A074140 -id:A074140 - cp's OEIS frontend

A074141 Sum of products of parts increased by 1 in all partitions of n.

1, 2, 7, 18, 50, 118, 301, 684, 1621, 3620, 8193, 17846, 39359, 84198, 181313, 383208, 811546, 1695062, 3546634, 7341288, 15207022, 31261006, 64255264, 131317012, 268336125, 545858260, 1110092387, 2250057282, 4558875555, 9213251118, 18613373708, 37529713890

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Replace each term in A036035 by the number of its divisors as in A074139; sequence gives sum of terms in the n-th row.

This is the sum of the number of submultisets of the multisets with n elements; a part of a partition is a frequency of such an element. - George Beck, Nov 01 2011

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products when parts are increased by 1 are 5,8,9,12,16 and their sum is a(4) = 50.

Alois P. Heinz, Table of n, a(n) for n = 0..3317

Cf. A006906, A036035, A074140, A256155.
Row sums of A074139 and of A079025 and of A079308 and of A238963.
Column k=2 of A261718.
Cf. A267008.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1) +(1+i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Sep 07 2014

Table[Plus @@ Times @@@ (IntegerPartitions[n] + 1), {n, 0, 28}] (* T. D. Noe, Nov 01 2011 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, (1+i) * b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

G.f.: 1/Product_{m>0} (1-(m+1)*x^m).
a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*(d+1)^(k/d).
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k+1)*S(n-k,k))+(n+1), S(n,n)=n+1, S(0,m)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 07 2014
a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1-(k+1)/2^k) = 18.56314656361011472747535423226928404842588594722907068201... = A256155. - Vaclav Kotesovec, Sep 11 2014, updated May 10 2021

More terms from Alford Arnold, Sep 17 2002

More terms, better description and formulas from Vladeta Jovovic, Vladimir Baltic, Nov 28 2002

A074139 Number of divisors of A036035(n,k).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64, 8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128, 9, 16, 21, 24, 25, 28, 36, 40, 45, 48, 48, 60, 64, 72, 81, 80, 96, 108, 128, 144, 192, 256

Offset: 0

Amarnath Murthy, Aug 28 2002

			Express A036035(n,k) by its prime signature; add one to each exponent, then multiply: 180 = (2^2)*(3^2)*(5^1) therefore the number of divisors is (2+1)*(2+1)*(1+1)= 18
From _Michel Marcus_, Nov 11 2015: (Start)
As an irregular triangle, whose n-th row has A000041(n) terms, sequence begins:
  1;
  2;
  3,  4;
  4,  6,  8;
  5,  8,  9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  ...
(End)

Alois P. Heinz, Rows n = 0..30, flattened
Byungchul Cha et al., An Investigation on Partitions with Equal Products, arXiv:1811.07451 [math.NT], 2018.

Row sums give A074141.

Cf. A036035, A074140.

tabf(nn) = {for (n=1, nn, forpart(p=n, print1(prod(k=1, #p, (1+p[k])), ", ")); print(););} \\ Michel Marcus, Nov 11 2015

T(n,k) = A000005(A036035(n,k)). - R. J. Mathar, Aug 28 2018

More terms from Alford Arnold, Sep 17 2002

Term ordering corrected by Alois P. Heinz, Aug 21 2019

A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1680, 1800, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 960, 1296, 1440

Offset: 0

Ray Chandler, Sep 04 2003

A025487 in a different order.

