A020486 -id:A020486 - cp's OEIS frontend

A158294 Averages of squares of divisors of numbers in A020486 (i.e., numbers m with property that average of squares of divisors is an integer: sigma_0(m) divides sigma_2(m)).

1, 5, 7, 13, 25, 61, 35, 85, 65, 145, 181, 91, 125, 265, 217, 205, 175, 421, 481, 305, 325, 685, 425, 841, 925, 427, 1105, 341, 817, 725, 595, 1405, 793, 905, 1741, 455, 1861, 1105, 2245, 1015, 1325, 2521, 2665, 1085, 1267, 1525, 3121, 3445, 875, 1885, 2105

Offset: 1

Jaroslav Krizek, Mar 15 2009

nonn

a(n) = A001157(A020486(n))/A000005(A020486(n)).

			a(12) = A001157(A020486(12))/A000005(A020486(12)) = A001157(20)/A000005(20) = 546/6 = 91.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000005, A001157, A020486.

a158294 n = (a001157 $ a020486 n) `div` (a000005 $ a020486 n)
-- Reinhard Zumkeller, Jan 15 2013

A140480 RMS numbers: numbers n such that root mean square of divisors of n is an integer.

Original entry on oeis.org

1, 7, 41, 239, 287, 1673, 3055, 6665, 9545, 9799, 9855, 21385, 26095, 34697, 46655, 66815, 68593, 68985, 125255, 155287, 182665, 242879, 273265, 380511, 391345, 404055, 421655, 627215, 730145, 814463, 823537, 876785, 1069895, 1087009, 1166399, 1204281, 1256489

Offset: 1

Ctibor O. Zizka, Jun 29 2008, Jul 11 2008

For any numbers, A and B, both appearing in the sequence, if gcd(A,B)=1, then A*B is also in the sequence. - Andrew Weimholt, Jul 01 2008
The primes in this sequence are the NSW primes (A088165). For the terms less than 2^31, the only powers greater than 1 appearing in the prime factorization of numbers are 3^3 and 13^2. It appears that all terms are +-1 (mod 8). See A224988 for even numbers. - T. D. Noe, Jul 06 2008, Apr 25 2013
A basis for this sequence is given by A002315. This can be considered as the convergents of quasiregular continued fractions or a special 6-ary numeration system (see A. S. Fraenkel) which gives the characterization of positions of some heap or Wythoff game. What is the Sprague-Grundy function of this game?
Sequence generalized: sigma_r-numbers are numbers n for which sigma_r(n)/sigma_0(n) = c^r. Sigma_r(n) denotes sum of r-th powers of divisors of n; c,r positive integers. This sequence are sigma_2-numbers, A003601 are sigma_1-numbers. In a weaker form we have sigma_r(n)/sigma_0(n) = c^t; t is an integer from <1,r>. - Ctibor O. Zizka, Jul 14 2008
The primes in this sequence are prime numerators with an odd index in A001333. The RMS values (A141812) of prime RMS numbers (this sequence) are prime Pell numbers (A000129) with an odd index. - Ctibor O. Zizka, Aug 13 2008
From Ctibor O. Zizka, Aug 30 2008: (Start)
The set of RMS numbers n could be split into subsets according to the number and form of divisors of n. By definition, RMS(n) = sqrt(sigma_2(n) / sigma_0(n)) should be an integer. Now consider some examples. For n prime number, n has 2 divisors [1,n] and we have to solve Pell's equation n^2 = 2*C^2 - 1; C positive integer. The solution is a prime n of the form u(i) = 6*u(i-1) - u(i-2), i >= 2, u(0)=1, u(1)=7, known as an NSW prime (A088165). For n = p_1*p_2, p_1 and p_2 primes, n has 4 divisors {1; p_1; p_2; p_1*p_2}. There are 2 possible cases. Firstly p^2 = (2*C)^2 - 1 which does not hold for any prime p; secondly p_1^2 = 2*C_1^2 - 1 and p_2^2 = 2*C_2^2 - 1; C_1 and C_2 positive integers.
The solution is that p_1 and p_2 are different NSW primes. If n = p^3, divisors of n are {1; p; p^2; p^3} and we have to solve the Diophantine equation (p^8 - 1)/(p - 1) = (2*C)^2. This equation has no solution for any prime p. RMS numbers n with 4 divisors are only of the form n = p_1*p_2, with p_1 and p_2 NSW primes. The general case is n = p_1*...*p_t, n has 2^t divisors, and for t >= 3, NSW primes are not the only solution. If some of the prime divisors are equals p_i = p_j = ... = p_k, the general case n = p_1*...*p_t is "degenerate" because of the multiplicity of prime factors and therefore n has fewer than 2^t divisors. (End)
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k = a(1) - 3, x = (1 + sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878, whose prime terms give A121534. a(1)=5 gives A001834, whose prime terms give A086386. a(1)=6 gives A030221, whose prime terms {29, 139, 3191, ...} are not a sequence on the OEIS. a(1)=7 gives A002315, whose prime terms give A088165. a(1)=8 gives A033890; the OEIS does not have its prime terms as a sequence (do there exist any prime terms?). a(1)=9 gives A057080, whose prime terms {71, 34649, 16908641, ...} are not a sequence in the OEIS. a(1)=10 gives A057081, whose prime terms {389806471, 192097408520951, ...} are not a sequence in the OEIS. - Ctibor O. Zizka, Sep 02 2008
16 of the first 1660 terms are even (the smallest is 2217231104). The first 16 even terms are all divisible by 30976. - Donovan Johnson, Apr 16 2013
All the 83 even terms up to 10^13 (see A224988) are divisible by 30976. - Giovanni Resta, Oct 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..7430 (terms < 10^13, first 455 terms from T. D. Noe, terms 456..1660 from Donovan Johnson)
A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279.
H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192. [_Ctibor O. Zizka_, Aug 30 2008]
Eric Weisstein's World of Math, Root Mean Square

