A074139 -id:A074139 - cp's OEIS frontend

A122172 Triangle read by rows relating A074139, A074141, A078436 and A079025.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 3, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 7, 4, 1, 1, 5, 10, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Alford Arnold, Aug 23 2006

A proper definition is needed for this sequence.
Are the row sums A074139(n) and the row lengths A000041(n)? - R. J. Mathar, May 08 2019 [Not exactly: see below. - M. F. Hasler, Jan 07 2024]
From M. F. Hasler, Jan 06 2024: (Start)
I get this triangle as T(n,k) = # { v in S(p_n), |v| = k }, where p_n is the n-th partition as listed in A036036 or A036037 (which has a nice table of the p's), and S(p) = {0, ..., p[1]} x ... x {0, ..., p[#p]}, the set of vectors v with 0 <= v[i] <= p[i] for all indices i from 1 to #p = number of parts in p.
Then the row sums are indeed the total number of elements in S(p_n) which is equal to the product (p[1]+1)*...*(p[#p]+1) which is also the number of divisors of the Heinz number of p (cf. A185974).
The row lengths are 1 + |p| = 1 + sum of all parts of p (corresponding to the possible values of |v| ranging from 0 to |p|), repeated A000041(|p|) times: A000041(0) = 1 row of length 0+1 for the partition () of 0, A000041(1) = 1 row of length 1+1 for partition (1) of 1; A000041(2) = 2 rows of length 2+1 for the two partitions (2) and (1,1) of 2;  A000041(3) = 3 rows of length 3+1 for the 3 partitions {(3), (2,1), (1,1,1)} of 3; etc. (End)

			The triangle begins:
  1
  1 1
  1 1 1
  1 2 1
  1 1 1 1
  1 2 2 1
  1 3 3 1
  1 1 1 1 1
  1 2 2 2 1
  1 2 3 2 1
  1 3 4 3 1
  1 4 6 4 1
  1 1 1 1 1 1
  1 2 2 2 2 1
  1 2 3 3 2 1
  1 3 4 4 3 1
  1 3 5 5 3 1
  1 4 7 7 4 1
  1 5 10 10 5 1

M. F. Hasler, Table of n, a(n) for n = 0..9172 (up to partitions of 15), Jan 07 2024

Cf. A036036 (partitions in A-S order), A036037 (the same, parts reversed), A185974 (corresponding Heinz numbers).

A122172_row(n, p=part(n))={my(c=Vec(0, vecsum(p)+1)); forvec(v=[[0, k]| k<-p], c[vecsum(v)+1]++); c} \\ instead of n one can directly give p as 2nd arg
/* helper function: n-th partition as listed in A036036, A036037 or A185974 */
part(n)={my(c, r=0); while(n >= c = numbpart(r), n -= c; r++); partitions(r)[n+1]}
for(n=0,5, forpart(p=n, print(A122172_row(, Vec(p))) )) \\ Illustration. \\ M. F. Hasler, Jan 06 2024

More terms from M. F. Hasler, Jan 07 2024

A122401 Subsequence of A074139 omitting values derived from partitions with a part of size 1.

Original entry on oeis.org

1, 3, 4, 5, 9, 6, 12, 7, 15, 16, 27, 8, 18, 20, 36, 9, 21, 24, 25, 45, 48, 81, 10, 24, 28, 30, 54, 60, 64, 108, 11, 27, 32, 35, 63, 36, 72, 75, 80, 135, 144, 243, 12, 30, 36, 40, 72, 42, 84, 90, 96, 162, 100, 180, 192, 324, 13, 33, 40, 45, 81, 48, 96, 49, 105, 112, 189, 108, 120, 216, 125, 225, 240, 405, 256, 432, 729, 14, 36, 44, 50, 90, 54

Offset: 0

Alford Arnold, Sep 01 2006

When viewed as a table, row sums are given by sequence A079274.

Corresponds to members of A036035 which are also powerful numbers (A001694).

			The two cyclic partitions of five are 5 and 3+2 yielding (5+1)=6 and (3+1)*(2+1) = 4*3 = 12
The array begins
1
(empty)
3
4
5 9
6 12
7 15 16 27
8 18 20 36

Cf. A122172 A079274 A122402, A316532.