			1;
2;
4,6;
8,12,30;
16,24,36,60,210;
32,48,72,120,180,420,2310;
64,96,144,216,240,360,840,900,1260,4620,30030;
128,192,288,432,480,720,1080,1680,1800,2520,6300,9240,13860,60060,510510;

Alois P. Heinz, Rows n = 0..26, flattened

Cf. A025487, A036035, A059901, A063008, A077569, A074140 (row sums), A328524.

b:= proc(n, i, l)
      `if`(n=0, [mul(ithprime(t)^l[t], t=1..nops(l))],
      `if`(i=1, b(0, 0, [l[], 1$n]), [b(n, i-1, l)[],
      `if`(i>n, [], b(n-i, i, [l[], i]))[]]))
    end:
T:= n-> sort(b(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 13 2012

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Join[{Product[Prime[t]^l[[t]], {t, 1, Length[l]}]}], If[i == 1, b[0, 0, Join[l, Table[1, {n}]]], Join[b[n, i - 1, l], If[i > n, {}, b[n - i, i, Append[l, i]]]]]];
T[n_] := Sort[b[n, n, {}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 06 2017, after Alois P. Heinz *)

A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 5, 4, 0, 1, 10, 19, 8, 0, 1, 17, 69, 65, 16, 0, 1, 28, 188, 410, 211, 32, 0, 1, 41, 496, 1726, 2261, 665, 64, 0, 1, 58, 1029, 7182, 14343, 11970, 2059, 128, 0, 1, 77, 2015, 20559, 93345, 112371, 61909, 6305, 256, 0, 1, 100, 3478, 54814, 360612, 1139166, 848506, 315850, 19171, 512, 0

Offset: 0

Alois P. Heinz, Apr 27 2021

			A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0].
A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0].
Square array A(n,k) begins:
  1,  1,    1,     1,      1,        1,        1, ...
  0,  2,    5,    10,     17,       28,       41, ...
  0,  4,   19,    69,    188,      496,     1029, ...
  0,  8,   65,   410,   1726,     7182,    20559, ...
  0, 16,  211,  2261,  14343,    93345,   360612, ...
  0, 32,  665, 11970, 112371,  1139166,  5827122, ...
  0, 64, 2059, 61909, 848506, 13379332, 89131918, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-4 give: A000007, A000079, A001047(n+1), A016273, A025931.
Rows n=0-2 give: A000012, A007504, A357251.
Main diagonal gives A332967.
Cf. A000040, A074140, A124960, A324939, A325054.

A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1,
     If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)

A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).

A(n,k) = A124960(n+k,k) for k >= 1.

A325054 Sum of all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i).

Original entry on oeis.org

1, 2, 10, 68, 640, 8372, 147820, 3321908, 90184300, 2857153892, 104146026820, 4363900557128, 209763325978480, 11462371025215112, 702793156696129600, 47649412958404240688, 3521160558576929028400, 280427910532671712997732, 23932837097476310995036900

Offset: 0

Alois P. Heinz, Sep 04 2019

nonn

			The compositions of 3 and their encodings are [1,1,1]-> 30, [1,2]-> 18, [2,1]-> 12, [3]-> 8.  The sum gives a(3) = 68.

Alois P. Heinz, Table of n, a(n) for n = 0..335
Wikipedia, Gödel's encoding

Row sums of A324939.

Cf. A000040, A074140.

b:= proc(n, j) option remember; `if`(n=0, 1,
      add(ithprime(j)^i*b(n-i, j+1), i=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 04 2019

b[n_, j_] := b[n, j] = If[n==0, 1, Sum[Prime[j]^i*b[n-i, j+1], {i, 1, n}]];
a[n_] := b[n, 1];
a /@ Range[0, 20] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)

A332626 Sum of least integers of prime signatures over all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 4, 20, 40, 152, 664, 1760, 5680, 24752, 138064, 356480, 1568320, 5886752, 32781664, 266420000, 726928960, 3135277952, 16299729664, 81402739520, 640678081600, 7084434124352, 18897678264064, 92846198695040, 464088929482240, 3347512310365952

Offset: 0

Alois P. Heinz, Feb 17 2020

nonn

			a(5) = 2^5 + 2^4*3^1 + 2^3*3^2 = 32 + 48 + 72 = 152.

Alois P. Heinz, Table of n, a(n) for n = 0..712
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Row sums of A328524.

