A057021 -id:A057021 - cp's OEIS frontend

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

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]

A069082 Numbers n such that sigma(n)/tau(n) has denominator 3.

Original entry on oeis.org

4, 9, 12, 25, 28, 52, 63, 75, 76, 84, 108, 117, 121, 124, 148, 156, 171, 172, 175, 180, 196, 228, 243, 244, 268, 279, 289, 292, 316, 325, 333, 363, 364, 372, 387, 388, 396, 412, 436, 441, 444, 475, 508, 516, 525, 529, 532, 549, 556, 588, 603, 604, 628, 652

Offset: 1

Benoit Cloitre, Apr 05 2002

Numbers such that the denominator of the fraction (sum of the divisors of n) divided by (the number of divisors of n) equals 3. - Harvey P. Dale, Jul 13 2012

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

Cf. A000005, A000203, A057021.

[n:n in [1..700]|Denominator(SumOfDivisors(n)/#Divisors(n)) eq 3]; // Marius A. Burtea, Sep 08 2019

Select[Range[700],Denominator[DivisorSigma[1,#]/DivisorSigma[0,#]] == 3&] (* Harvey P. Dale, Jul 13 2012 *)

n such that A057021(n)=3.

A069083 Numbers n such that sigma(n)/tau(n) has denominator 4.

Original entry on oeis.org

8, 40, 72, 104, 136, 200, 232, 296, 328, 360, 384, 392, 424, 488, 520, 584, 648, 680, 712, 776, 808, 872, 904, 936, 968, 1000, 1096, 1160, 1192, 1224, 1256, 1352, 1384, 1408, 1448, 1480, 1544, 1576, 1640, 1768, 1832, 1864, 1920, 1928, 1960, 2048, 2056

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..2100]|Denominator(SumOfDivisors(n)/#Divisors(n)) eq 4]; // Marius A. Burtea, Sep 08 2019

Select[Range[500], Denominator[DivisorSigma[1, #]/DivisorSigma[0, #]] == 4 &] (* Amiram Eldar, Sep 08 2019 *)

n such that A057021(n)=4.

A069084 Numbers n such that sigma(n)/tau(n) has denominator 5.

Original entry on oeis.org

16, 48, 80, 81, 112, 176, 208, 240, 272, 336, 368, 405, 496, 528, 560, 567, 592, 624, 625, 656, 688, 720, 752, 784, 816, 848, 880, 891, 976, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1232, 1328, 1360, 1377, 1456, 1488, 1536, 1552, 1584, 1616, 1620, 1648

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..1700]|Denominator(SumOfDivisors(n)/#Divisors(n)) eq 5]; // Marius A. Burtea, Sep 08 2019

Select[Range[500], Denominator[DivisorSigma[1, #]/DivisorSigma[0, #]] == 5 &] (* Amiram Eldar, Sep 08 2019 *)

n such that A057021(n)=5

A308051 Decimal expansion of lim_{m->oo} (sqrt(log(m))/m^2) Sum_{k=1..m} sigma(k)/d(k), where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

Original entry on oeis.org

3, 5, 6, 9, 0, 4, 9, 6, 5, 2, 4, 9, 9, 5, 7, 0, 7, 6, 1, 2, 2, 0, 0, 5, 3, 0, 2, 0, 1, 3, 9, 9, 6, 4, 5, 9, 1, 3, 6, 0, 6, 6, 6, 8, 2, 6, 2, 5, 7, 3, 8, 4, 4, 2, 9, 6, 8, 7, 8, 8, 0, 2, 0, 1, 2, 7, 7, 4, 3, 4, 4, 2, 1, 4, 1, 8, 7, 2, 1, 3, 8, 5, 5, 3, 2, 1, 5

Offset: 0

Amiram Eldar, May 10 2019

			0.35690496524995707612200530201399645913606668262573...

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

Paul T. Bateman, Paul Erdös, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer, in: Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol 899, Springer, Berlin, Heidelberg, 1981, pp. 197-220, alternative link.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 162.
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.

Cf. A000005, A000203, A057020, A057021.

$MaxExtraPrecision = 1000; m = 1000; f[x_] := Log[1 + x]/x/Sqrt[1 - x]; c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[(1/2/ Sqrt[Pi])*Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals (1/(2*sqrt(Pi))) * Product_{p prime} p^(3/2) * log(1 + 1/p) / sqrt(p-1).

A375080 a(n) is the numerator of ( Sum_{d|n} (n - d) )/tau(n).

Original entry on oeis.org

0, 1, 1, 5, 2, 3, 3, 17, 14, 11, 5, 22, 6, 8, 9, 49, 8, 23, 9, 13, 13, 13, 11, 33, 44, 31, 17, 56, 14, 21, 15, 43, 21, 41, 23, 233, 18, 23, 25, 115, 20, 30, 21, 30, 32, 28, 23, 178, 30, 69, 33, 107, 26, 39, 37, 41, 37, 71, 29, 46, 30, 38, 137, 321, 44, 48, 33, 47, 45, 52

Offset: 1

Stefano Spezia, Jul 29 2024

( Sum_{d|n} (n - d) )/tau(n) is the average distance between n and its divisor.

