A057020 -id:A057020 - 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

A057021 Denominator of (sum of divisors of n / number of divisors of n).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 3, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 7, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Jul 21 2000

a(n) = 1 when n is listed in A003601, a(n) > 1 when n is listed in A049642. - Alonso del Arte, Jan 31 2006

a(A069081(n)) = 2. - Bernard Schott, Sep 19 2019

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

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

Cf. A000005, A000203, A009205, A054025, A057020 (numerator), A057022, A069081.

import Data.Ratio ((%), denominator)
a057021 n = denominator $ a000203 n % a000005 n
-- Reinhard Zumkeller, Jan 06 2012

[Denominator(SumOfDivisors(n)/#Divisors(n)):n in [1..100]]; // Marius A. Burtea, Sep 08 2019

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

Denominator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* Alonso del Arte, Feb 24 2006 *)

a(n) = denominator(sigma(n)/numdiv(n)); \\ Michel Marcus, Apr 12 2016

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

a(n) = A057020(n)*A000005(n)/A000203(n) = A000005(n)/A009205(n).

A336838 Numerator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 2, 3, 13, 4, 6, 6, 10, 31, 8, 7, 13, 9, 12, 12, 121, 10, 62, 12, 52, 18, 14, 15, 30, 19, 18, 39, 26, 16, 24, 19, 182, 21, 20, 24, 403, 21, 24, 27, 40, 22, 36, 24, 91, 124, 30, 27, 363, 133, 38, 30, 39, 30, 78, 28, 60, 36, 32, 31, 52, 34, 38, 62, 1093, 36, 42, 36, 130, 45, 48, 37, 310, 40, 42, 57, 52, 42, 54, 42

Offset: 1

Antti Karttunen, Aug 07 2020

Ratio r(n) = a(n)/A336839(n) is multiplicative. For example r(3) = 3/1, r(4) = 13/3, thus r(12) = r(3)*r(4) = 13/1.

Conjecture: For all primes p with an odd exponent e, a(p^e) is a multiple of A048673(p). Note that q+1 is a divisor of (q+1)^e - sigma(q^e) = (q+1)^e - (1 + q + q^2 + ... + q^e) when e is odd, thus also A048673(p) = (q+1)/2 is, where q = A003961(p), thus the conjecture holds, unless the denominator (A336839) has enough prime factors of A048673(p).

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A003961, A003973, A048673, A057020, A336840, A336918.

Cf. A336839 (denominators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336838(n) = numerator(sigma(A003961(n))/numdiv(n));

a(n) = A057020(A003961(n)).
a(n) = numerator(A003973(n)/A000005(n)).
a(n) = A003973(n) / A336856(n) = A003973(n) / gcd(A000005(n), A003973(n)).
a(p) = A048673(p) for all primes p.
a(p^3) = 2*A048673(p)^3 - 2*A048673(p)^2 + A048673(p). [The denominator A336839(p^3) = 1 for all p]

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).

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

A322668 Numbers k such that (k, k+2) are not twin primes yet sigma(k+2)/d(k+2) - sigma(k)/d(k) = 1.

Original entry on oeis.org

1, 350, 6497, 12317, 133787, 181427, 404471, 439097, 485237, 501182, 549378, 1410119, 2696807, 6220607, 6827369, 6954767, 9770027, 10302419, 10449347, 10887977, 11014007, 16745387, 18959111, 25883519, 27334469, 39508037, 40311149, 40551617, 42561437, 44592209

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

A variation of A050507 with average of the divisors instead of their sum.

Amiram Eldar, Table of n, a(n) for n = 1..241 (terms below 10^10)

Cf. A000005 (number of divisors), A000203 (sum of divisors).

Cf. A050507, A056774, A057020, A057021, A238380.

f[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; aQ[n_] := f[n + 2] - f[n] ==  1 && !(PrimeQ[n] && PrimeQ[n + 2]); Select[Range[1000000], aQ]

isok(k) = !(isprime(k) && isprime(k+2)) && (sigma(k+2)/numdiv(k+2) - sigma(k)/numdiv(k) == 1); \\ Michel Marcus, Jan 22 2019

A348713 Numbers whose divisors can be partitioned into two disjoint sets with equal arithmetic mean.

Original entry on oeis.org

6, 20, 24, 30, 42, 48, 54, 56, 60, 66, 70, 72, 78, 84, 88, 90, 96, 102, 108, 114, 120, 126, 132, 135, 138, 140, 150, 156, 160, 168, 174, 180, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 228, 230, 234, 240, 246, 252, 258, 260, 264, 270, 273, 276

Offset: 1

Amiram Eldar, Oct 31 2021

nonn

The arithmetic mean of each of the two subsets is equal to the arithmetic mean of all the divisors of the number.

Also, numbers whose divisors can be partitioned into two disjoint sets with equal harmonic mean. This definition is equivalent since the harmonic mean of a subset {d_i} of the divisors of k is equal to k/, where is the arithmetic mean over the complementary divisors k/d_i.

			6 is a term since its set of divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets, {3} and {1, 2, 6}, whose arithmetic means are both 3.

Amiram Eldar, Table of n, a(n) for n = 1..872

Cf. A027750, A057020, A057021, A083207.

A347063 is a subsequence.

q[n_] := Module[{d = Divisors[n], nd, m, s, subs, ans = False}, nd = Length[d]; m = Plus @@ d/nd; subs = Subsets[d]; Do[s = subs[[k]]; If[0 < Length[s] < nd && Mean[s] == m, ans = True; Break[]], {k, 1, Length[subs]}]; ans]; Select[Range[300], q]

A348718 Numbers whose divisors can be partitioned into two disjoint sets without singletons whose arithmetic means are both integers.

Original entry on oeis.org

6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Amiram Eldar, Oct 31 2021

nonn

First differs from A343311 at n = 29.

Differs from A080257 which contains for example 8 and 128. - R. J. Mathar, Nov 03 2021

			6 is a term since its set of divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets {1, 3} and {2, 6} whose arithmetic means, 2 and 4 respectively, are both integers.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A003601, A027750, A057020, A057021, A083207, A343311, A348715.

amQ[d_] := IntegerQ @ Mean[d]; q[n_] := Module[{d = Divisors[n], nd, s, subs, ans = False}, nd = Length[d]; subs = Subsets[d]; Do[s = subs[[k]]; If[Length[s] > 1 && Length[s] <= nd/2 && amQ[s] && amQ[Complement[d, s]], ans = True; Break[]], {k, 1, Length[subs]}]; ans]; Select[Range[100], q]

cp's OEIS Frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

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A057021 Denominator of (sum of divisors of n / number of divisors of n).

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A336838 Numerator 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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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.

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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.

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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.

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