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.

A211777 Numbers n such that Sum_{d_A000010(x).

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 80, 81, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 189, 191, 193, 196, 197, 199, 211, 223, 227, 229, 233
Offset: 1

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Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

Union primes (A000040) and A211778.

Examples

			For n = 32 holds: 1/1 + 1/2 + 2/4 + 4/8 + 8/16 = 3 (integer).

Crossrefs

Cf: A066862 (numbers n such that Sum_{d | n} phi(d) / d is an integer).

Programs

  • Mathematica
    t = {}; Do[d2 = Sum[EulerPhi[d]/d, {d, Most[Divisors[n]]}]; If[IntegerQ[d2], AppendTo[t, n]], {n, 2, 233}]; t (* T. D. Noe, Apr 26 2012 *)
  • PARI
    is(n)=denominator(sumdiv(n,d,if(dCharles R Greathouse IV, Feb 21 2013

A211778 Composite numbers n such that Sum_{d_A000010(x).

Original entry on oeis.org

8, 12, 32, 80, 81, 128, 189, 196, 324, 336, 448, 512, 576, 768, 1600, 1936, 2025, 2048, 2187, 2500, 5292, 5324, 5616, 5780, 8192, 8748, 9477, 11264, 12096, 13520, 14400, 14800, 15552, 15625, 20736, 32768, 35721, 36864, 49152, 53248, 59049, 69696, 73575, 73872
Offset: 1

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

Complement of primes (A000040) with respect to A211777.

Examples

			For n = 32 holds: 1/1 + 1/2 + 2/4 + 4/8 + 8/16 = 3 (integer).

Crossrefs

Cf. A066862 (numbers n such that Sum_{d | n} phi(d) / d is an integer).

Programs

  • Mathematica
    t = {}; Do[If[! PrimeQ[n], d2 = Sum[EulerPhi[d]/d, {d, Most[Divisors[n]]}]; If[IntegerQ[d2], AppendTo[t, n]]], {n, 2, 10000}]; t (* T. D. Noe, Apr 26 2012 *)
  • PARI
    is(n)=denominator(sumdiv(n, d, if(dCharles R Greathouse IV, Mar 05 2013

Extensions

Extended by T. D. Noe, Apr 26 2012

A353264 a(n) is the least number k such that A018804(k)/k = n.

Original entry on oeis.org

1, 4, 15, 64, 48, 60, 144, 16384, 240, 1300, 1296, 960, 1008, 3564, 3840, 1073741824, 6000, 14580, 7056, 20800, 11520, 25500, 944784, 245760, 13104, 24948, 34560, 57024, 750000, 16380, 156816, 4611686018427387904, 102000, 364500, 46800, 233280, 134064, 174636
Offset: 1

Views

Author

Amiram Eldar, Apr 09 2022

Keywords

Comments

a(n) exist for all n>=1.
The solution k to A018804(k)/k = n is unique if and only if n is a power of 2: a(2^m) = 2^(2^(m+1)-2) = A051191(m+1).
The questions of existence and uniqueness are a part of a problem that was proposed during the Forty-Fifth International Mathematical Olympiad in Athens, Greece, July 7-19, 2004.

Examples

			a(2) = 4 since A018804(4)/4 = 8/4 = 2 and 4 is the least number with this property.

References

  • Dušan Djukić, Vladimir Janković, Ivan Matić, and Nikola Petrović, The IMO Compendium, A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004, Springer, New York, 2006. See Problem 25, pp. 331, 726-727.

Links

Crossrefs

Cf. A018804, A051191.
Subsequence of A066862.

Programs

  • Mathematica
    f[p_, e_] := (e*(p - 1)/p + 1); r[n_] := Times @@ (f @@@ FactorInteger[n]); a[n_] := Module[{k = 1}, While[r[k] != n, k++]; k]; Array[a, 15]
Showing 1-3 of 3 results.

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

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