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.

A007340 Numbers whose divisors' harmonic and arithmetic means are both integers.

Original entry on oeis.org

1, 6, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Intersection of A001599 and A003601.
The following are also in A046985: 1, 6, 672, 30240, 32760. Also contains multiply perfect (A007691) numbers. - Labos Elemer
The numbers whose average divisor is also a divisor. Ore's harmonic numbers A001599 without the numbers A046999. - Thomas Ordowski, Oct 26 2014, Apr 17 2022
Harmonic numbers k whose harmonic mean of divisors (A001600) is also a divisor of k. - Amiram Eldar, Apr 19 2022

Examples

			x = 270: Sigma(0, 270) = 16, Sigma(1, 270) = 720; average divisor a = 720/16 = 45 and integer 45 divides x, x/a = 270/45 = 6, but 270 is not in A007691.

References

  • G. L. Cohen, personal communication.
  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
  • N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • D. Wells, Curious and interesting numbers, Penguin Books, p. 124.

Links

Crossrefs

Intersection of A003601 and A001599.
Different from A090945.
Cf. A001600, A007691, A046985-A046987, A046999, A054025.

Programs

  • Haskell
    a007340 n = a007340_list !! (n-1)
a007340_list = filter ((== 0) . a054025) a001599_list
-- Reinhard Zumkeller, Dec 31 2013
  • Maple
    filter:= proc(n)
uses numtheory;
local a;
a:= sigma(n)/sigma[0](n);
type(a,integer) and type(n/a,integer);
end proc:
select(filter, [$1..2500000]); # Robert Israel, Oct 26 2014
  • Mathematica
    Do[ a = DivisorSigma[0, n]/ DivisorSigma[1, n]; If[IntegerQ[n*a] && IntegerQ[1/a], Print[n]], {n, 1, 2500000}] (* Labos Elemer *)
ahmQ[n_] := Module[{dn = Divisors[n]}, IntegerQ[Mean[dn]] && IntegerQ[HarmonicMean[dn]]]; Select[Range[2500000], ahmQ] (* Harvey P. Dale, Nov 16 2011 *)
  • PARI
    is(n)=my(d=divisors(n),s=vecsum(d)); s%#d==0 && #d*n%s==0 \\ Charles R Greathouse IV, Feb 07 2017

Formula

a = Sigma(1, x)/Sigma(0, x) integer and b = x/a also.

Extensions

More terms from Robert G. Wilson v, Oct 03 2002
Edited by N. J. A. Sloane, Oct 05 2008 at the suggestion of R. J. Mathar

A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101
Offset: 1

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Author

Jaroslav Krizek, Oct 31 2019

Keywords

Comments

Numbers m such that the arithmetic mean of the divisors of m is an integer but the harmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m)/A000005(m) is an integer but H(m) = m * A000005(m)/A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 4, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of H(m): 3/2, 5/3, 7/4, 11/6, 13/7, 7/3, 5/2, 17/9, 19/10, 20/7, 21/8, 22/9, ...
Complement of A007340 with respect to A003601.

Links

Crossrefs

Cf. A000005, A000203, A003601, A007340, A046999, A328945.

Programs

  • Magma
    [m: m in [1..10^5] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))];
  • Maple
    harm:= proc(S) local s; nops(S)/add(1/s,s=S) end proc:
filter:= proc(n) local S;
  S:= numtheory:-divisors(n);
  (convert(S,`+`)/nops(S))::integer and not harm(S)::integer
end proc:
select(filter, [$1..200]); # Robert Israel, May 04 2025
  • Mathematica
    Select[Range[100], Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 01 2019 *)

A353038 Unitary harmonic numbers (A006086) that are not unitary arithmetic numbers (A103826).

Original entry on oeis.org

90, 40682250, 81364500, 105773850, 423095400, 1798155450, 14385243600
Offset: 1

Views

Author

Amiram Eldar, Apr 19 2022

Keywords

Comments

There are 290 unitary harmonic numbers below 10^12, and only 7 of them are in this sequence.

Examples

			90 is in the sequence since its unitary divisors are {1, 2, 5, 9, 10, 18, 45, 90}, their harmonic mean, 4, is an integer, but their arithmetic mean, 45/2, is not.

Crossrefs

The unitary version of A046999.
Subsequence of A006086.
Cf. A006087, A103826.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], d, s}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); IntegerQ[n*d/s] && !IntegerQ[s/d]]; Select[Range[5*10^7], q]
Showing 1-3 of 3 results.

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

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