A046915 -id:A046915 - cp's OEIS frontend

A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 13, 15, 1, 1, 6, 28, 40, 31, 1, 1, 12, 31, 120, 121, 63, 1, 1, 8, 91, 156, 496, 364, 127, 1, 1, 12, 57, 600, 781, 2016, 1093, 255, 1, 1, 12, 112, 400, 3751, 3906, 8128, 3280, 511, 1, 1, 18, 117, 960, 2801, 22932, 19531, 32640

Offset: 1

Álvar Ibeas, Oct 30 2015

All the enumerated lattices have full rank k, since the quotient group is finite.
For m>=1, T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_nm x (C_m)^(k-1); and also, (C_nm)^(k-1) x C_m.
Also, number of subgroups of (C_n)^k isomorphic to C_n (and also, to (C_n)^{k-1}), cf. [Butler, Lemma 1.4.1].
T(n,k) is the sum of the divisors d of n^(k-1) such that n^(k-1)/d is k-free. Namely, the coefficient in n^(-(k-1)*s) of the Dirichlet series zeta(s) * zeta(s-1) / zeta(ks).
Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.
Columns are multiplicative functions.

			There are 7 = A160870(4,2) lattices of volume 4 in Z^2. Among them, only one (<(2,0), (0,2)>) gives the quotient group C_2 x C_2, whereas the rest give C_4. Hence, T(4,2) = 6 and T(1,2) = 1.
Array begins:
      k=1    k=2    k=3    k=4    k=5    k=6
n=1     1      1      1      1      1      1
n=2     1      3      7     15     31     63
n=3     1      4     13     40    121    364
n=4     1      6     28    120    496   2016
n=5     1      6     31    156    781   3906
n=6     1     12     91    600   3751  22932

Lynne M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., Vol. 112, No. 539, 1994.

Álvar Ibeas, First 100 antidiagonals, flattened
Jin Ho Kwak, Jang-Ho Chun, and Jaeun Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Examples 2 and 3]

Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).
Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
Main diagonal gives A344210.
Cf. A000203, A008683, A160870.

f[p_, e_, k_] := p^((k - 1)*(e - 1))*(p^k - 1)/(p - 1); T[n_, 1] = T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 08 2022 *)

T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).
T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).
Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).
If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{i=1..n} T(i, k) ~ c * n^k, where c = (1/k) * Product_{p prime} (1 + (p^(k-1)-1)/((p-1)*p^k)).
Sum_{i>=1} 1/T(i, k) = zeta(k-1)*zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(2*k-1)), for k > 2. (End)
T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^k). - Ridouane Oudra, Apr 03 2025

A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

Original entry on oeis.org

1, 87, 8299, 823081, 82256014, 8224740835, 822468118437, 82246711794796, 8224670422194237, 822467034112360628, 82246703352400266400, 8224670334323560419029, 822467033425357340138978, 82246703342420509396897774, 8224670334241228180927002517

Offset: 0

Rick L. Shepherd, Jul 02 2002

nonn

			For n=1, the sum of sigma(j) for j<=10 is 1+3+4+7+6+12+8+15+13+18=87, so a(1)=87 (=69+18=A049000(1)+A046915(1)).

P. L. Patodia, Seth Troisi and Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36 (terms a(0)-a(18) by P. L. Patodia and a(19)-a(24) by Seth Troisi)
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916

Compare with A049000. Note that a(n) = A049000(n) + A046915(n).

Cf. A000203 (sigma(n)), A072691 (Pi^2/12), A049000, A046915, A024916, A025281.

for(m=0,10,print1(sum(n=1,k=10^m,n*(k\n)),",")) \\ Improved by M. F. Hasler, Apr 18 2015

A072692(n)=A024916(10^n) \\ This is very efficient, using efficient code of A024916. - M. F. Hasler, Apr 18 2015

[(i, sum([d*(10**i//d) for d in range(1,10**i+1)])) for i in range(8)] # Seth A. Troisi, Jun 27 2010

from math import isqrt
def A072692(n): return -(s:=isqrt(m:=10**n))**2*(s+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

Asymptotic formula: a(n) ~ Pi^2/12 * 10^2n. See A072691 for Pi^2/12. Observe that A025281 also contains that constant in its asymptotic formula.

More terms from P L Patodia (pannalal(AT)usa.net), Jan 11 2008, Jun 25 2008

Corrected by N. J. A. Sloane, Jun 08 2008, following suggestions from Don Reble and David W. Wilson

A049000 Sum of sigma(j) for 1<=j<10^n, where sigma(j) is the sum of divisors of j.

Original entry on oeis.org

0, 69, 8082, 820741, 82231803, 8224494757, 822465638000, 82246686892516, 8224670172682646, 822467031614802290, 82246703327412473943, 8224670334073621455209, 822467033422857645316807, 82246703342395510922780776, 8224670334240978188556405240

Offset: 0

Enoch Haga and Jud McCranie

nonn

The ratio of successive terms approaches 100.

