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 1-6 of 6 results.

A072910 a(n) = sigma(A023883(n)) / tau(A023883(n)).

Original entry on oeis.org

1, 3, 6, 6, 7, 8, 9, 10, 9, 12, 12, 15, 14, 12, 14, 13, 18, 19, 18, 15, 18, 15, 20, 14, 24, 21, 18, 21, 24, 18, 24, 21, 27, 33, 30, 28, 28, 32, 36, 30, 21, 26, 27, 24, 27, 38, 30, 36, 35, 45, 36, 42, 39, 26, 44, 28, 40, 51, 30, 36, 28, 48, 54, 42, 45, 38, 31, 39, 36, 48, 60, 54
Offset: 1

Views

Author

Benoit Cloitre, Aug 10 2002

Keywords

Comments

Arithmetic means of divisors of nonprime arithmetic numbers (A023883). - Amiram Eldar, Jun 06 2020

Links

Crossrefs

Cf. A000005, A000203, A003601, A102187, A023883.

Programs

  • Mathematica
    meanDiv[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; Select[meanDiv /@ Select[Range[200], !PrimeQ[#] &], IntegerQ] (* Amiram Eldar, Jun 06 2020 *)
  • PARI
    f(x) = sigma(x)/numdiv(x);
apply(x->f(x), select(x->((denominator(f(x))==1) && !isprime(x)), [1..1000])) \\ Michel Marcus, Jun 06 2020

A003601 Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).

Original entry on oeis.org

1, 3, 5, 6, 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, 102, 103, 105
Offset: 1

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Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

Sometimes called arithmetic numbers.
Generalized (sigma_r)-numbers are numbers j for which sigma_r(j)/sigma_0(j) = c^r. Sigma_r(j) denotes the sum of the r-th powers of the divisors of j; c,r are positive integers. The numbers in this sequence are sigma_1-numbers; those in A140480 are sigma_2-numbers. - Ctibor O. Zizka, Jul 14 2008
{a(n)} = union A175678 and A175679 where A175678 = numbers m such that the arithmetic mean Ad(m) of divisors of m and the arithmetic mean Ah(m) of numbers h < m such that gcd(h,m) = 1 are both integers and A175679 = numbers m such that the arithmetic mean Ad(m) of the divisors of m and the arithmetic mean Ak(m) of the numbers k <= m are both integers. - Jaroslav Krizek, Aug 07 2010
All odd primes (A065091) are arithmetic numbers. - Wesley Ivan Hurt, Oct 04 2013
A069928(n) = number of arithmetic numbers not greater than n. - Reinhard Zumkeller, Jul 28 2014
A102187(n) divides a(n) for a(n) = 1, 6, 140, 270, 672, ... A007340. - Thomas Ordowski, Oct 24 2014
The quotients sigma(j)/tau(j) are in A102187. - Bernard Schott, Jun 07 2017

Examples

			Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - _Bernard Schott_, Jun 07 2017

References

  • R. K. Guy, Unsolved Problems in Number Theory, B2.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Complement is A049642.
Cf. A000005, A000203, A054025, A001599, A007340, A140480, A102187.
Cf. A245644, A245656, A069928. Nonprimes are in A023883.

Programs

  • GAP
    a:=Filtered([1..110],n->Sigma(n) mod Tau(n)=0);; Print(a); # Muniru A Asiru, Jan 25 2019
  • Haskell
    a003601 n = a003601_list !! (n-1)
a003601_list = filter ((== 1) . a245656) [1..]
-- Reinhard Zumkeller, Jul 28 2014, Dec 31 2013, Jan 06 2012
  • Maple
    with(numtheory); t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod tau(n) = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # corrected by Wesley Ivan Hurt, Oct 03 2013
  • Mathematica
    Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] (* Stefan Steinerberger, Apr 03 2006 *)
  • PARI
    is(n)=sigma(n)%numdiv(n)==0 \\ Charles R Greathouse IV, Jul 10 2012
  • Python
    from sympy import divisors, divisor_count
[n for n in range(1,10**5) if not sum(divisors(n)) % divisor_count(n)] # Chai Wah Wu, Aug 05 2014

Formula

a(n) ~ n. - Charles R Greathouse IV, Jul 10 2012
A245656(a(n)) = 1. - Reinhard Zumkeller, Jul 28 2014

Extensions

David W. Wilson, Oct 15 1996, points out that 30 was missing.
More terms from Stefan Steinerberger, Apr 03 2006

A048747 Factor composite n, divide sum of divisors by number of divisors; append n to sequence if quotient is integral and not previously seen.

Original entry on oeis.org

6, 14, 20, 21, 22, 27, 33, 38, 39, 45, 46, 49, 57, 62, 65, 85, 86, 87, 91, 93, 94, 99, 111, 116, 118, 123, 125, 129, 133, 134, 141, 142, 150, 158, 164, 166, 169, 183, 185, 188, 201, 206, 207, 213, 214, 215, 217, 219, 236, 237, 249, 254, 259, 261, 262, 267, 278
Offset: 1

Views

Author

Enoch Haga

Keywords

Examples

			a(3)=20; for n=20 the sum of divisors is 42; number of divisors is 6; 42/6=7. The quotient 7 is the mean divisor and this is the first time that 7 has occurred.

