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 1-10 of 10 results.

A280543 Expansion of 1/(1 - x - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 62, 123, 244, 483, 958, 1899, 3765, 7463, 14794, 29329, 58141, 115258, 228486, 452949, 897922, 1780031, 3528716, 6995293, 13867402, 27490602, 54497104, 108034531, 214166610, 424561814, 841647229, 1668473323, 3307565365, 6556885566, 12998306479, 25767716954, 51081672682
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 05 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into prime powers (1 included).

Examples

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

Links

Crossrefs

Cf. A000961, A001221, A023893, A280195.

Programs

  • Mathematica
    nmax = 37; CoefficientList[Series[1/(1 - x - Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

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

A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 5, 3, 11, 10, 13, 18, 19, 52, 30, 61, 77, 114, 109, 146, 260, 318, 341, 356, 631, 666, 927, 848, 1849, 1978, 2305, 2213, 3560, 4302, 4748, 5588, 6779, 13952, 9044, 15534, 16897, 25084, 20731, 29524, 34882, 49360, 50765, 55112, 106903, 83652, 128552, 106638
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(10) = 10 because we have [8, 2], [7, 3], [5, 3, 2], [5, 2, 3], [3, 7], [3, 5, 2], [3, 2, 5], [2, 8], [2, 5, 3] and [2, 3, 5].

Links

Crossrefs

Cf. A032020, A032021, A032022, A035942, A054685, A218396, A219107, A246655, A280195, A331843, A331844, A331845, A331846.

A284465 Number of compositions (ordered partitions) of n into prime power divisors of n (not including 1).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 36, 1, 2, 2, 56, 1, 90, 1, 201, 2, 2, 1, 4725, 2, 2, 20, 1085, 1, 15778, 1, 5272, 2, 2, 2, 476355, 1, 2, 2, 270084, 1, 302265, 1, 35324, 3910, 2, 1, 67279595, 2, 14047, 2, 219528, 1, 5863044, 2, 14362998, 2, 2, 1, 47466605656, 1, 2, 35662, 47350056, 2, 119762253, 1, 9479643
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2017

Keywords

Examples

			a(8) = 6 because 8 has 4 divisors {1, 2, 4, 8} among which 3 are prime powers > 1 {2, 4, 8} therefore we have [8], [4, 4], [4, 2, 2], [2, 4, 2], [2, 2, 4] and [2, 2, 2, 2].

Links

Crossrefs

Cf. A066882, A100346, A246655, A280195, A284289.

Programs

  • Maple
    F:= proc(n) local f,G;
      G:= 1/(1 - add(add(x^(f[1]^j),j=1..f[2]),f = ifactors(n)[2]));
      coeff(series(G,x,n+1),x,n);
end proc:
map(F, [$0..100]); # Robert Israel, Mar 29 2017
  • Mathematica
    Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[PrimePowerQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 68}]
  • Python
    from sympy import divisors, primefactors
from sympy.core.cache import cacheit
@cacheit
def a(n):
    l=[x for x in divisors(n) if len(primefactors(x))==1]
    @cacheit
    def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
    return b(n)
print([a(n) for n in range(71)]) # Indranil Ghosh, Aug 01 2017

Formula

a(n) = [x^n] 1/(1 - Sum_{p^k|n, p prime, k>=1} x^(p^k)).
a(n) = 1 if n is a prime.
a(n) = 2 if n is a semiprime.

A300704 Number of compositions (ordered partitions) of n into prime power parts (not including 1) that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 7, 2, 7, 5, 46, 2, 115, 20, 39, 16, 723, 16, 1819, 27, 559, 414, 11481, 16, 13204, 1763, 6450, 383, 181548, 172, 455646, 1326, 70476, 29809, 571110, 275, 7203906, 121535, 739513, 1703, 45380391, 7362, 113898438, 65049, 757426, 2009203, 717490902, 2304
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

Examples

			a(10) = 5 because we have [7, 3], [4, 3, 3], [3, 7], [3, 4, 3] and [3, 3, 4].

Links

Crossrefs

Cf. A246655, A280195, A284465, A300580, A300584, A300702, A300703, A300706.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:= proc(n) option
      remember; `if`(n=0, 1, add(`if`(nops(ifactors(j)[2])
       <>1 or irem(m, j)=0, 0, b(n-j)), j=2..n)) end; b(m)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 11 2018
  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && PrimePowerQ[k]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 48}]

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 16, 21, 26, 37, 47, 61, 84, 108, 143, 191, 249, 331, 437, 575, 763, 1004, 1326, 1754, 2311, 3055, 4036, 5323, 7033, 9288, 12257, 16193, 21379, 28223, 37278, 49212, 64984, 85815, 113297, 149614, 197551, 260839, 344439, 454795, 600517, 792958, 1047023, 1382519, 1825533, 2410456, 3182845
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

Comments

Number of compositions (ordered partitions) into odd prime powers (1 excluded).