			For n=1, the sum of sigma(j) for j<10 is 1+3+4+7+6+12+8+15+13=69, so a(1)=69.

Amiram Eldar, Table of n, a(n) for n = 0..36 (calculated from the b-file at A072692)

Cf. A000203, A046915.
Cf. A072691 (Pi^2/12).
Cf. A072692 (sum for 1<=j<=10^n and other links).

a(n) = A072692(n) - A046915(n) ~ Pi^2/12 * 10^(2*n). - Amiram Eldar, Feb 16 2020

One more term from Rick L. Shepherd, Jul 03 2002

More terms from Amiram Eldar, Feb 16 2020

A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 13, 7, 1, 1, 31, 40, 35, 6, 1, 1, 63, 121, 155, 31, 12, 1, 1, 127, 364, 651, 156, 91, 8, 1, 1, 255, 1093, 2667, 781, 600, 57, 15, 1, 1, 511, 3280, 10795, 3906, 3751, 400, 155, 13, 1, 1, 1023, 9841, 43435, 19531, 22932, 2801, 1395, 130, 18, 1

Offset: 1

Ralf Stephan, May 09 2007

Differs from sum of divisors of m^(n-1) in 4th column!

			Array starts:
1,1,1,1,1,1,1,1,1,
1,3,4,7,6,12,8,15,13,
1,7,13,35,31,91,57,155,130,
1,15,40,155,156,600,400,1395,1210,
1,31,121,651,781,3751,2801,11811,11011,
1,63,364,2667,3906,22932,19608,97155,99463,
1,127,1093,10795,19531,138811,137257,788035,896260,
1,255,3280,43435,97656,836400,960800,6347715,8069620,

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

Álvar Ibeas, First 100 antidiagonals, flattened
Michael Baake, Solution of the coincidence problem in dimensions d≤4, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
B. Gruber, Alternative formulas for the number of sublattices, Acta Cryst. A53 (1997) 807-808.
Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.

Rows include A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.
Columns include A000225, A003462, A006095, A003463, A160869, A023000, A006096, A006100, A046915.
Transposed array is A160870.

T[n_, m_] := If[m == 1, 1, Product[{p, e} = pe; (p^(e+j)-1)/(p^j-1), {pe, FactorInteger[m]}, {j, 1, n-1}]];
Table[T[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 10 2018 *)

T(n,m)=local(k,v);v=factor(m);k=matsize(v)[1];prod(i=1,k,prod(j=1,n-1,(v[i,1]^(v[i,2]+j)-1)/(v[i,1]^j-1)))

Dirichlet g.f. of n-th row: Product_{i=0..n-1} zeta(s-i).
If m is squarefree, T(n,m) = A000203(m^(n-1)). - Álvar Ibeas, Jan 17 2015
T(n, Product(p^e)) = Product(Gaussian_poly[e+n-1, e]p). - _Álvar Ibeas, Oct 31 2015

Edited by Charles R Greathouse IV, Oct 28 2009

A049030 Sum of sigma(j) for 1<=j<10^n, where sigma(j) = A048050(j) is the sum of the proper divisors >1 of j (excluding 1 and n).

Original entry on oeis.org

16, 3034, 320243, 32226805, 3224444759, 322465138002, 32246681892518, 3224670122682648, 322467031114802292, 32246703322412473945, 3224670334023621455211, 322467033422357645316809, 32246703342390510922780778, 3224670334240928188556405242

Offset: 1

Enoch Haga and Jud McCranie

			For n = 1, the sum of sigma(j), for j < 10 is 0 + 0 + 0 + 2 + 0 + 5 + 0 + 6 + 3 = 16, so a(1) = 16.

Amiram Eldar, Table of n, a(n) for n = 1..36 (calculated from the b-file at A049000)
Carlos Rivera, Problem 23.- Divisors (V) Aliquot sequences, Sociable Numbers, Prime Puzzles and Problems Connection.

Cf. A001065, A046915, A048050, A048995, A049000.

Cf. A072691 (Pi^2/12).

At a(3) = 320243, for example, take a(3) from A049000: 820741 - 500498 = 320243. Compute 500498 from 999*1000/2 = 499500, split evenly and reverse to 500499 - 1 = 500498. Add a 9 and 0 for each successive term.

a(n) = A049000(n) - 10^n * (10^n + 1) / 2 + 2 ~ (Pi^2/12 - 1/2) * 10^(2*n). - Amiram Eldar, Feb 16 2020

More terms from Amiram Eldar, Feb 16 2020

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A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

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A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

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A049000 Sum of sigma(j) for 1<=j<10^n, where sigma(j) is the sum of divisors of j.

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A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

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A049030 Sum of sigma(j) for 1<=j<10^n, where sigma(j) = A048050(j) is the sum of the proper divisors >1 of j (excluding 1 and n).

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