Cf. A000009, A025487, A074140.

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..30);

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, 0, If[n == 0, 1, b[n, i - 1, j] + Prime[j]^i b[n - i, Min[n - i, i - 1], j + 1]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 30] (* Jean-François Alcover, May 04 2020, after Maple *)

a(n) = Sum_{k=1..A000009(n)} A328524(n,k).

A227955 Triangle read by rows, T(n, k) = prime(1)^p(k,1)...prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n. The partitions of n are ordered in reversed lexicographic order read from left-to-right, starting with [1,1,...1] going down to [n].

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 8, 210, 60, 36, 24, 16, 2310, 420, 180, 120, 72, 48, 32, 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64, 510510, 60060, 13860, 6300, 9240, 2520, 1800, 1080, 1680, 720, 432, 480, 288, 192, 128, 9699690, 1021020, 180180, 69300, 44100

Offset: 0

Peter Luschny, Aug 01 2013

The sequence can be seen as an encoding of Young's lattice (see the links).
The ordering of Young's lattice is such that for two Young diagrams s, t, we have s <= t if and only if the Young diagram of s fits entirely inside the Young diagram of t (when the two diagrams are arranged so their lower-left corners coincide.) This order translates to our encoding as the divisibility relation. The number corresponding to s divides the number corresponding to t if and only if s <= t.
The partition corresponding to a number can be recovered as the exponents of the primes in the prime factorization of the number.

			For instance the partitions of 4 are ordered [1,1,1,1], [2,1,1,0], [2,2,0,0], [3,1,0,0], [4,0,0,0]. Consider the partition P = (3,2,1,1) written as a Young diagram (in French notation):
    [ ]
    [ ]
    [ ][ ]
    [ ][ ][ ]
Next replace the boxes at the bottom line by the sequence of primes and write the number of boxes in the same column as exponents; then multiply. 2^4*3^2*5^1 = 720. 720 will appear in line 7 of the triangle (because P is a partition of 7) at position 10 (because the sequence of exponents [4, 2, 1] is the 10th partition in the order of partitions which we assume).
[0]     1,
[1]     2,
[2]     6,    4,
[3]    30,   12,    8,
[4]   210,   60,   36,  24,  16,
[5]  2310,  420,  180, 120,  72,  48,  32,
[6] 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64.

Peter Luschny, Rows n = 0..25, flattened
Peter Luschny, Young's lattice (diagram)
Peter Luschny, Integer partition trees
Wikipedia, Young's lattice

Reversed rows: A036035, row sums: A074140.

with(combinat):
A227955_row := proc(n) local e, w, p;
p := [seq(ithprime(i), i=1..n)];
w := e -> mul(p[i]^e[nops(e)-i+1], i=1..nops(e));
seq(w(e), e = partition(n)) end:
seq(print(A227955_row(i)), i=0..8);

def A227955_row(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        L.append(mul(P[i]^p[i] for i in range(len(p))))
    return L[::-1]
for n in (0..8): A227955_row(n)

A228099 Triangle read by rows, T(n, k) = prime(1)^p(k,1)...prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n, additionally T(0,0) = 1. The partitions of n are ordered such that partitions of n into r parts appear in lexicographic order previous to the partitions of n into s parts if s < r. (Fenner-Loizou tree).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 8, 210, 60, 36, 24, 16, 2310, 420, 180, 120, 72, 48, 32, 30030, 4620, 1260, 840, 900, 360, 240, 216, 144, 96, 64, 510510, 60060, 13860, 9240, 6300, 2520, 1680, 1800, 1080, 720, 480, 432, 288, 192, 128, 9699690, 1021020, 180180, 120120

Offset: 0

Peter Luschny, Aug 10 2013

The partitions' representation (A228100) is a weakly decreasing list of parts.
The rows (read from left to right) are strongly decreasing. The T(n, 0) are the primorial numbers A002110(n). The right side of the triangle are the powers of 2,
T(n, A000041(n)) = A000079(n). The row sums are A074140.
The partition corresponding to a(n), n > 0, can be recovered as the exponents of the primes in the canonical prime factorization of a(n).