Examples

			a(10) = 3 because we have [7, 3], [5, 5] and [3, 7].

Links

Crossrefs

Cf. A001221, A023359, A061345, A246655, A280151, A280195.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 - Sum[Floor[1/PrimeNu[2 k - 1]] x^(2 k - 1), {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

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

A301501 Number of compositions (ordered partitions) of n into prime power parts (A246655) such that no two adjacent parts are equal (Carlitz compositions).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 6, 5, 12, 14, 22, 35, 44, 79, 99, 165, 228, 346, 516, 742, 1140, 1624, 2479, 3592, 5370, 7933, 11684, 17421, 25557, 38098, 56053, 83207, 122958, 181848, 269426, 397900, 589749, 871302, 1290349, 1908208, 2823440, 4178248, 6179602, 9146534, 13527806, 20019958
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2018

Keywords

Examples

			a(8) = 5 because we have [8], [5, 3], [3, 5], [3, 2, 3] and [2, 4, 2].

Links

Crossrefs

Cf. A003242, A246655, A280195, A301428, A301500.

Programs

  • Mathematica
    nmax = 46; CoefficientList[Series[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

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

A280605 Expansion of 1/(1 - Sum_{p prime, k>=2} x^(p^k)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0, 6, 5, 1, 0, 10, 10, 3, 0, 18, 23, 9, 2, 31, 46, 22, 6, 56, 94, 56, 19, 101, 184, 129, 50, 185, 364, 293, 134, 344, 708, 638, 332, 651, 1378, 1375, 805, 1265, 2665, 2901, 1878, 2503, 5161, 6057, 4306, 5061, 10005, 12488, 9653, 10384, 19461, 25556, 21319
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 06 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into proper prime powers (A246547).

Examples

			a(12) = 3 because we have [8, 4], [4, 8] and [4, 4, 4].

Links

Crossrefs

Cf. A246547, A280195, A280200, A280543, A280586.

Programs

  • Mathematica
    nmax = 67; CoefficientList[Series[1/(1 - Sum[Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
  • PARI
    x='x+O('x^68); Vec(1/(1 - sum(k=2, 67, sign(bigomega(k) - 1) * (1\omega(k)) * x^k))) \\ Indranil Ghosh, Apr 03 2017

Formula

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

A281852 Expansion of Sum_{p prime, i>=1} x^(p^i) / (1 - Sum_{p prime, j>=1} x^(p^j))^2.

Original entry on oeis.org

0, 1, 1, 3, 5, 9, 18, 29, 55, 91, 163, 274, 472, 798, 1349, 2275, 3804, 6380, 10614, 17685, 29318, 48584, 80296, 132506, 218329, 359139, 590092, 968120, 1586707, 2597349, 4247619, 6939353, 11326636, 18471726, 30099313, 49008929, 79739345, 129650164, 210661777, 342080831, 555153086, 900432434, 1459670289
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 31 2017

Keywords

Comments

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

Examples

			a(7) = 18 because we have [7], [5, 2], [4, 3], [3, 4], [3, 2, 2], [2, 5], [2, 3, 2], [2, 2, 3] and 1 + 2 + 2 + 2 + 3 + 2 + 3 + 3 = 18.

Links

Crossrefs

Cf. A121304, A246655, A280195.

Programs

  • Mathematica
    nmax = 43; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[i]] x^i, {i, 2, nmax}]/(1 - Sum[Floor[1/PrimeNu[j]] x^j, {j, 2, nmax}])^2, {x, 0, nmax}], x]]

Formula

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

A329098 Expansion of 1 / (1 + Sum_{p prime, k>=1} x^(p^k)).

Original entry on oeis.org

1, 0, -1, -1, 0, 1, 2, 0, -3, -3, 2, 5, 4, -4, -10, -5, 10, 16, 5, -20, -27, 0, 41, 38, -14, -73, -55, 46, 134, 63, -118, -219, -55, 252, 356, -11, -510, -527, 198, 951, 734, -644, -1702, -867, 1579, 2864, 764, -3415, -4609, 84, 6808, 6897, -2526, -12745, -9539, 8383
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

Crossrefs

Cf. A002121, A246655, A280195.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[1/(1 + Sum[Boole[PrimePowerQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[PrimePowerQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 55}]

Formula

G.f.: 1 / (1 + Sum_{k>=1} x^A246655(k)).

A369221 Number of compositions (ordered partitions) of n into prime power parts (not including 1) not greater than sqrt(n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 5, 7, 9, 12, 16, 21, 28, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 14823, 22741, 34888, 53524, 82114, 125976, 193267, 296502, 454881, 697859, 1070626, 1642509, 2519868, 3865875, 5930862, 9098878, 13959114, 21415483, 32854729, 50404337, 77328204
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 16 2024

Keywords

Crossrefs

Cf. A280195, A364526.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k, {k, 1, Floor[Sqrt[n]]}]), {x, 0, n}], {n, 0, 45}]
Showing 1-10 of 10 results.

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