A064987 -id:A064987 - cp's OEIS frontend

A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.

1, 5, 9, 20, 20, 48, 35, 76, 72, 110, 77, 204, 104, 196, 210, 288, 170, 405, 209, 480, 378, 440, 299, 816, 425, 598, 594, 868, 464, 1200, 527, 1104, 858, 986, 910, 1800, 740, 1216, 1170, 1960, 902, 2184, 989, 1980, 1890, 1748, 1175, 3216, 1470, 2475, 1938, 2704, 1484, 3456, 2090

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

nonn

Dirichlet convolution of natural numbers (A000027) with triangular numbers (A000217).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000027, A000203, A000217, A007437, A007503, A034715, A038040, A064987, A309732.

with(numtheory): seq(n*(tau(n)+sigma(n))/2, n=1..30); # Ridouane Oudra, Nov 28 2019

nmax = 55; CoefficientList[Series[Sum[k x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j, j (j + 1)/2, j, n], {n, 1, 55}]
Table[n (DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 55}]

a(n)=sumdiv(n,d,binomial(n/d+1,2)*d); \\ Andrew Howroyd, Aug 14 2019

a(n)=n*(numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Apr 19 2021

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k/(1 - x^k)^2.
a(n) = n * (d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-1) * (zeta(s-2) + zeta(s-1))/2.
a(n) = Sum_{k=1..n} k*tau(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

A327165 Numbers n that have a divisor d such that sigma(d)*d is equal to n.

Original entry on oeis.org

1, 6, 12, 28, 30, 56, 72, 117, 120, 132, 180, 182, 306, 336, 360, 380, 496, 552, 672, 702, 775, 792, 840, 870, 992, 1080, 1092, 1406, 1440, 1568, 1584, 1680, 1722, 1836, 1892, 2016, 2160, 2184, 2256, 2280, 2793, 2862, 3276, 3312, 3510, 3540, 3600, 3672, 3696, 3782, 3960, 4032, 4556, 4560, 4650, 5096, 5112, 5220

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Numbers n for which A327153(n) > 0.
Sequence A064987 sorted into ascending order, with duplicates removed.
All even terms of A000396 occur here.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000203, A000396, A064987, A327153, A327599 (subsequence of odd terms).

isA327165(n) = { fordiv(n, d, if(n==d*sigma(d),return(1))); (0); };

A327165list(up_to) = { my(res = List()); for(i = 1, sqrtint(up_to), c = i*sigma(i); if(c <= up_to, listput(res, c))); listsort(res, 1); Vec(res); }; \\ From David A. Corneth, Sep 18 2019

A009242 a(n) = lcm(n, sigma(n)).

Original entry on oeis.org

1, 6, 12, 28, 30, 12, 56, 120, 117, 90, 132, 84, 182, 168, 120, 496, 306, 234, 380, 420, 672, 396, 552, 120, 775, 546, 1080, 56, 870, 360, 992, 2016, 528, 918, 1680, 3276, 1406, 1140, 2184, 360, 1722, 672, 1892, 924, 1170, 1656, 2256, 1488, 2793, 4650, 1224, 2548

Offset: 1

David W. Wilson

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

Cf. A009194, A017665, A064987.

A009242[n_]:=LCM[DivisorSigma[1,n],n]; Array[A009242,100] (* Enrique Pérez Herrero, Aug 25 2011 *)

a(n) = lcm(n, sigma(n)); \\ Amiram Eldar, Mar 27 2024

a(n) = n*A017665(n). - Enrique Pérez Herrero, Aug 25 2011

a(n) = A064987(n)/A009194(n). - Amiram Eldar, Mar 27 2024

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 3, 4, 5, 4, 3, 4, 5, 6, 6, 6, 4, 5, 6, 7, 6, 6, 4, 5, 6, 7, 8, 8, 6, 8, 5, 6, 7, 8, 9, 8, 9, 8, 5, 6, 7, 8, 9, 10, 10, 9, 8, 10, 6, 7, 8, 9, 10, 11, 10, 9, 8, 10, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 10, 12, 7, 8, 9, 10, 11, 12, 13, 12, 12, 12, 10, 12, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 06 2006

An equivalent definition: Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127093 from the right. That is, T(n,m) = Sum_{j=m..n} A000012(n,j)*A127093(j,m) = Sum_{j=m..n} A127093(j,m) = m*floor(n/m) = m*A010766(n,m). - Gary W. Adamson, Jan 05 2007

The number of parts k in the triangle is A000203(k) hence the sum of parts k is A064987(k). - Omar E. Pol, Jul 05 2014

			Triangle begins:
{1},
{2, 2},
{3, 2, 3},
{4, 4, 3, 4},
{5, 4, 3, 4, 5},
{6, 6, 6, 4, 5, 6},
{7, 6, 6, 4, 5, 6, 7},
{8, 8, 6, 8, 5, 6, 7, 8},
{9, 8, 9, 8, 5, 6, 7, 8, 9},
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A074025, A093960.

