cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A007503 Number of subgroups of dihedral group: sigma(n) + d(n).

Original entry on oeis.org

2, 5, 6, 10, 8, 16, 10, 19, 16, 22, 14, 34, 16, 28, 28, 36, 20, 45, 22, 48, 36, 40, 26, 68, 34, 46, 44, 62, 32, 80, 34, 69, 52, 58, 52, 100, 40, 64, 60, 98, 44, 104, 46, 90, 84, 76, 50, 134, 60, 99, 76, 104, 56, 128, 76, 128, 84, 94, 62, 180, 64, 100, 110, 134
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

Essentially first differences of A257644. - Franklin T. Adams-Watters, Nov 05 2015
Write D_{2n} as , then the subgroups are of the form for d|n or for d|n and 0 <= r < d. There are d(n) subgroups of the first type and sigma(n) subgroups of the second type. - Jianing Song, Jul 21 2022
Inverse Möbius transform of n+1 (see Arndt formula). - Wesley Ivan Hurt, Jul 05 2025

Examples

			a(4) = 10 since D_8 = <a, x | a^4 = x^2 = 1, x*a*x = a^(-1)> has 10 subgroups. The 6 subgroups {e}, {e,a^2}, {e,a,a^2,a^3}, {e,a^2,x,a^2*x}, {e,a^2,a*x,a^3*x} and D_8 are normal, and the 4 subgroups {e,x}, {e,a*x}, {e,a^2*x} and {e,a^3*x} are not. - _Jianing Song_, Jul 21 2022

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000005, A000203, A037852 (number of normal subgroups).
Cf. A027750, A257644 (cumulative sums, start=1).

Programs

Formula

G.f.: Sum_{k>=1} 1/(1-x^k)^2. - Benoit Cloitre, Apr 21 2003
G.f.: Sum_{i>=1} (1+i)*x^i/(1-x^i). - Jon Perry, Jul 03 2004
a(n) = Sum_{d|n} tau(p^d), where tau is A000005 and p any prime. - Enrique Pérez Herrero, Apr 14 2012
a(n) = Sum_{d divides n} d+1. - Joerg Arndt, Apr 14 2013
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(1+1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
a(n) = A000005(n) + A000203(n). - Omar E. Pol, Aug 19 2019
a(n) = A348219(n) + A386438(n). - Wesley Ivan Hurt, Jul 21 2025

A366983 a(n) = Sum_{k=1..n} (k+1) * floor(n/k).

Original entry on oeis.org

2, 7, 13, 23, 31, 47, 57, 76, 92, 114, 128, 162, 178, 206, 234, 270, 290, 335, 357, 405, 441, 481, 507, 575, 609, 655, 699, 761, 793, 873, 907, 976, 1028, 1086, 1138, 1238, 1278, 1342, 1402, 1500, 1544, 1648, 1694, 1784, 1868, 1944, 1994, 2128, 2188, 2287, 2363, 2467
Offset: 1

Views

Author

Seiichi Manyama, Oct 30 2023

Keywords

Crossrefs

Partial sums of A007503.
Cf. A006218, A366984, A366985.
Cf. A024916, A257644.

Programs

  • PARI
    a(n) = sum(k=1, n, (k+1)*(n\k));
  • Python
    from math import isqrt
def A366983(n): return -(s:=isqrt(n))*(s*(s+4)+5)+sum(((q:=n//w)+1)*(q+(w<<1)+4) for w in range(1,s+1))>>1 # Chai Wah Wu, Oct 31 2023

Formula

a(n) = A006218(n) + A024916(n).
G.f.: 1/(1-x) * Sum_{k>0} (1/(1-x^k)^2 - 1) = 1/(1-x) * Sum_{k>0} (k+1) * x^k/(1-x^k).
a(n) = A257644(n) - 1. - Hugo Pfoertner, Oct 31 2023

A264100 Sum of the lengths of the arithmetic progressions in {1,2,3,...,n}, including trivial arithmetic progressions of lengths 1 and 2.

Original entry on oeis.org

0, 1, 4, 12, 26, 50, 82, 130, 188, 265, 358, 473, 602, 765, 944, 1151, 1386, 1657, 1948, 2284, 2642, 3048, 3490, 3972, 4480, 5056, 5666, 6322, 7022, 7784, 8578, 9452, 10360, 11337, 12366, 13453, 14592, 15831, 17110, 18453, 19856, 21357, 22902, 24551
Offset: 0

Views

Author

Gionata Neri, Nov 03 2015

Keywords

Comments

Conjecture: the second differences give A007503(n+1), the sum of the divisors (A000203) plus the number of divisors (A000005) of n+1.
The first differences trivially are the total length of such sequences that end in n+1. Mapping each sequence to a different sequence by adding 1 to each term, we see that the second differences are the number of sequences up to n+2 that include both 1 and n+2. For each divisor d of n+1, there is a single such sequence of length d+1 (with increment (n+1)/d). The second difference is then sum_{d|n+1} d+1, which is sigma(n+1) + tau(n+1), as claimed. - Franklin T. Adams-Watters, Nov 05 2015

Examples

			For n = 3 the arithmetic progressions are (1), (2), (3), (1, 2), (1, 3), (2, 3), (1, 2, 3) and the respective lengths are (1), (1), (1), (2), (2), (2), (3), so a(3) = 1 + 1 + 1 + 2 + 2 + 2 + 3 = 12.
The first difference at 2, sequences ending with 3, are (3), (1, 3), (2, 3), and (1, 2, 3), total length 8 = 12-4. The second difference at 2, sequences starting with 1 and ending with 4 are (1, 4) and (1, 2, 3, 4), total length 6 = 26 - 2*12 +4.

Crossrefs

Cf. A000005, A000203, A007503, A257644.

Programs

  • PARI
    vector(50, n, n--; n + sum(k=2, n, k*floor((n-1)/(k-1))*(2*n-(k-1)*floor((n+k-2)/(k-1)))/2)) \\ Altug Alkan, Nov 04 2015

Formula

a(n) = n + Sum_{k=2..n} k*floor((n-1)/(k-1))*(2*n-(k-1)*floor((n+k-2)/(k-1)))/2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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