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 1-9 of 9 results.

A224344 Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 5, 1, 3, 8, 5, 5, 13, 13, 1, 7, 23, 27, 7, 11, 39, 52, 25, 1, 17, 65, 99, 66, 9, 27, 106, 186, 151, 41, 1, 40, 177, 340, 323, 133, 11, 61, 293, 608, 666, 358, 61, 1, 92, 482, 1076, 1330, 867, 236, 13, 142, 781, 1894, 2581, 1971, 737, 85, 1
Offset: 0

Views

Author

Alois P. Heinz, May 23 2013

Keywords

Examples

			A(5,1) = 8: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [3,1,1], [1,3,1], [1,1,3], [5].
Triangle T(n,k) begins:
   1;
   1;
   1,   1;
   1,   3;
   2,   5,   1;
   3,   8,   5;
   5,  13,  13,   1;
   7,  23,  27,   7;
  11,  39,  52,  25,   1;
  17,  65,  99,  66,   9;
  27, 106, 186, 151,  41,  1;
  40, 177, 340, 323, 133, 11;
  ...

Links

Crossrefs

Column k=0 gives: A052284.
Row sums are: A011782.
Row lengths are: A008619.
T(floor(n/2)) = A093178(n).
T(2n,n-1) = A001844(n-1) for n>0.
Cf. A000040, A000079, A004526, A102291, A121303, A222656.

Programs

  • Maple
    T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)->x+y, %,
      [`if`(isprime(j), 0, NULL), T(n-j)], 0) od; %[] fi
    end:
seq(T(n), n=0..16);
  • Mathematica
    zip[f_, x_List, y_List, z_] :=  With[{m = Max[Length[x], Length[y]]},  Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; T[n_] := T[n] =  Module[{j, pc}, If[n == 0, {1}, pc = {}; For[j = 1, j <= n, j++, pc = zip[Plus, pc, Join[If[PrimeQ[j], {0}, {}], T[n-j]], 0]]; pc]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

Formula

Sum_{k=1..floor(n/2)} k * T(n,k) = A102291(n).

A309561 Total sum of prime parts in all compositions of n.

Original entry on oeis.org

0, 0, 2, 7, 16, 44, 102, 244, 554, 1247, 2772, 6111, 13334, 28916, 62302, 133557, 285020, 605869, 1283362, 2710008, 5706546, 11986171, 25118500, 52529339, 109643310, 228455907, 475250388, 987177924, 2047710144, 4242128909, 8777675002, 18142184432, 37458037658
Offset: 0

Views

Author

Alois P. Heinz, Aug 07 2019

Keywords

Examples

			a(4) = 16: 1111, (2)11, 1(2)1, 11(2), (2)(2), (3)1, 1(3), 4.

Links

Crossrefs

Cf. A000040, A001787, A073118, A102291, A336579.

Programs

  • Maple
    a:= proc(n) option remember; add(a(n-j) +j*
      `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);
  • Mathematica
    a[n_] := a[n] = Sum[a[n-j]+j*If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}];
a /@ Range[0, 33] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=1} prime(k)*x^prime(k)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.27326606442562679135064648817419092073886899135... - Vaclav Kotesovec, Aug 18 2019

A336632 Number of prime parts, counted without multiplicity, in all compositions of n.

Original entry on oeis.org

0, 0, 1, 3, 6, 15, 33, 74, 160, 344, 731, 1544, 3237, 6753, 14022, 29009, 59819, 123010, 252341, 516560, 1055476, 2153115, 4385889, 8922556, 18131000, 36805009, 74643126, 151255021, 306267833, 619719217, 1253191291, 2532750315, 5116124712, 10329574480
Offset: 0

Views

Author

Alois P. Heinz, Jul 28 2020

Keywords

Examples

			a(4) = 0 + 1 + 1 + 1 + 1 + 1 + 1 + 0 = 6: 1111, 11(2), 1(2)1, (2)11, (2)2, 1(3), (3)1, 4.