Cf. A002315, A001653, A001834, A001835, A001599, A000005, A000040, A003601, A010052, A001157, A020486, A158294, A224988.

a140480 n = a140480_list !! (n-1)
a140480_list = filter
    ((== 1) . a010052 . (\x -> a001157 x `div` a000005 x)) a020486_list
-- Reinhard Zumkeller, Jan 15 2013

rmsQ[n_] := IntegerQ[Sqrt[DivisorSigma[2, n]/DivisorSigma[0, n]]]; m = 160000; sel1 = Select[8*Range[0, m]+1, rmsQ]; sel7 = Select[8*Range[m]-1, rmsQ]; Union[sel1, sel7] (* Jean-François Alcover, Aug 31 2011, after T. D. Noe's comment *)
Select[Range[1300000],IntegerQ[RootMeanSquare[Divisors[#]]]&] (* Harvey P. Dale, Mar 24 2016 *)

More terms from T. D. Noe and Andrew Weimholt, Jul 01 2008

A046871 Numbers k such that sigma_2(k) divides sigma_4(k).

Original entry on oeis.org

1, 4, 9, 16, 20, 25, 36, 48, 49, 64, 81, 100, 121, 144, 162, 169, 180, 196, 225, 245, 256, 289, 324, 361, 400, 432, 441, 484, 500, 529, 576, 605, 625, 648, 676, 729, 784, 841, 900, 931, 961, 980, 1024, 1089, 1156, 1200, 1225, 1280, 1296, 1369, 1444, 1521

Offset: 1

Labos Elemer

nonn

sigma_2(k) is the sum of the squares of the divisors of k (A001157).

sigma_4(k) is the sum of the 4th powers of the divisors of k (A001159).

			k = a(8) = 48 of which divisor power sums for powers 0, 1, 2, 3, 4 are 10, 124, 3410, 131068, 5732210, respectively. Here sigma_2(k) = 3410 and sigma_4(k) = 3410*1681.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A001157, A001159, A020487, A020486, A046841.

