A057021 -id:A057021 - cp's OEIS frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 14, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 16, 18, 17, 68, 21, 26

Offset: 1

Floor van Lamoen, Jul 24 2001

nonn

Similar to A027709, which is minimal perimeter of polyomino of n cells, or equivalently, minimal perimeter of rectangle of area at least n and with integer sides. Present sequence is minimal semiperimeter of rectangle with area exactly n and with integer sides. - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 03 2002
Semiperimeter b+d, d >= b, of squarest (smallest d-b) integral rectangle with area bd = n. That is, b = largest divisor of n <= sqrt(n), d = smallest divisor of n >= sqrt(n). a(n) = n+1 iff n is noncomposite (1 or prime). - Daniel Forgues, Nov 22 2009
From Juhani Heino, Feb 05 2019: (Start)
Basis for any thickness "frames" around the minimal area. Perimeter can be thought as the 0-thick frame, it is obviously 2a(n). Thickness 1 is achieved by laying unit tiles around the area, there are 2(a(n)+2) of them. Thickness 2 comes from the second such layer, now there are 4(a(n)+4) and so on. They all depend only on a(n), so they share this structure:
Every n > 1 is included. (For different thicknesses, every integer that can be derived from these with the respective formula. So, the perimeter has every even n > 2.)
For each square n > 1, a(n) = a(n-1).
a(1), a(2) and a(6) are the only unique values - the others appear multiple times.
(End)
Gives a discrete Uncertainty Principle. A complex function on an abelian group of order n and its Discrete Fourier Transform must have at least a(n) nonzero entries between them. This bound is achieved by the indicator function on a subgroup of size closest to sqrt(n). - Oscar Cunningham, Oct 10 2021
Also two times the median divisor of n, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The version for mean instead of median is A057020/A057021. Other doubled medians of multisets are: A360005 (prime indices), A360457 (distinct prime indices), A360458 (distinct prime factors), A360459 (prime factors), A360460 (prime multiplicities), A360555 (0-prepended differences). - Gus Wiseman, Mar 18 2023

			Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.

T. D. Noe, Table of n, a(n) for n = 1..10000
Roy Meshulam, An uncertainty inequality for finite abelian groups, arXiv:math/0312407 [math.CO], 2003.
Roy Meshulam, An uncertainty inequality for finite abelian groups, European Journal of Combinatorics, 27 (2006) 63-67.

Cf. A027709, A033676, A033677, A092510, A162348.
Positions of odd terms are A139710.
Positions of even terms are A139711.
A000005 counts divisors, listed by A027750.
A000975 counts subsets with integer median.
Cf. A003601, A026424, A057020/A057021, A325347, A359893, A359901.

A063655 := proc(n)
    local i,j;
    for i from floor(sqrt(n)) to 1 by -1 do
        j := floor(n/i) ;
        if i*j = n then
            return i+j;
        end if;
    end do:
end proc:
seq(A063655(n), n=1..80); # Winston C. Yang, Feb 03 2002

Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 2*Sqrt[n], d[[len/2]] + d[[1 + len/2]]], {n, 100}] (* T. D. Noe, Mar 06 2012 *)
Table[2*Median[Divisors[n]],{n,100}] (* Gus Wiseman, Mar 18 2023 *)

A063655(n) = { my(c=1); fordiv(n,d,if((d*d)>=n,if((d*d)==n,return(2*d),return(c+d))); c=d); (0); }; \\ Antti Karttunen, Oct 20 2017

from sympy import divisors
def A063655(n):
    d = divisors(n)
    l = len(d)
    return d[(l-1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019

a(n) = A033676(n) + A033677(n).
a(n) = A162348(2n-1) + A162348(2n). - Daniel Forgues, Sep 29 2014
a(n) = Min_{d|n} (n/d + d). - Ridouane Oudra, Mar 17 2024

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001

A057020 Numerator of (sum of divisors of n / number of divisors of n).

