Displaying 1-10 of 13 results found.
Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<= A000009(n).
+10
31
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187
COMMENTS
The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.
EXAMPLE
The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
1;
2;
3;
5, 6;
7, 10;
11, 14, 15;
13, 21, 22, 30;
17, 26, 33, 35, 42;
19, 34, 39, 55, 66, 70;
23, 38, 51, 65, 77, 78, 105, 110;
29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
...
Corresponding triangle of strict integer partitions begins:
0
(1)
(2)
(3) (21)
(4) (31)
(5) (41) (32)
(6) (42) (51) (321)
(7) (61) (52) (43) (421)
(8) (71) (62) (53) (521) (431)
(9) (81) (72) (63) (54) (621) (432) (531). - Gus Wiseman, Feb 23 2018
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)
CROSSREFS
Last elements of rows give: A246868.
a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).
+10
25
1, 2, 3, 11, 17, 40, 86, 153, 283, 547, 1069, 1737, 3238, 5340, 9574, 17251, 27897, 45845, 78601, 126725, 207153, 353435, 550422, 881454, 1393870, 2239938, 3473133, 5546789, 8762663, 13341967, 20676253, 31774563, 48248485, 74174759, 111904363, 170184798
COMMENTS
Sum of all squarefree numbers whose prime indices sum to n. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, May 09 2019
FORMULA
a(n) = [x^n] Product_{k>=1} 1+prime(k)*x^k. - Alois P. Heinz, Sep 05 2014 a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = prime(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 10 2020
EXAMPLE
Form a product from the primes: (1 + 2*x) * (1 + 3*x^2) * (1 + 5*x^3) * ...* (1 + prime(n)*x^n) * ... Multiplying out gives 1 + 2*x + 3*x^2 + 11*x^3 + ..., so the sequence begins 1, 2, 3, 11, ....
Let f(m) = prime(m). Using the strict partitions of n (see A000009), we get: a(1) = f(1) = 2,
a(2) = f(2) = 3,
a(3) = f(3) + f(1)*f(2) = 5 + 2*3 = 11,
a(4) = f(4) + f(1)*f(3) = 7 + 2*5 = 17,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 11 + 2*7 + 3*5 = 40,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 13 + 2*11 + 3*7 + 2*3*5 = 86,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 17 + 2*13 + 3*11 + 5*7 + 2*3*7 = 153. (End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
end:
a:= n-> b(n$2):
MATHEMATICA
nn=40; Take[Rest[CoefficientList[Expand[Times@@Table[1+Prime[n]x^n, {n, nn}]], x]], nn] (* Harvey P. Dale, Jul 01 2012 *)
CROSSREFS
Cf. A000009, A005117, A015723, A022629, A056239, A066189, A112798, A145519, A147541, A325504, A325506, A325537.
a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n-k+1), where k = [ n/2 ], p = A000040, the primes.
+10
16
0, 6, 10, 29, 43, 94, 128, 231, 279, 484, 584, 903, 1051, 1552, 1796, 2489, 2823, 3784, 4172, 5515, 6091, 7758, 8404, 10575, 11395, 14076, 15174, 18339, 19667, 23414, 24906, 29437, 31089, 36500, 38614, 44731, 47071, 54198, 56914, 65051, 68371, 77402, 81052, 91341
COMMENTS
This is the sum of distinct squarefree semiprimes with prime indices summing to n + 1. A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, Dec 05 2020
EXAMPLE
The sequence of sums begins (n > 1):
6 = 6
10 = 10
29 = 14 + 15
43 = 22 + 21
94 = 26 + 33 + 35
128 = 34 + 39 + 55
231 = 38 + 51 + 65 + 77
279 = 46 + 57 + 85 + 91
(End)
MATHEMATICA
f[n_] := Block[{primeList = Prime@ Range@ n}, Total[ Take[ primeList, Floor[n/2]]*Reverse@ Take[ primeList, {Floor[(n + 3)/2], n}]]]; Array[f, 44] (* Robert G. Wilson v, Apr 07 2014 *)
PROG
(Haskell)
a025129 n = a025129_list !! (n-1)
a025129_list= f (tail a000040_list) [head a000040_list] 1 where
f (p:ps) qs k = sum (take (div k 2) $ zipWith (*) qs $ reverse qs) :
f ps (p : qs) (k + 1)
CROSSREFS
The nonsquarefree version is A024697 (shifted right). Row sums of A338905 (shifted right). A332765 is the greatest among these squarefree semiprimes.
A006881 lists squarefree semiprimes.
A014342 is the self-convolution of the primes.
A056239 is the sum of prime indices of n.
A339194 sums squarefree semiprimes grouped by greater prime factor.
Cf. A001221, A005117, A062198, A098350, A168472, A320656, A338900, A338901, A338904, A339114, A339116.
Triangle read by rows: T1[n,k;x] := Sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).
+10
5
2, 3, 4, 5, 6, 8, 7, 19, 12, 16, 11, 29, 38, 24, 32, 13, 68, 85, 76, 48, 64, 17, 94, 181, 170, 152, 96, 128, 19, 177, 326, 443, 340, 304, 192, 256, 23, 231, 683, 787, 886, 680, 608, 384, 512, 29, 400, 1066, 1780, 1817, 1772, 1360, 1216, 768, 1024, 31, 484, 1899, 3119
COMMENTS
Let p(n; m_1, m_2, m_3, ..., m_n) denote a partition of integer n in exponential representation, i.e., the m_i are the counts of parts i and satisfy 1*m_1 + 2*m_2 + 3*m_3 + ... + n*m_n = n.
