k=1..n} 2^k/k) == 0 (mod n^2) if and only if n is prime. See my formula below. Cf. A332786. - Thomas Ordowski, Mar 02 2020

Examples

			The initial values of the sum are 2, 4, 20/3, 32/3, 256/15, 416/15, 4832/105, 8192/105, 42496/315, 74752/315, 1467392/3465, 2650112/3465, 62836736/45045, 115552256/45045, 42790912/9009, 79691776/9009, 2535587840/153153, 4766040064/153153, 170851041280/2909907, ...

References

  • A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see p. 278.

Links

Crossrefs

Cf. A087910. The denominators are A229726 (repeated).

Programs

  • Mathematica
    Join[{0},Accumulate[Table[2^n/n,{n,30}]]//Numerator] (* Harvey P. Dale, Oct 28 2018 *)
  • PARI
    a(n) = numerator(sum(k=1, n, 2^k/k)); \\ Michel Marcus, Mar 07 2020

Formula

a(n) = numerator(Sum_{k=1..n} (2^k-2)/k + Sum_{k=1..n} 2/k). This formula is a heuristic of my conjecture in the comments section. Cf. A330718. - Thomas Ordowski, Mar 02 2020

Extensions

a(0) corrected by A.H.M. Smeets, Mar 06 2020

A332786 a(n) = numerator(-1/n + Sum_{k=1..n} 2^(k-1)/k).

Original entry on oeis.org

0, 3, 3, 61, 25, 137, 343, 32663, 2357, 74689, 66671, 5299069, 2416531, 115545821, 106974277, 637525199, 74575583, 1588674349, 4496071973, 3234136824109, 1535024393629, 5843920343363, 5575228585159, 1961561381531581, 93953561866435, 9016382638527647, 2888981280567587, 200248741591132607, 96525489421136333
Offset: 1

Views

Author

Thomas Ordowski, Feb 24 2020

Keywords

Comments

If p > 3 is a prime, then p^2 | a(p).
Does the above statement follow from Wolstenholme's theorem?
If p is a Wolstenholme prime (A088164), then p^3 | a(p).
However, it should be noted that also 7^3 | a(7).
Conjecture: there are no pseudoprimes m such that m^2 | a(m).
Is 7^2 the only weak pseudoprime (i.e., a composite m such that m | a(m))?

Examples

			a(5) = numerator(-1/5 + 1/1+2/2+4/3+8/4+16/5) = numerator(128/15 - 1/5) = numerator(25/3) = 25.

Links

Crossrefs

Cf. A001008, A088164, A108866, A330718.

Programs

  • Maple
    f:= proc(n) local k; numer(-1/n + add(2^(k-1)/k,k=1..n)) end proc:
map(f, [$1..30]); # Robert Israel, Sep 15 2024
  • Mathematica
    n = 30; Numerator[Accumulate @ Table[(2^(k-1))/k, {k, 1, n}] - 1/Range[n]] (* Amiram Eldar, Feb 24 2020 *)
  • PARI
    a(n) = numerator(-1/n + sum(k=1, n, 2^(k-1)/k)); \\ Michel Marcus, Feb 24 2020

Formula

a(n) = numerator(-2/n + S(n))/2 for odd n and a(n) = numerator(-2/n + S(n)) for even n, where S(n) = Sum_{k=1..n} 2^k/k, see A108866 / A229726.
a(n) = numerator(Sum_{k=1..n} (2^(k-1)-1)/k + Sum_{k=1..n-1} 1/k), see A330718 / A330719 and A001008 / A002805.

Extensions

More terms from Amiram Eldar, Feb 24 2020

A229727 Numerator of Sum_{k=1..2n+1} 2^k/k.

Original entry on oeis.org

2, 20, 256, 4832, 42496, 1467392, 62836736, 42790912, 2535587840, 170851041280, 3070050172928, 256460544016384, 4697678227177472, 52001663422038016, 5598478396465086464, 647670579588837146624, 2427980359983876276224, 9138219784535816536064, 1277040945143933271277568
Offset: 0

Views

Author

N. J. A. Sloane, Sep 28 2013

Keywords

Examples

			2, 20/3, 256/15, 4832/105, 42496/315, 1467392/3465, 62836736/45045, 42790912/9009, 2535587840/153153, 170851041280/2909907, 3070050172928/14549535, 256460544016384/334639305, 4697678227177472/1673196525, 52001663422038016/5019589575, ...

Crossrefs

Cf. A229726, A108866.

Programs

  • Mathematica
    Take[Numerator[Accumulate[Table[2^k/k,{k,51}]]],{1,-1,2}] (* Harvey P. Dale, Jun 28 2021 *)
Showing 1-3 of 3 results.

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

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