Cf. also A024916 (row sums), A127093, A127094, A127096, A127097, A127098, A127099, A038040, A000203, A126988, A127013, A127057.

Flat(List([1..10],n->List([1..n],m->n-(n mod m)))); # Muniru A Asiru, Oct 12 2018

seq(seq(n-modp(n,m),m=1..n),n=1..13); # Muniru A Asiru, Oct 12 2018

a = Table[Table[n - Mod[n, m], {m, 1, n}], {n, 1, 20}]; Flatten[a]

for(n=1,9,for(m=1,n,print1(n-n%m", "))) \\ Charles R Greathouse IV, Nov 07 2011

Edited by N. J. A. Sloane, Jul 05 2014 at the suggestion of Omar E. Pol, who observed that A127095 (Gary W. Adamson, with edits by R. J. Mathar) was the same as this sequence.

A126832 Ramanujan numbers (A000594) read mod 5.

Original entry on oeis.org

1, 1, 2, 3, 0, 2, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 1, 2, 0, 0, 2, 2, 2, 0, 0, 2, 0, 3, 0, 0, 2, 1, 4, 1, 0, 1, 1, 0, 4, 0, 2, 2, 2, 1, 0, 2, 1, 2, 3, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 4, 1, 3, 4, 0, 2, 0, 2, 1, 0, 0, 2, 4, 0, 0, 1, 2, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 4, 1, 0, 2, 1, 3, 4, 0, 2, 2, 2, 0, 0

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 166-167.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
R. P. Bambah and S. Chowla, Congruence properties of Ramanujan’s function tau(n), Bull. Amer. Math. Soc. 53 (1947), 950-955.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000203, A000594, A064987.

Cf. this sequence (mod 5^1), A126833 (mod 5^2), A126834 (mod 5^3), A126835 (mod 5^4).

Mod[RamanujanTau@ #, 5] & /@ Range@ 105 (* Michael De Vlieger, Apr 26 2016 *)

a(n) = n*sigma(n) % 5; \\ Amiram Eldar, Jan 05 2025

from sympy import divisor_sigma
def A126832(n): return n*divisor_sigma(n)%5 # Chai Wah Wu, Aug 24 2023

a(n) = n*sigma(n) mod 5. - Michel Marcus, Apr 26 2016. See also the Hardy reference, p. 166, (10.5.2), with a proof. - Wolfdieter Lang, Feb 03 2017

A249670 a(n) = A017665(n)*A017666(n).

Original entry on oeis.org

1, 6, 12, 28, 30, 2, 56, 120, 117, 45, 132, 21, 182, 84, 40, 496, 306, 78, 380, 210, 672, 198, 552, 10, 775, 273, 1080, 2, 870, 60, 992, 2016, 176, 459, 1680, 3276, 1406, 570, 2184, 36, 1722, 112, 1892, 231, 390, 828, 2256, 372, 2793, 4650, 408, 1274, 2862

Offset: 1

Michel Marcus, Nov 03 2014

nonn

If n is a k-multiperfect, then a(n) = k.

Allan C. Wechsler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)).
Cf. A017665/A017666 (abundancy of n).
Cf. A009194 (gcd(n, sigma(n))), A064987 (n*sigma(n)).
Cf. A000396, A005820, A027687, A046060, A046061.

a249670 n = div (n * s) (gcd n s ^ 2)
 where s = sum (filter (\k -> mod n k == 0) [1..n])
-- Allan C. Wechsler, Mar 31 2023

a249670[n_Integer] := Numerator[DivisorSigma[-1, n]]*Denominator[DivisorSigma[-1, n]]; a249670 /@ Range[80] (* Michael De Vlieger, Nov 10 2014 *)

a(n) = my(ab = sigma(n)/n); numerator(ab)*denominator(ab);

a(n) = A064987(n)/A009194(n)^2.
a(A000396(n)) = 2 (perfect).
a(A005820(n)) = 3 (tri-perfect).
For p prime, a(p) = p*(p+1).

A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 1026, 59052, 1050628, 9765630, 60587352, 282475256, 1075843080, 3486961557, 10019536380, 25937424612, 62041684656, 137858491862, 289819612656, 576679982760, 1101663313936, 2015993900466, 3577622557482, 6131066257820, 10260044315640

Offset: 0

Seiichi Manyama, Feb 10 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 25 2018

D. H. Lehmer shows that a(n) == tau(n) (mod 7) for n > 0, where tau is Ramanujan's tau function (A000594). Furthermore, if n == 3, 5, 6 (mod 7) then a(n) == tau(n) (mod 49). See the Wikipedia link below. - Jianing Song, Aug 12 2020

			a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Congruences for the tau function.