Original entry on oeis.org

1, 3, 2, 7, 3, 3, 4, 15, 13, 9, 6, 14, 7, 6, 6, 31, 9, 13, 10, 7, 8, 9, 12, 15, 31, 21, 10, 28, 15, 9, 16, 21, 12, 27, 12, 91, 19, 15, 14, 45, 21, 12, 22, 14, 13, 18, 24, 62, 19, 31, 18, 49, 27, 15, 18, 15, 20, 45, 30, 14, 31, 24, 52, 127, 21, 18, 34, 21, 24, 18

Offset: 1

Henry Bottomley, Jul 21 2000

Numerator of arithmetic mean of the divisors of n. - Jaroslav Krizek, Apr 26 2010

The average order of a(n)/A057021(n) is asymptotic to n/sqrt(log(n)); see the Bateman et al. link or the Sutantyo link. - Charles R Greathouse IV, May 17 2012

			a(12) = 14 since the 6 factors of 12 are 1, 2, 3, 4, 6 and 12 and 1 + 2 + 3 + 4 + 6 + 12 = 28 and 28/6 = 14/3.

V. I. Arnold, Dynamics, Statistics, and Projective Geometry of Galois Fields, Cambridge University Press, Cambridge, 2011, p. 78.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Arnold, Number-theoretical turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics, Journal of Mathem. Fluid Mechanics 7 (2005), pp. S4-S50.
Paul T. Bateman, Paul Erdős, Carl Pomerance, and Ernst G. Straus, The arithmetic mean of the divisors of an integer in Analytic Number Theory (1980), pp. 197-220.
Marcin Mazur and Bogdan V. Petrenko, Representations of analytic functions as infinite products and their application to numerical computations, The Ramanujan Journal, Vol. 34, No. 1 (2014), pp. 129-141; arXiv preprint, arXiv:1202.1335 [math.NT], 2012.
Daniel Sutantyo, Elementary and Analytic Methods in Number Theory, M.S. thesis (Macquarie University, 2007), chapter 3. [Wayback Machine link]

Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022, A308051.

import Data.Ratio ((%), numerator)
a057020 n = numerator $ a000203 n % a000005 n
-- Reinhard Zumkeller, Jan 06 2012

with(numtheory): seq(numer(sigma(n)/tau(n)), n=1..70) ; # Zerinvary Lajos, Jun 04 2008

Numerator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* Alonso del Arte, Feb 24 2006 *)
Table[Numerator[DivisorSigma[1,n]/DivisorSigma[0,n]],{n,100}] (* Harvey P. Dale, Dec 19 2023 *)

a(n)=numerator(sigma(n)/numdiv(n)) \\ Charles R Greathouse IV, May 17 2012

[numerator(sigma(n, 1)/sigma(n, 0)) for n in range(1, 71)] # Stefano Spezia, Jul 18 2025

a(n) = A057021(n) * A000203(n)/A000005(n) = A000203(n)/A009205(n) = (A057022(n) + A054025(n)/A000005(n)) * A057021(n).

Sum_{k=1..n} a(k)/A057021(k) ~ c * n^2 /sqrt(log(n)), where c = A308051. - Amiram Eldar, Apr 15 2025

A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 07 2020

Also denominator of A336841(n) / A000005(n).

All terms are odd because A336932(n) = A007814(A003973(n)) >= A295664(n) for all n.

Cf. A000005, A000225, A003961, A003973, A007814, A052940, A057021, A295664, A336840, A336841, A336856, A336931, A336932.
Cf. A336918 (positions of 1's), A336919 (of terms > 1).
Cf. A336837 and A336838 (numerators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336839(n) = denominator(sigma(A003961(n))/numdiv(n));

a(n) = denominator(A003973(n)/A000005(n)).
a(n) = d(n)/A336856(n) = d(n)/gcd(d(n),A003973(n)) = d(n)/gcd(d(n),A336841(n)), where d(n) is the number of divisors of n, A000005(n).
a(n) = A057021(A003961(n)).
For all primes p, and e >= 0, a(A000225(e)) = a(p^((2^e) - 1)) = 1. [See A336856]
It seems that for all odd primes p, and with the exponents e=5, 11, 17 or 23 (at least these), a(p^e) = 1.
It seems that a(27^((2^n)-1)) = A052940(n-1) for all n >= 1.

A057022 a(n) = floor((sum of divisors of n) / (number of divisors of n)), or floor(sigma_1(n)/sigma_0(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 4, 4, 6, 4, 7, 6, 6, 6, 9, 6, 10, 7, 8, 9, 12, 7, 10, 10, 10, 9, 15, 9, 16, 10, 12, 13, 12, 10, 19, 15, 14, 11, 21, 12, 22, 14, 13, 18, 24, 12, 19, 15, 18, 16, 27, 15, 18, 15, 20, 22, 30, 14, 31, 24, 17, 18, 21, 18, 34, 21, 24, 18

Offset: 1

Henry Bottomley, Jul 21 2000

nonn

Floor of mean of divisors of n. - Jon E. Schoenfield, Dec 24 2016

			a(4)=2 since the 3 divisors of 4 are 1, 2 and 4 and floor((1 + 2 + 4)/3) = floor(7/3) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A009205, A054025, A057020, A057021.

