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User: Andrew Ivashenko

Andrew Ivashenko has authored 3 sequences.

A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

0, 0, 0, 1, 0, 1, 2, 1, 2, 4, 2, 4, 6, 4, 6, 9, 6, 9, 12, 9, 12, 16, 12, 16, 20, 16, 20, 25, 20, 25, 30, 25, 30, 36, 30, 36, 42, 36, 42, 49, 42, 49, 56, 49, 56, 64, 56, 64, 72, 64, 72, 81, 72, 81, 90, 81, 90, 100, 90, 100, 110, 100, 110, 121, 110, 121, 132

Offset: 0

Andrew Ivashenko, Mar 19 2019

Index entries for linear recurrences with constant coefficients, signature (0,1,2,0,-2,-1,0,1).

Cf. A000041, A002620, A103221, A114209, A321202.

Cf. A008133, A069905, A008611.

a:=[0,0,0,1,0,1,2,1];; for n in [9..80] do a[n]:=a[n-2]+2*a[n-3] -2*a[n-5]-a[n-6]+a[n-8]; od; a; # G. C. Greubel, Apr 03 2019

R:=PowerSeriesRing(Integers(), 80); [0,0,0] cat Coefficients(R!( x^3/((1-x^2)*(1-x^3)^2) )); // G. C. Greubel, Apr 03 2019

LinearRecurrence[{0,1,2,0,-2,-1,0,1}, {0,0,0,1,0,1,2,1}, 80] (* G. C. Greubel, Apr 03 2019 *)
Table[(6n(2+n)-5-27(-1)^n+8(4+3n)Cos[2n Pi/3]-8Sqrt[3]n Sin[2n Pi/3])/216,{n,0,66}] (* Stefano Spezia, Apr 21 2022 *)

my(x='x+O('x^80)); concat([0,0,0], Vec(x^3/((1-x^2)*(1-x^3)^2))) \\ G. C. Greubel, Apr 03 2019

(x^3/((1-x^2)*(1-x^3)^2)).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

a(n+2) = A321202(n) - A114209(n+1).
a(3n+1) = A002620(n+2).
a(3n+2) = A002620(n+1).
a(3n+3) = A002620(n+2).
G.f.: x^3/((1+x)*(1+x+x^2)^2*(1-x)^3). - Alois P. Heinz, Mar 19 2019
a(n) = a(n-2) + 2*a(n-3) - 2*a(n-5) - a(n-6) + a(n-8). - G. C. Greubel, Apr 03 2019
a(n) = (6*n*(2 + n) + 8*(4 + 3*n)*cos(2*n*Pi/3) - 8*sqrt(3)*n*sin(2*n*Pi/3) - 5 - 27*(-1)^n)/216. - Stefano Spezia, Apr 21 2022
From Ridouane Oudra, Nov 24 2024: (Start)
a(n) = (7*n/2 - 7*n^2/2 - 9*floor(n/2) + (6*n+4)*floor(2*n/3) + 4*floor(n/3))/18.
a(n) = A008133(n) - A069905(n-1).
a(n) = A002620(A008611(n)). (End)

More terms from Alois P. Heinz, Mar 19 2019

A282572 Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).

Original entry on oeis.org

1, 3, 7, 9, 15, 21, 27, 31, 45, 49, 63, 81, 93, 105, 127, 135, 147, 189, 217, 225, 243, 255, 279, 315, 343, 381, 405, 441, 465, 511, 567, 651, 675, 729, 735, 765, 837, 889, 945, 961, 1023, 1029, 1143, 1215, 1323, 1395, 1519, 1533, 1575, 1701, 1785, 1905, 1953, 2025, 2047, 2187, 2205, 2295, 2401

Offset: 1

Andrew Ivashenko, Feb 18 2017

nonn

Odd orders of finite abelian groups that appear as the group of units in a commutative ring (Chebolu and Lockridge, see A296241). - Jonathan Sondow, Dec 15 2017

Actually, the Chebolu and Lockridge paper states that this sequence gives all odd numbers that are possible numbers of units in a (commutative or non-commutative) ring (Ditor's theorem). Concretely, if k = (2^(e_1)-1)*(2^(e_2)-1)*...(2^(e_r)-1) is a term, let R = (F_2)^s X F_(2^(e_1)) X F_(2^(e_2)) X ... X F_(2^(e_r)) for s >= 0, then the number of units in R is k. - Jianing Song, Dec 23 2021

			63 = 1*3^3*7, 81 = 1*3^4, 93 = 1*3*31, 105 = 1*7*15, 41013 = 1*3^3*7^2*31.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Sunil K. Chebolu and Keir Lockridge. How many units can a commutative ring have?, arXiv preprint arXiv:1701.02341 [math.AC], 2017. See Th. 8.

Cf. A000225, A056652, A296241.

Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

d:= 15: # for terms < 2^d
N:= 2^d:
S:= {1}:
for m from 2 to d do
  r:= 2^m-1;
  k:= ilog[r](N);
  V:= S;
  for i from 1 to k do
    V:= select(`<`, map(`*`, V, r), N);
    S:= S union V
  od;
od:
sort(convert(S, list)); # Ridouane Oudra, Sep 14 2021

lmt = 2500; a = b = Array[2^# - 1 &, Floor@ Log2@ lmt]; k = 2; While[k < Length@ a, e = 1; While[e < Floor@ Log[ a[[k]], lmt], b = Union@ Join[b, Select[ a[[k]]^e*b, # < 1 + lmt &]]; e++]; k++]; b (* Robert G. Wilson v, Feb 23 2017 *)

forstep(x=1,1000000,2, t=x; forstep(n=20,2,-1, m=2^n-1; while(t%m==0, t=t\m)); if(t==1, print1(x,","))) \\ Dmitry Petukhov, Feb 23 2017

More terms from Michel Marcus, Feb 23 2017

Definition changed by David A. Corneth, Mar 12 2017

A282534 Integers that are powers of Mersenne numbers A000225 (i.e., of the form (2^n - 1)^m).

Original entry on oeis.org

1, 3, 7, 9, 15, 27, 31, 49, 63, 81, 127, 225, 243, 255, 343, 511, 729, 961, 1023, 2047, 2187, 2401, 3375, 3969, 4095, 6561, 8191, 16129, 16383, 16807, 19683, 29791, 32767, 50625, 59049, 65025, 65535, 117649, 131071, 177147, 250047, 261121, 262143, 524287, 531441, 759375

Offset: 1

Andrew Ivashenko, Feb 18 2017

nonn

The cardinality of the set of subsets in a multiset excluding empty subsets.

			3 = (2^2-1), 7 = (2^3-1), 9 = (2^2-1)^2, 81 = (2^2-1)^4, 1070599167 = (2^10-1)^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000225, A056652.

mx = 10^6; Union@ Flatten@ {1, #^Range[Log[#, mx]] & /@ (2^ Range[2, Log2@ mx] -1)} (* Giovanni Resta, Mar 08 2017 *)

ismn(n) = n++; n == 2^valuation(n,2);
isok(n) = ismn(n) || (ispower(n,,&m) && ismn(m)); \\ Michel Marcus, Feb 18 2017

More terms from Michel Marcus, Feb 18 2017

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User: Andrew Ivashenko

A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

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A282572 Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).

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A282534 Integers that are powers of Mersenne numbers A000225 (i.e., of the form (2^n - 1)^m).

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Examples

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Crossrefs

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Mathematica

PARI

Extensions