Let p(n, k; m_1, m_2, m_3, ..., m_n) be the partitions of n into exactly k parts; these are further constrained by m_1 + m_2 + m_3 + ... + m_n = k.
Then the triangle is given by T1[n,k;x] := Sum_{all p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i is the i-th prime number ( A000040). 2nd column (4, 6, 19, 29, 68, 94, 177, ...) is A024697.
EXAMPLE
Triangle starts:
2;
3, 4;
5, 6, 8;
7, 19, 12, 16;
11, 29, 38, 24, 32;
13, 68, 85, 76, 48, 64;
...
MAPLE
g:= proc(n, i) option remember; `if`(n=0 or i=1, (2*x)^n,
expand(add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2)):
MATHEMATICA
g[n_, i_] := g[n, i] = If[n==0 || i==1, (2 x)^n, Expand[Sum[g[n-i*j, i-1]*(Prime[i]*x)^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
EXTENSIONS
Reference to more terms etc. changed to make it version independent by Tilman Neumann, Sep 02 2009
Sum over all partitions lambda of n into 3 distinct parts of Product_{i:lambda} prime(i).
+10
2
30, 42, 136, 293, 551, 892, 1765, 2570, 4273, 6747, 9770, 13958, 21206, 28280, 39702, 54913, 72227, 94682, 127095, 160046, 206119, 263581, 327790, 406354, 512372, 616764, 754412, 921169, 1100165, 1314196, 1584835, 1854384, 2191013, 2590565, 3006512, 3495086
MAPLE
g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
, j=0..min(1, n/i)))), x, 4), polynom)
end:
a:= n-> coeff(g(n$2), x, 3):
seq(a(n), n=6..60);
MATHEMATICA
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[g[n - i j, i - 1] (Prime[i] x)^j, {j, 0, Min[1, n/i]}]]];
a[n_] := Coefficient[g[n, n], x, 3];
Sum over all partitions lambda of n into 4 distinct parts of Product_{i:lambda} prime(i).
+10
2
210, 330, 852, 1826, 4207, 6595, 13548, 21479, 38905, 59000, 95953, 142843, 231431, 324152, 487361, 683227, 1003028, 1347337, 1907811, 2541970, 3526314, 4597020, 6194948, 7969172, 10618000, 13401580, 17424498, 21875750, 28102737, 34685941, 43856482, 53791587
MAPLE
g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
, j=0..min(1, n/i)))), x, 5), polynom)
end:
a:= n-> coeff(g(n$2), x, 4):
seq(a(n), n=10..60);
MATHEMATICA
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[g[n - i j, i - 1] (Prime[i] x)^j, {j, 0, Min[1, n/i]}]]];
a[n_] := Coefficient[g[n, n], x, 4];
Sum over all partitions lambda of n into 5 distinct parts of Product_{i:lambda} prime(i).
+10
2
2310, 2730, 7860, 15606, 35594, 67255, 120061, 201324, 364479, 592991, 1004771, 1530056, 2444073, 3691392, 5610179, 8334486, 12213775, 17529361, 25187765, 35345858, 49999364, 68516285, 94223007, 127478773, 172613052, 230362430, 305639795, 401637665, 527011287
MAPLE
g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
, j=0..min(1, n/i)))), x, 6), polynom)
end:
a:= n-> coeff(g(n$2), x, 5):
seq(a(n), n=15..60);
MATHEMATICA
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[g[n - i j, i - 1] (Prime[i] x)^j, {j, 0, Min[1, n/i]}]]];
a[n_] := Coefficient[g[n, n], x, 5];
Sum over all partitions lambda of n into 6 distinct parts of Product_{i:lambda} prime(i).
+10
2
30030, 39270, 90300, 177930, 381222, 722434, 1477619, 2309879, 4194446, 6846481, 11667593, 18212397, 30309561, 45149226, 70722044, 105790662, 160115543, 232478684, 346845682, 489561123, 709058342, 994019962, 1405076982, 1932862089, 2705315737, 3653574123
MAPLE
g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
, j=0..min(1, n/i)))), x, 7), polynom)
end:
a:= n-> coeff(g(n$2), x, 6):
seq(a(n), n=21..60);
Sum over all partitions lambda of n into 7 distinct parts of Product_{i:lambda} prime(i).
+10
2
510510, 570570, 1436820, 2655870, 5532330, 9757518, 19659886, 34710965, 58356321, 96541978, 161476211, 256683013, 419693431, 647984259, 1021626403, 1536889595, 2332063802, 3443800806, 5133970767, 7443724123, 10827942578, 15520714599, 22052126419, 30994058608
MAPLE
g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
, j=0..min(1, n/i)))), x, 8), polynom)
end:
a:= n-> coeff(g(n$2), x, 7):
seq(a(n), n=28..60);
Sum over all partitions lambda of n into 8 distinct parts of Product_{i:lambda} prime(i).
+10
2
9699690, 11741730, 27927900, 49533330, 98525490, 170218830, 325872714, 562212782, 1032566057, 1629661685, 2724030632, 4284584225, 6990871609, 10713813287, 17001782121, 25600766613, 39614085330, 58088625761, 87187552970, 126762441906, 186103726454, 266554756593
MAPLE
g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
, j=0..min(1, n/i)))), x, 9), polynom)
end:
a:= n-> coeff(g(n$2), x, 8):
seq(a(n), n=36..60);
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