Cf. A003601.

a057022 n = a000203 n `div` a000005 n
-- Reinhard Zumkeller, Jan 06 2012

Floor[Table[Total[Divisors[n]]/Length[Divisors[n]],{n,20}]] (* Daniel Jolly, Nov 15 2014 *)
Table[Floor[DivisorSigma[1,n]/DivisorSigma[0,n]],{n,70}] (* Harvey P. Dale, Jan 14 2015 *)

a(n) = sigma(n)\numdiv(n); \\ Michel Marcus, Nov 15 2014

a(n) = (A000203(n) - A054025(n))/A000005(n).

A069081 Numbers n such that sigma(n)/tau(n) has denominator 2.

Original entry on oeis.org

2, 10, 18, 24, 26, 32, 34, 50, 58, 74, 82, 88, 90, 98, 106, 120, 122, 130, 146, 152, 160, 170, 178, 194, 202, 216, 218, 226, 234, 242, 250, 274, 288, 290, 298, 306, 312, 314, 338, 344, 346, 362, 370, 386, 394, 408, 410, 416, 440, 442, 458, 466, 472, 482, 490

Offset: 1

Benoit Cloitre, Apr 05 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A057021.

[n:n in [1..500]|Denominator(SumOfDivisors(n)/#Divisors(n)) eq 2]; // Marius A. Burtea, Sep 08 2019

Select[Range[500], Denominator[DivisorSigma[1, #]/DivisorSigma[0, #]] == 2 &] (* Amiram Eldar, Sep 08 2019 *)

n such that A057021(n)=2.

A348968 a(n) = gcd(n, A099377(n)), where A099377(n) is the numerator of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 18, 19, 20, 21, 22, 23, 8, 25, 26, 27, 1, 29, 10, 31, 32, 11, 34, 35, 36, 37, 38, 39, 8, 41, 7, 43, 22, 45, 23, 47, 24, 49, 50, 17, 52, 53, 18, 55, 56, 57, 58, 59, 30, 61, 31, 63, 64, 65, 11, 67, 68, 23, 35, 71, 72, 73, 74, 75, 38, 77, 26, 79, 80, 81, 82, 83, 3

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

Cf. A057021, A099377, A348510, A348969.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A099377(n) = { my(d=divisors(n)); numerator(#d/sum(k=1, #d, 1/d[k])); }; \\ From A099377
A348968(n) = gcd(n, A099377(n));

a(n) = gcd(n, A099377(n)) = gcd(n, A348510(n)) = gcd(A099377(n), A348510(n)).
a(n) = n / A348969(n).
a(n) = A099377(n) / A057021(n). [Apparently, holds at least up to n = 2^25]

A348969 a(n) = n / gcd(n, A099377(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 28, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 6, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 28, 1, 1, 3, 1, 1, 3, 7, 4, 1, 2, 5, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

Cf. A057021, A099377, A348510, A348968.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A099377(n) = { my(d=divisors(n)); numerator(#d/sum(k=1, #d, 1/d[k])); }; \\ From A099377
A348969(n) = (n / gcd(n, A099377(n)));

a(n) = n / A348968(n) = n / gcd(n, A099377(n)).

A175678 Numbers m such that the arithmetic mean Ad(m) of the divisors of m and the arithmetic mean Ah(m) of the numbers h < m such that gcd(h,m) = 1 are both integer.

Original entry on oeis.org

1, 6, 14, 20, 22, 30, 38, 42, 44, 46, 54, 56, 60, 62, 66, 68, 70, 78, 86, 92, 94, 96, 102, 110, 114, 116, 118, 126, 132, 134, 138, 140, 142, 150, 154, 158, 164, 166, 168, 174, 182, 184, 186, 188, 190, 198, 204, 206, 210, 212, 214, 220, 222, 224, 230, 236, 238, 246, 248, 254, 258

Offset: 1

Jaroslav Krizek, Aug 07 2010

nonn

Numbers m such that Ad(m) = A000203(m)/A000005(m) = A057020(m)/A057021(m) and Ah(m) = A023896(m)/A000010(m) = A026741(m)/A040001(m-1) are both integers.
{a(n)} = 1 and the even arithmetic numbers from A003601.
Subsequence of A003601.
Union of {a(n)} and A175679 = A003601 (arithmetic numbers).

			a(2) = 6, Ad(6) = (1+2+3+6)/4 = 3, Ah(6) = (1+5)/2 = 3, Ad(6) and Ah(6) are both integer.