			The six-th row is:
[1, 1, 1, 1, 1, 1] -> 30030
[2, 1, 1, 1, 1] -> 4620
[2, 2, 1, 1] -> 1260
[3, 1, 1, 1] -> 840
[2, 2, 2] -> 900
[3, 2, 1] -> 360
[4, 1, 1] -> 240
[3, 3] -> 216
[4, 2] -> 144
[5, 1] -> 96
[6] -> 64

D. E. Knuth: The Art of Computer Programming. Generating all combinations and partitions, vol. 4, fasc. 3, 7.2.1.4, exercise 10.

Peter Luschny, Rows n = 0..25, flattened
T. I. Fenner, G. Loizou, A binary tree representation and related algorithms for generating integer partitions, The Computer J. 23(4), 332-337 (1980).
Peter Luschny, Integer Partition Trees

Cf. A228100.

b:= proc(n, i) b(n, i):= `if`(n=0 or i=1, [[1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [i, x[]], b(n-i, i)))[]])
    end:
T:= n-> map(h-> mul(ithprime(j)^h[j], j=1..nops(h)), sort(b(n$2),
        proc(x, y) local i; if nops(x)<>nops(y) then return
        nops(x)>nops(y) else for i to nops(x) do if x[i]<>y[i]
        then return x[i]Alois P. Heinz, Aug 13 2013

b[n_, i_] := If[n == 0 || i == 1, {Array[1&, n]}, Join[b[n, i-1], If[i>n, {}, Map[Function[x, Prepend[x, i]], b[n-i, i]]]]]; T[n_] := Map[Function[h, Times @@ ((Prime /@ Range[Length[h]])^h)], Sort[b[n, n], Which[Length[#1] > Length[#2], True, Length[#1] < Length[#2], False, True, OrderedQ[#1, #2]]&]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Maple *)

from collections import deque
def Partitions_Fenner_Loizou(n):
    p = ([], 0, n)
    queue = deque()
    queue.append(p)
    yield p
    while len(queue) > 0 :
        (phead, pheadLen, pnum1s) = queue.popleft()
        if pnum1s != 1 :
            head = phead[:pheadLen] + [2]
            q = (head, pheadLen + 1, pnum1s - 2)
            if 1 <= q[2] : queue.append(q)
            yield q
        if pheadLen == 1 or (pheadLen > 1 and \
                    (phead[pheadLen - 1] != phead[pheadLen - 2])) :
            head = phead[:pheadLen]
            head[pheadLen - 1] += 1
            q = (head, pheadLen, pnum1s - 1)
            if 1 <= q[2] : queue.append(q)
            yield q
def A228099_row(n):
    if n == 0: return [1]
    L = []
    P = primes_first_n(n)
    for p in Partitions_Fenner_Loizou(n):
        e = p[0] + [1 for i in range(p[2])]
        c = mul(P[i]^e[i] for i in range(len(e)))
        L.append(c)
    return L
for n in (0..7): A228099_row(n)

cp's OEIS Frontend

A074141 Sum of products of parts increased by 1 in all partitions of n.

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A074139 Number of divisors of A036035(n,k).

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A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

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A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A325054 Sum of all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i).

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A332626 Sum of least integers of prime signatures over all partitions of n into distinct parts.

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A227955 Triangle read by rows, T(n, k) = prime(1)^p(k,1)*...*prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n. The partitions of n are ordered in reversed lexicographic order read from left-to-right, starting with [1,1,...1] going down to [n].

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A227955 Triangle read by rows, T(n, k) = prime(1)^p(k,1)...prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n. The partitions of n are ordered in reversed lexicographic order read from left-to-right, starting with [1,1,...1] going down to [n].