A122401_row := proc(n)
    local e, a,L;
    L := [] ;
    for e in ListTools[Reverse](partition(n)) do
        if member(1,e) then
            ;
        else
            a := 1;
            for p in e do
                a := a*(p+1) ;
            end do:
            L := [op(L),a] ;
        end if;
    end do:
    L ;
end proc:
seq(A122401_row(i), i=0..15); # R. J. Mathar, Aug 28 2018

Extended by R. J. Mathar, Aug 28 2018

A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

Original entry on oeis.org

0, 1, 3, 7, 15, 29, 54, 95, 163, 270, 439, 696, 1088, 1669, 2530, 3780, 5591, 8173, 11845, 17000, 24215, 34210, 48008, 66895, 92660, 127554, 174651, 237830, 322297, 434625, 583524, 779972, 1038356, 1376787, 1818755, 2393775, 3139812, 4104433, 5348375, 6947545, 8998201, 11620313, 14965126, 19220569

Offset: 0

Thomas Wieder, Sep 11 2006

nonn

Consider a fragmentation process of an n-object which consists of n unlabeled elements (= 1-parts). By definition the n-object can scatter into up to n m-parts where an m-part consists of 1 up to n elements. A 4-object can split up for example into 4 1-parts which corresponds to the integer partition [1,1,1,1], or it can, for example, rest unfragmented which corresponds to [4]. Since the number of integer partitions of n=4 equals 5, there are 5 n=4-fragmentation processes.
Now we ask for the probability of getting an m-part after an n-fragmentation. Think of a Greek statue which had been broken into n parts and covered by earth. We could find several m-parts, in the most lucky case we would find all m-parts which add up to m_1+m_2+...+m_n=n. Then the statue could be restored.
For example for n=4 we could ask for the probability prob(n=4,m=2) of just a single 2-part. We have 2 cases for a 2-part and we have 15 cases in total, thus prob(n=4,m=2)=2/15 (the 2 cases come from [1,1,2] and [2,2]). The chances to find the two 2-parts from the [2,2]-fragmentation are 1/15 only. The chances to find the n=4-object unsplitted are also 1/15 only.
This sequence is generated over the unordered partitions; for example, when n = 4 there are 1+3+2+5+4 = 15 cases. If we allow a null case for each of the five partitions then we have 15+5 = 20 which is A000712(4). - Alford Arnold, Dec 12 2006
Number of partitions into two kinds of parts with the first kind of parts used in each partition. - Joerg Arndt, Jun 21 2011

			a(n=4) = 15 because the possible combinations of all five integer partitions of n=4 are: [1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1], [2], [1, 1], [1, 2], [1, 1, 2], [2], [2, 2], [1], [3], [1, 3], [4].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A000079, A000262.
Cf. A000712, A074139, A074141.
Cf. A048574, A144064.

a122768 n = a122768_list !! n
a122768_list = 0 : f (tail a000041_list) [1] where
   f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

A122768 := proc(n::integer) local i,j,prttnlst,prttn,ZahlTeile,H; prttnlst:=partition(n); H := NULL; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); for j from 1 to ZahlTeile do H := H,op(choose(prttn,j)); od; od; print(n,H,nops([H])); end proc;
A000712 := proc(n) option remember ; add(combinat[numbpart](k)*combinat[numbpart](n-k),k=0..n) ; end: A000041 := proc(n) combinat[numbpart](n) ; end: A122768 := proc(n::integer) RETURN( A000712(n)-A000041(n)) ; end: for n from 0 to 80 do printf("%d,",A122768(n)) ; od: # R. J. Mathar, Aug 25 2008
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      k*numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n,2)-b(n,1):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 31 2017

1/QPochhammer[x]^2 - 1/QPochhammer[x] + O[x]^50 // CoefficientList[#, x]& (* Jean-François Alcover, Feb 05 2017, after Joerg Arndt *)

x='x+O('x^66); /* that many terms */
Vec(1/eta(x)^2-1/eta(x)) /* show terms (omitting initial zero) */
/* Joerg Arndt, Jun 21 2011 */

from sympy import npartitions
def A122768(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(1,n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) + npartitions(n) if n else 0 # Chai Wah Wu, Sep 25 2023

G.f.: 1/P(x)^2 - 1/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011
With sum_i^P(n) = the sum over all P(n) integer partitions of n, sum_j^p(i) = the sum over all p(i) parts of the i-th integer partition, prttn(i) = the i-th partition whereat prttn(i) is a list, choose(L,k) = construct the list LC of combinations of a list L (see Maple), |LC| = number of elements of list LC (=Maple's nops command) we have a(n) = sum_i^P(n) sum_j^p(i) |choose(prttn,j)|
a(n) = A000712(n) - A000041(n). - Alford Arnold, Dec 12 2006
a(n) = A144064(n,2)-A144064(n,1). - Alois P. Heinz, Mar 31 2017
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/12 + 45/(16*Pi))/sqrt(3*n)). - Vaclav Kotesovec, Mar 31 2017

Extended by R. J. Mathar, Aug 25 2008

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

Original entry on oeis.org

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

A079025 Triangular array read by rows: column sums of frequency distributions associated with number of divisors of least prime signatures.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 6, 6, 3, 5, 12, 16, 12, 5, 7, 20, 32, 32, 20, 7, 11, 35, 65, 79, 65, 35, 11, 15, 54, 113, 160, 160, 113, 54, 15, 22, 86, 199, 318, 371, 318, 199, 86, 22, 30, 128, 323, 573, 756, 756, 573, 323, 128, 30, 42, 192, 523, 1013, 1485, 1683, 1485, 1013, 523, 192, 42

Offset: 0

Alford Arnold, Feb 01 2003

Row sums of the triangular table is sequence A074141. The left column and the main diagonal are the partition numbers A000041.