Cf. A000203, A000005, A057020, A057021.
CF. A023896, A000010, A026741, A040001.
Cf. A003601, A175679.

Ad(n) = sigma(n)/numdiv(n);
Ah(n) = if(n<2, n>0, n*eulerphi(n)/2)/eulerphi(n);
isok(k) = (type(Ad(k)) == "t_INT") && (type(Ah(k)) == "t_INT"); \\ Michel Marcus, Oct 09 2023

More terms from Michel Marcus, Oct 09 2023

A175679 Numbers m such that arithmetic mean Ad(m) of divisors of m and arithmetic mean Ak(m) of numbers 1 <= k <= m are both integer.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141

Offset: 1

Jaroslav Krizek, Aug 07 2010

nonn

Numbers m such that Ad(m) = A000203(m) / A000005(m) = A057020(m) / A057021(m) and Ak(m) = A000217(m) / A000027(m) = A145051(m) / A040001(m) are both integer.
Subsequence of A003601: {a(n)} = odd arithmetic numbers from A003601.
{a(n)} union A175678 = A003601 (arithmetic numbers).
From Robert G. Wilson v, Aug 09 2010: (Start)
All terms are odd because the second criterion is equivalent to n|T(n), where T(n) is the n-th triangular number, A000217(n).
Terms that are not prime are 1, 15, 21, 27, 33, 35, 39, 45, 49, 51, 55, 57, 65, 69, 77, 85, ..., .
Odd integers that are not terms: 9, 25, 63, 75, 81, 117, 121, 171, 175, 225, 243, 279, 289, ..., . (End)

			a(4) = 7, Ad(7) = (1+7)/2 = 4, Ak(7) = (1+2+3+4+5+6+7)/7 = 4, Ad(7) and Ak(7) are both integer.

fQ[n_] := OddQ@n && Mod[DivisorSigma[1, n], DivisorSigma[0, n]] == 0; Select[ Range@ 142, fQ] (* Robert G. Wilson v, Aug 09 2010 *)

More terms from Robert G. Wilson v, Aug 09 2010

A176801 a(n) is the smallest natural number m such that product of arithmetic mean of the divisors of n and arithmetic mean of the divisors of m is an integer.

Original entry on oeis.org

1, 3, 1, 5, 1, 1, 1, 7, 5, 3, 1, 2, 1, 1, 1, 19, 1, 3, 1, 1, 1, 1, 1, 3, 5, 3, 1, 2, 1, 1, 1, 3, 1, 3, 1, 17, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 19, 1, 3, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 13, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 2, 2, 1, 1, 1, 19, 19, 3, 1, 2, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 17, 1, 1, 1, 7, 1

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

a(n) = 1 for infinitely many n.
a(n) = 1 for numbers from A003601: a(A003601(n)) = 1.
a(n) = 1 iff A057021(n) = 1.
Not all terms are 1's or primes. For example, a(128) = 21. - Antti Karttunen, Dec 24 2018

			For n = 12; b(12) = 14/3, a(n) = 2 because b(2) = 3/2; 14/3 * 3/2 = 7 (integer).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005/A000203 or A057020/A057021: arithmetic mean.

Cf. A003601, A245656.

A176801(n) = { my(am=(sigma(n)/numdiv(n))); for(i=1, oo, if(1==denominator(am*(sigma(i)/numdiv(i))), return(i))); }; \\ Antti Karttunen, Dec 24 2018

More terms from Antti Karttunen, Dec 24 2018

cp's OEIS Frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

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A057020 Numerator of (sum of divisors of n / number of divisors of n).

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A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

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A057022 a(n) = floor((sum of divisors of n) / (number of divisors of n)), or floor(sigma_1(n)/sigma_0(n)).

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A069081 Numbers n such that sigma(n)/tau(n) has denominator 2.

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A348968 a(n) = gcd(n, A099377(n)), where A099377(n) is the numerator of the harmonic mean of the divisors of n.

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A348969 a(n) = n / gcd(n, A099377(n)).

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