Links

Crossrefs

Gives first occurrence of terms in A023883.
Cf. A000005, A000203, A048748.

Programs

  • Mathematica
    (Select[{Mean[Divisors[#]], #}& /@ Select[Range[300], CompositeQ], IntegerQ[#[[1]]]&] // Sort // Split[#, #1[[1]] == #2[[1]]&]&)[[All, 1]][[All, 2]] // Sort (* Jean-François Alcover, Oct 31 2017 *)

Extensions

Title clarified by Sean A. Irvine, Jul 03 2021

A048749 Factor n, divide sum of aliquot divisors by number of aliquot divisors; append n to sequence if quotient is integral and not previously seen.

Original entry on oeis.org

6, 15, 30, 33, 44, 49, 51, 69, 81, 87, 114, 117, 120, 123, 124, 141, 159, 164, 170, 177, 213, 244, 249, 252, 267, 270, 276, 282, 284, 303, 320, 321, 339, 345, 366, 393, 404, 411, 427, 447, 462, 501, 511, 513, 519, 524, 529, 534, 537, 570, 573, 590, 591, 604
Offset: 1

Views

Author

Enoch Haga

Keywords

Examples

			a(2)=15; for n=15 the sum of aliquot divisors is 9; number of aliquot divisors is 3; 9/3=3. The quotient 3 is the mean aliquot divisor and this is the first time that 3 has occurred.

Links

Crossrefs

Cf. A001065, A023883, A032741, A048747.

Programs

  • Mathematica
    (Select[{Mean[Most[Divisors[#]]], #}& /@ Select[Range[300], CompositeQ], IntegerQ[#[[1]]]&] // Sort // Split[#, #1[[1]] == #2[[1]]&]&)[[All, 1]][[All, 2]] // Sort (* Amiram Eldar, Sep 06 2019 after Jean-François Alcover at A048747 *)

A230605 Composite numbers n such that number of non-divisors of n divides sum of non-divisors of n.

Original entry on oeis.org

4, 18, 20, 24, 432, 588, 692945344
Offset: 1

Views

Author

Jaroslav Krizek, Nov 07 2013

Keywords

Comments

a(7) > 10^5.
Composite numbers n such that A231167(n) mod A049820(n) = 0. Complement of primes (A000040) with respect to A140826. Intersection of A002808 and A140826.
The terms 20 and 432 are the only composite numbers < 10^5 that are also terms of A023883.
No more terms up to 10^8. - Michel Marcus, Nov 09 2013
a(8) > 10^11. - Donovan Johnson, Nov 12 2013
a(8) > 4*10^12. - Giovanni Resta, Feb 11 2014

Examples

			20 is in sequence because A024816(20) mod A049820(20) = 168 mod 14 = 0.

Crossrefs

Cf. A231167, A000040, A049820, A024816.

Programs

  • PARI
    isok(n) = (nnd = n - numdiv(n)) && !isprime(n) && !((n*(n+1)/2-sigma(n)) % nnd); \\ Michel Marcus, Nov 09 2013

Extensions

a(7) from Donovan Johnson, Nov 10 2013

A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer.

Original entry on oeis.org

232, 2152, 3240, 3560, 3944, 6516, 17908, 22504, 23716, 26172, 32360, 34344, 36584, 37736, 43300, 45612, 48204, 55080, 55912, 60520, 61480, 69352, 73084, 78184, 79056, 79300, 96552, 104168, 105832, 106088, 125356, 130432, 133864, 140040, 149992, 163764, 168424, 172840, 176360, 183204
Offset: 1

Views

Author

Bernard Schott, Oct 14 2019

Keywords

Comments

If sigma(m)/tau(m) is a square (m is in A144695) then sigma(m)*tau(m) is also a square (m is in A327830), but the converse is false (see 232 in the Example section). This sequence consists of these counterexamples.
It seems that all terms are even. - Marius A. Burtea, Oct 15 2019

Examples

			sigma(232) = 450 and tau(232) = 8, so sigma(232)*tau(232) = 450*8 = 3600 = 60^2 and sigma(232)/tau(232) = 450/8 = 225/4 is not an integer, hence 232 is a term.

Crossrefs

Equals A144695 \ A327830.
Similar to A327624 with sigma(m) and phi(m).
Cf. A003601 (sigma(m)/tau(m) is an integer), A023883 (sigma(m)/tau(m) is an integer and m is nonprime).
Cf. A000005 (tau), A000203 (sigma).

Programs

  • Magma
    [k:k in [1..200000] | not IsIntegral(a/b) and IsSquare(a*b) where a is DivisorSigma(1,k) where b is #Divisors(k)]; // Marius A. Burtea, Oct 15 2019
  • Maple
    filter:= u -> sigma(u)/tau(u) <> floor(sigma(u)/tau(u)) and issqr(sigma(u)*tau(u)) : select(filter, [$1..100000]);
  • Mathematica
    sQ[n_] := IntegerQ@Sqrt[n]; aQ[n_] := sQ[(d = DivisorSigma[0, n]) * (s = DivisorSigma[1, n])] && !sQ[s/d]; Select[Range[2*10^5], aQ] (* Amiram Eldar, Oct 15 2019 *)
  • PARI
    isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ Michel Marcus, Oct 15 2019
Showing 1-6 of 6 results.

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