T(n,k) is the total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of k and m runs through the set of least numbers whose prime signature is a partition of n. - Alois P. Heinz, Aug 23 2019

			The seven least integers associated with prime signatures 5, 41, 32, 311, 221, 2111, 11111 (partitions of 5) are 32, 48, 72, 120, 180, 420 and 2310 (see A036035).  The corresponding numbers of divisors 6, 10, 12, 16, 18, 24 and 32 (see A074139) can be refined with the following frequency distributions D(p,s), which counts how many divisors of the entry of A036035 have a sum of prime exponents s, 0<=s<=n:
  1  1  1  1  1 1
  1  2  2  2  2 1
  1  2  3  3  2 1
  1  3  4  4  3 1
  1  3  5  5  3 1
  1  4  7  7  4 1
  1  5 10 10  5 1 , therefore the column sums are:
  7 20 32 32 20 7 , which is row 5 of the triangle.
Triangle T(n,k) begins:
    1
    1   1
    2   3    2
    3   6    6    3
    5  12   16   12    5
    7  20   32   32   20     7
   11  35   65   79   65    35    11
   15  54  113  160  160   113    54    15
   22  86  199  318  371   318   199    86    22
   30 128  323  573  756   756   573   323   128   30
   42 192  523 1013 1485  1683  1485  1013   523  192   42
   56 275  803 1683 2701  3405  3405  2701  1683  803  275   56
   77 399 1237 2776 4822  6662  7413  6662  4822 2776 1237  399  77
  101 556 1826 4366 8144 12205 14901 14901 12205 8144 4366 1826 556 101
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000041, A006128, A309691, A309693, A309919, A309920, A309921, A309922, A309923, A309924, A309925.
Row sums give A074141.
T(2n,n) gives A309915.
Cf. A036035, A074139, A079474, A087443.

A079025 := proc(n,k)
    local psig ,d,a;
    a := 0 ;
    for psig in A036035_row(n) do
        for d in numtheory[divisors](psig) do
            if numtheory[bigomega](d) = k then
                a := a+1 ;
            end if:
        end do:
    end do:
    a ;
end proc:
for n from 0 to 13 do
    for k from 0 to n do
        printf("%d ",A079025(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Aug 28 2018
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
      b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 22 2019

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)]*b[n - i, Min[n - i, i]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

A074140 Sum of least integers of prime signatures over all partitions of n.

Original entry on oeis.org

1, 2, 10, 50, 346, 3182, 38770, 609290, 11226106, 250148582, 7057182250, 216512001950, 7903965900226, 321552174623162, 13779150603234010, 644574260638821590, 33968684108427733426, 1994885097404292104942, 121496572792097514728530, 8114030083731371137603190

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Old name was: Sum of terms in n-th group in A036035.

a(n) is also the sum of terms in n-th row of A063008, A087443 or A227955.

			a(6) = 64+96+144+216+240+360+900+840+1260+4620+30030 = 38770.

Peter Luschny and Alois P. Heinz, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Cf. A036035, A063008, A074139, A074141, A025487, A087443, A227955, A332626.

b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, ithprime(j)^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013

b[n_, i_, j_] := b[n, i, j] = If[n == 0, 1, If[i<1, 0, b[n, i-1, j]+If[i>n, 0, Prime[j]^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

def A074140(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        m = mul(P[i]^pi for i, pi in enumerate(p))
        L.append(m)
    return add(L)
[A074140(n) for n in (0..20)]  # Peter Luschny, Aug 02 2013

More terms from Alford Arnold, Sep 10 2002
a(10)-a(12) from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Sep 30 2004
a(12) corrected by Peter Luschny, Aug 03 2013
New name from Alois P. Heinz, Aug 03 2013

A238963 Number of divisors of A063008(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, 20, 16, 24, 32, 27, 36, 48, 64, 8, 14, 18, 24, 20, 30, 40, 32, 36, 48, 64, 54, 72, 96, 128, 9, 16, 21, 28, 24, 36, 48, 25, 40, 45, 60, 80, 48, 64, 72, 96, 128, 81, 108, 144, 192, 256, 10, 18, 24, 32, 28, 42, 56, 30, 48, 54, 72, 96, 50, 60, 80, 90, 120, 160, 64, 96, 128, 108, 144, 192, 256, 162, 216, 288, 384, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

Equivalent to A074139 but using canonical order.

			Triangle begins:
  1;
  2;
  3,  4;
  4,  6,  8;
  5,  8,  9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283

Row sums are A074141.