Links

Crossrefs

Cf. A000040, A102291, A336579.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((h-> [0, `if`(j>0 and isprime(i),
       h[1], 0)]+h)(b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
     If[i < 1, 0, Sum[Function[h, {0, If[j > 0 && PrimeQ[i],
     h[[1]], 0]} + h][b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

A284942 Expansion of Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 8, 19, 46, 107, 244, 547, 1213, 2665, 5807, 12567, 27042, 57899, 123428, 262115, 554750, 1170538, 2463154, 5170462, 10829234, 22635087, 47223412, 98353299, 204519549, 424665001, 880581806, 1823667221, 3772341661, 7794697759, 16089424392, 33178906531, 68357928558
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2017

Keywords

Comments

Total number of squarefree parts in all compositions (ordered partitions) of n.

Examples

			a(4) = 19 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 0 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 19.

Links

Crossrefs

Cf. A005117, A008683, A011782, A059570, A097941, A097979, A102291, A281573, A284943.

Programs

  • Maple
    a:= proc(n) option remember; add(`if`(numtheory[
      issqrfree](j), ceil(2^(n-j-1)), 0)+a(n-j), j=1..n)
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019
  • Mathematica
    nmax = 33; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k (1 - x)^2/(1 - 2 x)^2, {k, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^34); Vec(sum(k=1, 34, moebius(k) ^2*x^k*(1 - x)^2/(1 - 2*x)^2)) \\ Indranil Ghosh, Apr 06 2017

Formula

G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2.

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

Original entry on oeis.org

0, 1, 3, 8, 20, 47, 110, 251, 564, 1251, 2750, 5994, 12978, 27934, 59825, 127565, 270959, 573575, 1210466, 2547562, 5348385, 11203292, 23419629, 48865346, 101782870, 211670094, 439548898, 911515214, 1887865266, 3905400206, 8070139762, 16658958223, 34355273843
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2017

Keywords

Comments

Total number of prime power parts (1 excluded) in all compositions (ordered partitions) of n.

Examples

			a(5) = 20 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 4], [1, 3, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 3], [1, 1, 2, 1], [1, 1, 1, 2], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 0 = 20.

Links

Crossrefs

Cf. A011782, A059570, A073335, A097941, A097979, A102291, A246655, A284942.

Programs

  • Maple
    b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019
  • Mathematica
    nmax = 33; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[k]] x^k (1 - x)^2/(1 - 2 x)^2, {k, 2, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^34); concat([0], Vec(sum(k=2, 34, (1\omega(k))*x^k*(1 - x)^2/(1 - 2*x)^2))) \\ Indranil Ghosh, Apr 06 2017

Formula

G.f.: Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

A309535 Total number of square parts in all compositions of n.

Original entry on oeis.org

0, 1, 2, 5, 13, 30, 69, 156, 348, 769, 1682, 3653, 7884, 16924, 36160, 76944, 163137, 344770, 726533, 1527052, 3202076, 6700096, 13992080, 29167936, 60703424, 126141953, 261754114, 542448645, 1122778124, 2321317916, 4794159168, 9891365008, 20388823360
Offset: 0

Views

Author

Alois P. Heinz, Aug 06 2019

Keywords

Examples

			a(4) = 13: (1)(1)(1)(1), (1)(1)2, (1)2(1), 2(1)(1), 22, (1)3, 3(1), (4).

Links

Crossrefs

Cf. A000196, A000290, A045623, A073336, A102291.

