A335106
Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.
Original entry on oeis.org
0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 0, 2, 3, 2, 1, 1, 1, 2, 3, 3, 1, 0, 3, 4, 3, 1, 1, 1, 2, 4, 4, 2, 1, 0, 3, 5, 5, 2, 1, 1, 1, 3, 5, 5, 3, 2, 0, 3, 6, 7, 3, 2, 1, 1, 1, 3, 7, 7, 4, 3, 1, 0, 4
Offset: 0
A000607(10) = 5 and the prime partitions of 10 are: (2,2,2,2,2), (2,2,3,3), (2,3,5), (5,5) and (3,7). Thus G(10) = {2,3,5,5,7}, and consequently row 10 is [1,1,2,1]. In the table below, for n >= 2, 0 is used to indicate when prime(k) is not in G(n) and is less than the greatest member of G(n), otherwise the entry for prime(k) not in G(n) is left empty. For n >= 2 the sum of entries in the n-th row is |G(n)| = A000607(n). Triangle T(n,k) begins:
0;
0;
1;
0, 1;
1;
0, 1, 1;
1, 1;
0, 1, 1, 1;
1, 1, 1;
0, 2, 1, 1;
1, 1, 2, 1;
0, 2, 2, 1, 1;
1, 2, 2, 2;
0, 2, 3, 2, 1, 1;
1, 2, 3, 3, 1;
0, 3, 4, 3, 1, 1;
1, 2, 4, 4, 2, 1;
0, 3, 5, 5, 2, 1, 1;
...
-
Flatten@ Block[{nn = 22, t}, t = Block[{s = {Prime@ PrimePi@ nn}}, KeySort@ Merge[#, Identity] &@ Join[{0 -> {}, 1 -> {}}, Reap[Do[If[# <= nn, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1] ] ] &@Total[s], {i, Infinity}]][[-1, -1]] ] ]; Array[Function[p, If[! IntegerQ@ First@ p, {0}, Array[Count[p, Prime@ #] &, PrimePi@ Max@ p]]]@ Map[Max, t[[#]]] &, Max@ Keys@ t]] (* Michael De Vlieger, May 23 2020 *)
row[0]={0}; row[k_] := Join[If[OddQ@k, {0}, {}], Last /@ Tally@ Sort[ First /@ IntegerPartitions[k, All, Prime@ Range@ PrimePi@ k]]]; Join @@ Array[row, 20, 0] (* Giovanni Resta, May 31 2020 *)
A140952
Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)).
Original entry on oeis.org
1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 31, 35, 39, 43, 48, 53, 58, 64, 70, 77, 84, 91, 100, 108, 117, 127, 137, 148, 159, 172, 184, 198, 212, 227, 243, 259, 277, 295, 314, 334, 355, 377, 400, 424, 449, 475, 502, 531, 560
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1).
-
M := Matrix(28, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 15, 19, 20, 28])) then 1 elif j=1 and member(i, [8, 9, 13, 17, 25, 26]) then -1 else 0 fi):
a:= n-> (M^(n))[1,1]:
seq(a(n), n=0..50);
-
CoefficientList[Series[1/Times@@(1-x^Prime[Range[5]]),{x,0,70}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1},{1,0,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,28,31,35,39,43,48},70] (* Harvey P. Dale, Jun 18 2021 *)
A140953
Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)*(1-x^13)).
Original entry on oeis.org
1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19, 21, 25, 28, 32, 36, 41, 46, 52, 58, 65, 72, 80, 89, 98, 109, 119, 132, 144, 158, 173, 189, 206, 224, 244, 264, 287, 310, 336, 362, 391, 421, 453, 487, 523, 561, 601, 644, 688, 736, 785, 838, 893
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (0,1,1,0, 0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1).
-
M := Matrix(41, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 19, 20, 21, 22, 30, 38, 39])) then 1 elif j=1 and member(i, [8, 9, 16, 17, 24, 25, 32, 33, 41]) then -1 else 0 fi):
a:= n -> (M^(n))[1,1]:
seq(a(n), n=0..50);
-
CoefficientList[Series[1/Times@@Table[1-x^p,{p,Prime[Range[6]]}],{x,0,60}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1},{1,0,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,16,19,21,25,28,32,36,41,46,52,58,65,72,80,89,98,109,119,132,144,158,173,189,206},70] (* Harvey P. Dale, Dec 05 2022 *)
Showing 1-3 of 3 results.
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