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.

A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

Original entry on oeis.org

18, 27, 30, 45, 63, 64, 72, 99, 105, 112, 117, 144, 153, 160, 162, 165, 171, 195, 207, 225, 243, 252, 255, 261, 279, 285, 288, 294, 320, 333, 336, 345, 352, 360, 369, 387, 396, 405, 416, 423, 435, 441, 465, 468, 477, 490, 504, 531, 544, 549, 555, 567, 576
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

A special case of a bound on d(n) by Erdős and Suranyi (1960) was used to get a limit: a = x/d(x) > 0.5*sqrt(x) and below 4194304 a computer test shows these values did not occur as x = a*d(x). For larger x this is impossible since if d(x) < sqrt(x), then x/d(x) > sqrt(4194304) = 2048 > the given terms.
A051521(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
This sequence contains all numbers of the form k = 9p, p prime (i.e., k = 18, 27, 45, 63, 99, ...). - Jianing Song, Nov 25 2018

Examples

			No natural number x exists for which x = 18*d(x), so 18 is a term.

References

  • P. Erdős and J. Suranyi, Selected Topics in Number Theory, Tankonyvkiado, Budapest, 1960 (in Hungarian).
  • P. Erdős and J. Suranyi, Selected Topics in Number Theory, Springer, New York, 2003 (in English).

Links

Crossrefs

Cf. A000005, A033950, A036761, A036762, A036764, A051278, A051279, A051280, A051521.

Programs

  • Haskell
    a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011
  • Maple
    with(numtheory): A036763 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return NULL: fi: od: return n: end: seq(A036763(n),n=1..100); # Nathaniel Johnston, May 04 2011
  • Mathematica
    noSolQ[n_] := !AnyTrue[Range[4*n^2], # == DivisorSigma[0, n*#]& ];
Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* Jean-François Alcover, Jan 30 2018 *)

Extensions

Definition corrected by N. J. A. Sloane, May 18 2022 at the suggestion of David James Sycamore.

A036762 The integer values of x/d(x) in order of magnitude of x in A033950.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 4, 5, 7, 5, 6, 8, 7, 11, 8, 13, 9, 16, 11, 17, 19, 13, 10, 23, 17, 25, 19, 29, 12, 31, 14, 23, 16, 37, 41, 43, 29, 15, 31, 47, 24, 22, 53, 49, 37, 32, 25, 26, 59, 20, 61, 41, 21, 43, 67, 28, 47, 71, 73, 25, 34, 125, 79, 53, 40, 83, 28, 38, 59
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			If n=63, then x=625 and d(x) = 5 divides x. The quotient is 125 = a(63).

Links

Crossrefs

Cf. A000005, A033950, A036761, A036763, A039819.

Programs

  • Maple
    with(numtheory): A033950 := proc(n) option remember: local k: if(n=1)then return 1: else k:=procname(n-1)+1: do if(type(k/tau(k),integer))then return k: fi: k:=k+1: od: fi: end: A036762 := proc(n) return A033950(n)/tau(A033950(n)): end: seq(A036762(n),n=1..70); # Nathaniel Johnston, May 04 2011
  • Mathematica
    Select[Table[n/DivisorSigma[0, n], {n, 708}], IntegerQ] (* Michael De Vlieger, Jul 04 2016 *)

A036764 If n can be expressed as m/d(m) for some m, where d(m) is the number of divisors of m (A000005), then a(n) is the smallest such m, otherwise a(n) = 0.

Original entry on oeis.org

1, 8, 9, 36, 40, 72, 56, 80, 108, 180, 88, 240, 104, 252, 360, 128, 136, 0, 152, 480, 504, 396, 184, 384, 225, 468, 0, 560, 232, 0, 248, 448, 792, 612, 1260, 864, 296, 684, 936, 640, 328, 1680, 344, 880, 0, 828, 376, 1152, 441, 1800, 1224, 1040, 424, 972, 1980
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

If a(n) = q (say) is not zero, then x = q*d(x) has only a finite number of solutions. See A036763 for the numbers which cannot be expressible as m/d(m) for some m.
a(9p) = 0 for all primes p. - Jianing Song, Nov 25 2018

Examples

			If q=25 then 25*9 = 225, 25*18 = 450 and 25*24 = 600 so that d(225), d(450), d(600) are 9, 18, 24, respectively. The smallest is 225. Thus a(25)=225.

Links

Crossrefs

Cf. A000005, A033950, A036761-A036763, A051521.

Programs

  • Maple
    with(numtheory): A036764 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return p: fi: od: return 0: end: seq(A036764(n),n=1..40); # Nathaniel Johnston, May 04 2011

Extensions

Additional comments from Asher Auel, May 17 2001
Showing 1-3 of 3 results.

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

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