Cf. A000005, A000041, A063008, A074139.

b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> numtheory[tau](mul(ithprime(i)
        ^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 24 2020

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := DivisorSigma[0, #]&[Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n]];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

\\ here b(n) is A000005.
b(n)={numdiv(n)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 24 2020

def A238963row(n):
    return list(product(t + 1 for t in p) for p in Partitions(n))
print([A238963row(n) for n in range(10)])  # Peter Luschny, Dec 11 2023

T(n, k) = A000005(A063008(n,k)).

Trow(n) = List_{p in Partitions(n)} (Product_{t in p}(t + 1)). # Peter Luschny, Dec 11 2023

Offset corrected by Andrew Howroyd, Mar 24 2020

A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=23, p(3)=23*5,...).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 8, 36, 30, 1, 16, 216, 900, 210, 1, 32, 1296, 27000, 44100, 2310, 1, 64, 7776, 810000, 9261000, 5336100, 30030, 1, 128, 46656, 24300000, 1944810000, 12326391000, 901800900, 510510, 1, 256, 279936, 729000000, 408410100000

Offset: 1

Alford Arnold, Jan 15 2003

In the expansion of [1+x+x^2+...+x^(r-s)]^s, the x^n coefficient states how many factors of a(r,s) have n prime factors.

As a square array A(n,k) n>=0 k>=1 read by descending antidiagonals, A(n,k) when n>=1 is the least common period over the positive integers of the occurrence of the first n prime numbers as the k-th least operand in the respective integers' prime factorizations (written without exponents). - Peter Munn, Jan 25 2017

			Triangle starts
  1;
  1,  2;
  1,  4,    6;
  1,  8,   36,    30;
  1, 16,  216,   900,   210;
  1, 32, 1296, 27000, 44100, 2310;
  ...

Cf. A002110, A007318, A008287, A027907, A036035, A061742, A074139.

a(2n,n) gives A181555.

p:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*p(n-1)) end:
a:= (r, s)-> p(s)^(r-s):
seq(seq(a(r, s), s=0..r-1), r=0..10);  # Alois P. Heinz, Aug 22 2019

p[0] = 1; p[s_] := p[s] = Prime[s] p[s-1];
a[r_, s_] := p[s]^(r-s);
Table[a[r, s], {r, 0, 10}, {s, 0, r-1}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Edited by Don Reble, Nov 02 2005

A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

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, 60, 64, 72, 80, 81, 96, 108, 128, 144, 192, 256

Offset: 0

Amiram Eldar, Jun 16 2022

First differs from A074139 at the 8th row.
The n-th row begins with n+1, which corresponds to powers of primes, and ends with 2^n, which corresponds to squarefree numbers.
The n-th row contains the distinct values of the n-th row of A238963.

			Table begins:
  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;
  ...
Numbers k with Omega(k) = 2 are either of the form p^2 with p prime, or of the form p1*p2 with p1 and p2 being distinct primes. The corresponding numbers of divisors are 3 and 4, respectively. Therefore the second row is {3, 4}.

Amiram Eldar, Table of n, a(n) for n = 0..18645 (rows 0..32, flattened)

Cf. A000005, A001222, A036035, A063008, A074139, A238963, A355027 (row lengths).

row[n_] := Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[row, 9, 0] // Flatten

row(n) = { my (m=Map()); forpart(p=n, mapput(m,prod(k=1, #p, 1+p[k]),0)); Vec(m) } \\ Rémy Sigrist, Jun 17 2022

A122979 Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Abramowitz and Stegun order), that include the maximum element.

Original entry on oeis.org

2, 4, 7, 8, 21, 45, 16, 58, 84, 200, 500, 32, 152, 293, 748, 1184, 3220

Offset: 1

Franklin T. Adams-Watters, Sep 21 2006

After a(18) - for partition [1^5] - the sequence continues ?, 64, 384, 938, 1238, 2520, 5591, ?, ?, ?, ?, ?, 128.

			For a(5), partition [2,1], the lattice consists of the 6 pairs (i,j) where 0<=i<=2 and 0<=j<=1, with (i,j) <= (i',j') iff i<=i' and j<=j'. {(2,1), (2,0), (0,1), (0,0)} is one distributive sublattice.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A122977, A122980, A122981, A036036, A074139.

cp's OEIS Frontend

A122172 Triangle read by rows relating A074139, A074141, A078436 and A079025.

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A122401 Subsequence of A074139 omitting values derived from partitions with a part of size 1.

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A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

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

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Maxima

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A079025 Triangular array read by rows: column sums of frequency distributions associated with number of divisors of least prime signatures.

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A074140 Sum of least integers of prime signatures over all partitions of n.

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Sage

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A238963 Number of divisors of A063008(n,k).

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A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=23, p(3)=23*5,...).