Programs

  • Maple
    a:= proc(n) option remember; add(a(n-j)+
      `if`(issqr(j), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);
  • Mathematica
    CoefficientList[Series[(EllipticTheta[3, 0, x]-1)*(1-x)^2/(2*(1-2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)
Table[Sum[If[k == n, 1, (2^(n - k - 2)*(3 + n - k))] * If[IntegerQ[Sqrt[k]], 1, 0], {k, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Aug 18 2019 *)

Formula

G.f.: Sum_{k>=1} x^(k^2)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = (EllipticTheta[3, 0, 1/2] - 1)/8 = 0.1411171034014846448336823185681189155765645674... - Vaclav Kotesovec, Aug 18 2019, updated Mar 17 2024
a(n) = Sum_{k=1..A000196(n)} A045623(n-k^2). - Gregory L. Simay, Jun 07 2021

A309536 Total number of triangular numbers in all compositions of n.

Original entry on oeis.org

0, 1, 2, 6, 14, 33, 77, 174, 389, 860, 1885, 4098, 8853, 19020, 40668, 86593, 183698, 388421, 818892, 1721884, 3611968, 7560337, 15793474, 32932549, 68556300, 142495004, 295754816, 613039248, 1269137729, 2624393922, 5421024773, 11186523404, 23061994524
Offset: 0

Views

Author

Alois P. Heinz, Aug 06 2019

Keywords

Examples

			a(4) = 14: (1)(1)(1)(1), 2(1)(1), (1)2(1), (1)(1)2, 22, (3)(1), (1)(3), 4.

Links

Crossrefs

Cf. A000217, A102291, A263235.

Programs

  • Maple
    a:= proc(n) option remember; add(a(n-j)+
      `if`(issqr(8*j+1), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);
  • Mathematica
    CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)) - 1)*((1 - x)^2/(1 - 2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)

Formula

G.f.: Sum_{k>=1} x^(k*(k+1)/2)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.1604081401637884665734606925563573585565153844... - Vaclav Kotesovec, Aug 18 2019

A309537 Total number of Fibonacci parts in all compositions of n.

Original entry on oeis.org

0, 1, 3, 8, 19, 46, 106, 241, 541, 1198, 2629, 5724, 12380, 26625, 56978, 121413, 257740, 545308, 1150272, 2419856, 5078336, 10633921, 22222338, 46353669, 96525324, 200686620, 416645184, 863834256, 1788756288, 3699688128, 7643727360, 15776156928, 32529718272
Offset: 0

Views

Author

Alois P. Heinz, Aug 06 2019

Keywords

Links

Crossrefs

Cf. A000045, A102291, A124091, A144115.

Programs

  • Maple
    a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)
      or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);
  • Mathematica
    a[n_] := a[n] = Sum[a[n - j] + With[{t = 5 j^2}, If[IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4], Ceiling[2^(n - j - 1)], 0]], {j, 1, n}];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=2} x^Fibonacci(k)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - Vaclav Kotesovec, Aug 18 2019
c = A124091/4 - 3/8. - Vaclav Kotesovec, Mar 17 2024

A309538 Total number of factorial parts in all compositions of n.

Original entry on oeis.org

0, 1, 3, 7, 17, 40, 93, 210, 469, 1036, 2268, 4928, 10640, 22848, 48832, 103936, 220416, 465920, 982016, 2064384, 4329472, 9060352, 18923520, 39452672, 82116609, 170655746, 354156549, 734003212, 1519386652, 3141533760, 6488588432, 13388218688, 27598521024
Offset: 0

Views

Author

Alois P. Heinz, Aug 06 2019

Keywords

Links

Crossrefs

Cf. A000142, A102291.

Programs

  • Maple
    g:= proc(n) local i; 1; for i from 2 do
      if n=% then 1; break elif n<% then 0; break fi;
      %*i od; g(n):=%
    end:
a:= proc(n) option remember; add(a(n-j)+
      `if`(g(j)=1, ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);
  • Mathematica
    g[n_] := g[n] = Module[{i, p = 1}, For[i = 2, True, i++, If[n == p, p = 1; Break[], If[nJean-François Alcover, Jan 10 2023, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=1} x^(k!)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.1914062649011611938476562500000000001880790961... - Vaclav Kotesovec, Aug 18 2019
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