[n: n in [1..1600] | IsZero(DivisorSigma(4,n) mod DivisorSigma(2,n))]; // Bruno Berselli, Apr 10 2013

Select[Range@ 1600, Divisible[DivisorSigma[4, #], DivisorSigma[2, #]] &] (* Michael De Vlieger, May 20 2017 *)

isok(n) = !(sigma(n, 4) % sigma(n, 2)); \\ Michel Marcus, May 21 2017

A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

Original entry on oeis.org

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Ctibor O. Zizka, Sep 19 2008

A000203(n)/A000005(n) = c^2. Generalized sigma-sequences are sequences of numbers n for which sigma_r(n)/sigma_s(n) = c^t . Sigma_i(n) denotes sum of i-th powers of divisors of n; c,r,s,t positive integers. This sequence has r=1,s=0,t=2, sequence A003601 has r=1,s=0,t=1, sequence {1,21,53,85,102,110,127,217,431,....} has r=1,s=0,t=3, sequence A020487 has r=2,s=1,t=1, sequence A020486 has r=2,s=0,t=1, sequence A140480 has r=2,s=0,t=2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor function

Cf. A000005, A000203, A140480, A003601, A020487, A020486.

A000005 := proc(n) numtheory[tau](n) ; end: A000203 := proc(n) numtheory[sigma](n) ; end: isA144695 := proc(n) local s ; s := A000005(n) ; if s <> 0 then issqr(A000203(n)/s) ; else false ; fi; end: for n from 1 to 5000 do if isA144695(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Select[Range[1125], IntegerQ @ Sqrt[DivisorSigma[1, #]/DivisorSigma[0, #]] &] (* Amiram Eldar, Nov 20 2019 *)

isok(m) = my(f=factor(m), q=sigma(f)/numdiv(f)); issquare(q) && !frac(q); \\ Michel Marcus, Mar 15 2022

More terms from R. J. Mathar, Sep 20 2008

A091428 Numbers k such that abs(A092673(k)) = 1.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 12, 13, 15, 17, 19, 20, 21, 23, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 51, 52, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 71, 73, 76, 77, 79, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 116, 119, 123, 124, 127, 129

Offset: 1

Jon Perry, Mar 02 2004

nonn

The asymptotic density of this sequence is 5/Pi^2 = 0.506605... . - Amiram Eldar, Jan 11 2023

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

Disjoint union of A056911 and A081770.
Cf. A092673, A359592 (characteristic function).
Distinct from A020486.

Select[Range[130], Abs[MoebiusMu[#] - If[EvenQ[#], MoebiusMu[#/2], 0]] == 1 &] (* Amiram Eldar, Jan 11 2023 *)

A277553 Numbers k such that the sum of the divisors of k is divisible by the number of divisors of k, and the sum of the squares of the divisors of k is divisible by the sum of the divisors of k.

Original entry on oeis.org

1, 20, 49, 169, 361, 500, 605, 961, 980, 1025, 1369, 1445, 1700, 1849, 2645, 3380, 3721, 4205, 4352, 4489, 4693, 5329, 6241, 7220, 8228, 8281, 8405, 9409, 9425, 10609, 11045, 11849, 11881, 12493, 12500, 14045, 14580, 14641, 15125, 16129, 17405, 17689, 18785

Offset: 1

Harvey P. Dale, Oct 19 2016

nonn

Numbers k such that A000005(k) divides A000203(k), and A000203(k) divides A001157(k).

			1369 has 3 divisors which sum to 1407; 1407 is divisible by 3; the sum of the squares of the divisors of 1369 is 1875531 which is divisible by 1407; so 1369 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000203, A001157.

Intersection of A003601 and A020487. See also A020486.

Select[Range[50000],Divisible[DivisorSigma[1,#],DivisorSigma[0,#]] && Divisible[ DivisorSigma[2,#], DivisorSigma[1,#]]&]

isok(k) = {my(f = factor(k), d = numdiv(f), s = sigma(f), s2 = sigma(f, 2)); !(s % d) && !(s2 % s);} \\ Amiram Eldar, Jan 25 2025

A046839 Numbers k such that the number of divisors of k divides the sum of cubes of divisors of k.

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, 64, 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

Offset: 1

Labos Elemer

nonn

The first 42 terms agree with A003601 but a(43) = 64 is not a term in A003601.

			64 is a term since it has 7 divisors, and sigma_3(64) = 299593 = 7 * 42799 is divisble by 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A000005, A001158.

Cf. A003601, A020486, A046840.

