A121304 -id:A121304 - cp's OEIS frontend

A121303 Triangle read by rows: T(n,k) is the number of compositions of n into k primes (i.e., ordered sequences of k primes having sum n; n>=2, k>=1).

1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 2, 3, 0, 2, 3, 1, 0, 2, 4, 4, 0, 3, 6, 6, 1, 1, 0, 6, 8, 5, 0, 2, 9, 13, 10, 1, 1, 2, 6, 16, 15, 6, 0, 3, 6, 22, 25, 15, 1, 0, 2, 10, 24, 36, 26, 7, 0, 4, 9, 22, 50, 45, 21, 1, 1, 0, 12, 32, 65, 72, 42, 8, 0, 4, 12, 34, 70, 106, 77, 28, 1, 1, 2, 12, 40, 90, 150

Offset: 2

Emeric Deutsch, Aug 06 2006

Row n has floor(n/2) terms.
Sum of terms in row n = A023360(n).
T(n,1) = A010051(n) (characteristic function of primes); T(n,2) = A073610(n); T(n,3) = A098238(n).
Sum_{k=1..floor(n/2)} k*T(n,k) = A121304(n).

			T(9,3) = 4 because we have [2,2,5], [2,5,2], [5,2,2] and [3,3,3].
Triangle starts:
  1;
  1;
  0, 1;
  1, 2;
  0, 1, 1;
  1, 2, 3;
  0, 2, 3, 1;
  0, 2, 4, 4;
  ...

Alois P. Heinz, Rows n = 2..200, flattened

Cf. A010051, A023360, A073610, A098238, A121304, A224344.

T(n^2,n) gives A341459.

G:=1/(1-t*sum(z^ithprime(i),i=1..30))-1: Gser:=simplify(series(G,z=0,25)): for n from 2 to 21 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 2 to 21 do seq(coeff(P[n],t,j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(numtheory):
b:= proc(n) option remember; local j; if n=0 then [1]
      else []; for j to pi(n) do zip((x, y)->x+y, %,
      [0, b(n-ithprime(j))[]], 0) od; % fi
    end:
T:= n-> subsop(1=NULL, b(n))[]:
seq(T(n), n=2..20);  # Alois P. Heinz, May 23 2013

nn=20;a[x_]:=Sum[x^Prime[n],{n,1,nn}];CoefficientList[Series[1/(1-y a[x]),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Nov 08 2013 *)

G.f.: 1/(1 - t*Sum_{i>=1} z^prime(i)).

A281812 Expansion of Sum_{i>=1} mu(i)^2x^i / (1 - Sum_{j>=1} mu(j)^2x^j)^2, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 8, 19, 44, 99, 218, 473, 1012, 2144, 4504, 9395, 19482, 40189, 82534, 168829, 344145, 699334, 1417146, 2864510, 5776889, 11626101, 23353272, 46827677, 93747221, 187399328, 374092162, 745817021, 1485138398, 2954041789, 5869650947, 11651500427, 23107388495, 45787040997, 90652188078, 179340159228

Offset: 1

Ilya Gutkovskiy, Jan 30 2017

nonn

Total number of parts in all compositions (ordered partitions) of n into squarefree parts (A005117).

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

Index entries for sequences related to compositions

Cf. A005117, A008683, A121304, A280194.

nmax = 36; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i, {i, 1, nmax}]/(1 - Sum[MoebiusMu[j]^2 x^j, {j, 1, nmax}])^2, {x, 0, nmax}], x]]

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

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

Ilya Gutkovskiy, Jan 31 2017

nonn

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

			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.

Index entries for sequences related to compositions

Cf. A121304, A246655, A280195.

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]]

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

A281853 Expansion of Sum_{k>=2} x^prime(k) / (1 - Sum_{k>=2} x^prime(k))^2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 4, 3, 6, 10, 8, 19, 22, 26, 48, 53, 78, 112, 136, 205, 264, 354, 504, 639, 890, 1204, 1568, 2173, 2868, 3826, 5192, 6839, 9214, 12295, 16296, 21894, 28996, 38624, 51552, 68230, 90930, 120715, 159988, 212728, 281696, 373574, 495312, 655365, 868510, 1149161, 1520020, 2011591, 2658416, 3514446

Offset: 1

Ilya Gutkovskiy, Jan 31 2017

nonn

Total number of parts in all compositions (ordered partitions) of n into odd primes (A065091).

			a(11) = 10 because we have [11], [5, 3, 3], [3, 5, 3], [3, 3, 5] and 1 + 3 + 3 + 3 = 10.

Index entries for sequences related to compositions

Cf. A002124, A065091, A121304.

nmax = 55; Rest[CoefficientList[Series[Sum[x^Prime[k], {k, 2, nmax}]/(1 - Sum[x^Prime[k], {k, 2, nmax}])^2, {x, 0, nmax}], x]]

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

cp's OEIS Frontend

A121303 Triangle read by rows: T(n,k) is the number of compositions of n into k primes (i.e., ordered sequences of k primes having sum n; n>=2, k>=1).

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A281812 Expansion of Sum_{i>=1} mu(i)^2*x^i / (1 - Sum_{j>=1} mu(j)^2*x^j)^2, where mu() is the Moebius function (A008683).

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

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A281853 Expansion of Sum_{k>=2} x^prime(k) / (1 - Sum_{k>=2} x^prime(k))^2.

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A281812 Expansion of Sum_{i>=1} mu(i)^2x^i / (1 - Sum_{j>=1} mu(j)^2x^j)^2, where mu() is the Moebius function (A008683).