Cf. A000005, A000203, A057020, A057021 (denominator).

a[n_]:=Numerator[n-DivisorSigma[1,n]/DivisorSigma[0,n]];  Array[a,70]

from math import prod
from fractions import Fraction
from sympy import factorint
def A375080(n):
    f = factorint(n).items()
    return (n-Fraction(prod((p**(e+1)-1)//(p-1) for p, e in f),prod(e+1 for p,e in f))).numerator # Chai Wah Wu, Jul 30 2024

a(n) = numerator((n - sigma(n))/tau(n)).
a(n) = numerator(n - A000203(n)/A000005(n)).
a(n) = numerator(n - A057020(n)/A057021(n)).

A346644 Least k >= 1 such that sigma(k)/tau(k) has denominator n or zero if no k exists.

Original entry on oeis.org

1, 2, 4, 8, 16, 450, 64, 128, 36, 162, 1024, 1800, 4096, 1458, 144, 32768, 65536, 54450, 262144, 405000, 576, 118098, 4194304, 28800, 1296, 1062882, 900, 5832, 268435456, 115200, 1073741824, 2147483648, 9216, 86093442, 5184, 217800, 68719476736, 774840978, 102400

Offset: 1

Benoit Cloitre, Jul 26 2021

nonn

Conjecture: k always exists.

Cf. A057020, A057021, A069081.

seq[max_] := Module[{s = Table[0,{max}], c = 0, n = 1}, While[c < max, d = Denominator[DivisorSigma[1,n]/DivisorSigma[0,n]]; If[d <= max && s[[d]] == 0, c++; s[[d]] = n]; n++]; s]; seq[22] (* Amiram Eldar, Jul 26 2021 *)

a(n)=if(n<0, 0, t=1; while(denominator(sigma(t)/numdiv(t))!=n, t++); t)

a(29)-a(36) from Amiram Eldar, Jul 26 2021
a(37) from David A. Corneth, Jul 26 2021
a(38)-a(39) from Jinyuan Wang, Jul 26 2021

A347077 Numbers m such that sigma(m) / tau(m) = sigma(m - 1) / tau(m - 1) + sigma(m + 1) / tau(m + 1).

Original entry on oeis.org

15063, 18519, 49841, 137607, 179943, 203345, 412763, 421307, 517334, 881851, 1102204, 2003233, 2831435, 3869018, 17378593, 76645063, 107594182, 118012619, 190791881, 418588841, 447287713, 475734745, 632799289, 661709127, 664171759, 900701138, 998754443, 1756922665

Offset: 1

Jaroslav Krizek, Aug 15 2021

nonn

Numbers m such that A057020(m) / A057021(m) = A057020(m - 1) / A057021(m - 1) + A057020(m + 1) / A057021(m + 1).

Corresponding values of fractions sigma(m) / tau(m): 5022, 6174, 7128, 45870, 59982, 31008, 111132, 106680, 99636, 220948, 163044, 263160, 449712, 726864, 2278152, ...

			sigma(15063) / tau(15063) = sigma(15062) / tau(15062) + sigma(15064) / tau(15064); 20088 / 4 = 23976 / 8 + 32400 / 16; 5022 = 2997 + 2025.

Cf. A000005 (tau), A000203 (sigma), A057020, A057021.

[m: m in [2..10^6] | (&+Divisors(m) / #Divisors(m)) eq (&+Divisors(m - 1) / #Divisors(m - 1)) + (&+Divisors(m + 1) / #Divisors(m + 1))]

r[n_] := Divide @@ DivisorSigma[{1, 0}, n]; s = {}; r1 = r[1]; r2 = r[2]; Do[r3 = r[n]; If[r2 == r1 + r3, AppendTo[s, n - 1]]; r1 = r2; r2 = r3, {n, 3, 4*10^6}]; s (* Amiram Eldar, Aug 16 2021 *)
Flatten[Position[Partition[Table[DivisorSigma[1,n]/DivisorSigma[0,n],{n,900000}],3,1],?(#[[2]] == #[[1]]+#[[3]]&),1,Heads->False]]+1 (* The program generates the first 10 terms of the sequence. *) (* _Harvey P. Dale, Apr 19 2024 *)

a(16)-a(18) from Jon E. Schoenfield, Aug 15 2021

a(19)-a(28) from Amiram Eldar, Aug 16 2021

cp's OEIS Frontend

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

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

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

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A069082 Numbers n such that sigma(n)/tau(n) has denominator 3.

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Magma

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A069083 Numbers n such that sigma(n)/tau(n) has denominator 4.

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Magma

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A069084 Numbers n such that sigma(n)/tau(n) has denominator 5.

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Magma

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A308051 Decimal expansion of lim_{m->oo} (sqrt(log(m))/m^2) Sum_{k=1..m} sigma(k)/d(k), where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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A375080 a(n) is the numerator of ( Sum_{d|n} (n - d) )/tau(n).

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