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), this sequence (phi_{10, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A280869 (E_6^2), A282102 (E_2*E_4*E_6).
Cf. A013957, A013668, A196292.

Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

G.f.: phi_{10, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (3*A008411(n) + 2*A280869(n) - 5*A282102(n))/1584.
If p is a prime, a(p) = p^10 + p = A196292(p).
a(n) = n*A013957(n) for n > 0, where A013957(n) is sigma_9(n), the sum of the 9th powers of the divisors of n. - Seiichi Manyama, Feb 18 2017
Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020
From Amiram Eldar, Oct 30 2023: (Start)
Dirichlet g.f.: zeta(s-1)*zeta(s-10).
Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)

A337873 Numbers m such that the equation m = k*sigma(k) has more than one solution.

Original entry on oeis.org

336, 5952, 10080, 27776, 44352, 60480, 61152, 97536, 102816, 127680, 178560, 185472, 196560, 260400, 292320, 333312, 455168, 472416, 578592, 635712, 758016, 785664, 833280, 961632, 1083264, 1179360, 1189440, 1270752, 1330560, 1530816, 1717632, 1815072, 1821312, 1834560

Offset: 1

Bernard Schott, Sep 27 2020

The map k -> k*sigma(k) = m is not injective (A064987), this sequence lists in increasing order the integers m that have several preimages.
These terms m satisfy A327153(m) > 1.
If 2^p-1 and 2^r-1 are distinct Mersenne primes (A000668), then k = (2^p-1)* 2^(r-1) and q = (2^r-1) * 2^(p-1) satisfy k*sigma(k) = q*sigma(q) = m = (2^p-1) * (2^r-1) * 2^(p+r-1) [see examples a(1) and a(2)].
The multiplicativity of sigma(k) ensures an infinity of solutions and thus of terms m [see example a(3)].

			For a(1): 12 * sigma(12) = 14 * sigma(14) = 336 with p=2 and r=3.
For a(2): 48 * sigma(48) = 62 * sigma(62) = 5952 with p=2 and r=5.
For a(3): 60 * sigma(60) = 70 * sigma(70) = 10080 with 60/12 = 70/14 = 5.
a(16) = 333312 is the smallest term with 3 preimages because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.

David A. Corneth, Table of n, a(n) for n = 1..10000
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Cf. A000203, A000668, A064987.
Cf. A327153. Subsequence of A327165.
Cf. A212490, A337874 (preimages), A337875 (primitive terms).

m = 2*10^6; v = Table[0, {m}]; Do[i = n*DivisorSigma[1, n]; If[i <= m, v[[i]]++], {n, 1, Floor@Sqrt[m]}]; Position[v, ?(# > 1 &)] // Flatten (* _Amiram Eldar, Sep 28 2020 *)

upto(n) = {m = Map(); res = List(); n = sqrtint(n); for(i = 1, n, c = i*sigma(i); if(mapisdefined(m, c), listput(res, c); mapput(m, c, mapget(m, c) + 1) , mapput(m, c, 1); ) ); listsort(res, 1); select(x -> x <= (n+1)^2, res) } \\ David A. Corneth, Sep 27 2020

isok(m) = {my(nb=0); fordiv(m, d, if (d*sigma(d) == m, nb++; if (nb>1, return(1)));); return (0);} \\ Michel Marcus, Sep 29 2020

More terms from David A. Corneth, Sep 27 2020

A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 5, 12, 7, 15, 13, 15, 11, 28, 13, 21, 20, 31, 17, 39, 19, 35, 28, 33, 23, 60, 25, 39, 40, 49, 29, 60, 31, 63, 44, 51, 35, 91, 37, 57, 52, 75, 41, 84, 43, 77, 65, 69, 47, 124, 49, 75, 68, 91, 53, 120, 55, 105, 76, 87, 59, 140, 61, 93, 91, 127, 65, 132

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A003586, A064987, A072044, A072045, A072079.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385138(n).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.
In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.
Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (3/4)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.

A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 14, 16, 18, 16, 18, 18, 20, 24, 24, 24, 24, 24, 31, 28, 27, 32, 30, 36, 32, 32, 36, 36, 48, 36, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 48, 57, 62, 54, 56, 54, 54, 72, 64, 60, 60, 60, 72, 62, 64, 72, 64, 84, 72, 68, 72

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A007310, A013661, A064987, A072044, A072045, A186099.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p > 3, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p > 3, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385137(n).
Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.

cp's OEIS Frontend

A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.

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A327165 Numbers n that have a divisor d such that sigma(d)*d is equal to n.

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A009242 a(n) = lcm(n, sigma(n)).

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A123229 Triangle read by rows: T(n, m) = n - (n mod m).

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A126832 Ramanujan numbers (A000594) read mod 5.

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A249670 a(n) = A017665(n)*A017666(n).

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A282254 Coefficients in q-expansion of (3*E_4^3 + 2*E_6^2 - 5*E_2*E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.