[n: n in [1..110] | IsZero(DivisorSigma(3, n) mod NumberOfDivisors(n))]; // Bruno Berselli, Apr 11 2013

Select[Range[103], Divisible[DivisorSigma[3, #], DivisorSigma[0, #]] &] (* Jayanta Basu, Jun 29 2013 *)

isok(n) = sigma(n, 3) % numdiv(n) == 0; \\ Michel Marcus, May 13 2018

A046840 Numbers k such that the number of divisors of k divides the sum of the 4th powers of the divisors of k.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 100, 101, 103, 105, 107, 108, 109, 111

Offset: 1

Labos Elemer

nonn

A020486 is very similar to this sequence, but it does not include the following values below 1000 (which this sequence does include): {16, 80, 81, 176, 304, 324, 400, 405, 464, 496, 656, 784, 880, 891, 944, 976}.

			k = 16 is a term since it has 5 divisors, and sigma_4(16) = 69905 is divisible by 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A000005, A001159.

Cf. A003601, A020486, A046839.

[n: n in [1..120] | IsZero(DivisorSigma(4, n) mod NumberOfDivisors(n))]; // Bruno Berselli, Apr 11 2013

Select[Range[120], Divisible[DivisorSigma[4, #], DivisorSigma[0, #]] &] (* Amiram Eldar, Mar 17 2025 *)

isok(n) = sigma(n, 4) % numdiv(n) == 0; \\ Michel Marcus, May 13 2018

A046841 Numbers whose sum of divisors divides their sum of cubes of divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Labos Elemer

nonn

A005117 (squarefree numbers) is a subsequence. - Ivan Neretin, Dec 20 2017

			2 is a term because 1 + 8 = 9 is divisible by 1 + 2 = 3.
208 is a term: The power sums of divisors for k = 0, 1, 2, 3 are as follows: 10, 434, 54970, 10288838, and sigma(1,208) = 434 divides sigma(3,208) = 10288838 = 434*23707.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A000203, A001158, A005117.

Cf. A003601, A020486.

with(numtheory):
for n to 100 do
 if (type(sigma[3](n)/sigma[1](n), integer)) then print(n) end if;
end do; # Peter Bala, Jan 12 2025

Select[Range@ 85, Divisible[DivisorSigma[3, #], DivisorSigma[1, #]] &] (* Michael De Vlieger, Aug 01 2017 *)

isA046841(n)=sigma(n,3)%sigma(n,1)==0 \\ Michael B. Porter, Apr 07 2010

A158298 Denominators of averages of squares of the divisors of n.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 1, 1, 2, 1, 5, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 3, 7, 1, 2, 1, 1, 1, 2, 1, 12, 1, 2, 1, 1, 1, 2, 1, 5, 5, 2, 1, 1, 1, 2

Offset: 1

Jaroslav Krizek, Mar 15 2009

Average of squares of the divisors of n = A001157(n)/A000005(n).

Iff a(n) = 1, n is in A020486.

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

Cf. A001157, A000005, A020486, A158299 (for numerators).

Array[Denominator[DivisorSigma[2, #]/DivisorSigma[0, #]] &, 100] (* Amiram Eldar, Jul 15 2019 *)

a(n) = denominator(sigma(n, 2)/numdiv(n)) \\ Michel Marcus, Jun 13 2013

Data corrected and extended by Amiram Eldar, Jul 15 2019

cp's OEIS Frontend

A158294 Averages of squares of divisors of numbers in A020486 (i.e., numbers m with property that average of squares of divisors is an integer: sigma_0(m) divides sigma_2(m)).

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Haskell

A140480 RMS numbers: numbers n such that root mean square of divisors of n is an integer.

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Haskell

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A046871 Numbers k such that sigma_2(k) divides sigma_4(k).

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Magma

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A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

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Maple

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A091428 Numbers k such that abs(A092673(k)) = 1.

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Mathematica

A277553 Numbers k such that the sum of the divisors of k is divisible by the number of divisors of k, and the sum of the squares of the divisors of k is divisible by the sum of the divisors of k.

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Mathematica

PARI

A046839 Numbers k such that the number of divisors of k divides the sum of cubes of divisors of k.

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Magma

Mathematica

PARI

A046840 Numbers k such that the number of divisors of k divides the sum of the 4th powers of the divisors of k.

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