A320656 -id:A320656 - cp's OEIS frontend

A340655 Number of twice-balanced factorizations of n.

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 2, 0, 2, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
  2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
  3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
        2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
        2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
                 3*4*30    4*8*15    4*5*45    3*3*5*56
                 3*6*20    5*8*12    5*6*30    3*3*7*40
                 3*8*15    6*8*10    5*9*20    3*5*7*24
                 4*5*18    2*12*20   2*10*45   2*2*2*315
                 5*6*12    4*10*12   2*15*30   2*2*3*210
                 2*10*18             2*18*25   2*2*5*126
                 2*12*15             3*10*30   2*3*3*140
                 3*10*12             3*12*25
                                     3*15*20
                                     5*10*18
                                     5*12*15

The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340652.
The balanced version is A340653.
The cross-balanced version is A340654.
Positions of zeros are A340656.
Positions of nonzero terms are A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, A340608.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],#=={}||Length[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]

A338901 Position of the first appearance of prime(n) as a factor in the list of squarefree semiprimes.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 9, 11, 13, 17, 18, 21, 23, 25, 29, 31, 34, 36, 40, 42, 45, 47, 50, 52, 56, 58, 61, 64, 67, 70, 76, 78, 81, 82, 86, 89, 93, 97, 100, 104, 106, 107, 112, 113, 116, 118, 125, 129, 133, 134, 135, 139, 141, 147, 150, 154, 159, 160, 165, 167, 169

Offset: 1

Gus Wiseman, Nov 16 2020

nonn

The a(n)-th squarefree semiprime is the first divisible by prime(n).

After a(1) = 1, these are the positions of even terms in the list of all squarefree semiprimes A006881.

Amiram Eldar, Table of n, a(n) for n = 1..10000

A001358 lists semiprimes, with odds A046315 and evens A100484.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A115392 is the not necessarily squarefree version.
A166237 gives the first differences of squarefree semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898 gives prime indices of semiprimes, with differences A176506.
A338899 gives prime indices of squarefree semiprimes, differences A338900.
A338912 and A338913 give the prime indices of semiprimes.
Cf. A001221, A001222, A002100, A056239, A065516, A112798, A167171, A320891, A320911, A338903, A338905.

rs=First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Position[rs,i][[1,1]],{i,Union@@rs}]

A006881(a(n)) = A100484(n).

A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 6, 8, 12, 16, 21, 28, 37, 49, 64, 80, 104, 135, 169, 216, 268, 341, 420, 527, 654, 809, 991, 1218, 1488, 1828, 2213, 2687, 3262, 3934, 4754, 5702, 6849, 8200, 9819, 11693, 13937, 16562, 19659, 23262, 27577, 32493, 38341, 45112, 53059, 62265

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a loop-graphical partition (A339656, A339658).

			The a(2) = 1 through a(10) = 16 partitions:
  (11)  (21)  (22)  (32)    (33)    (43)    (44)    (54)      (55)
              (31)  (41)    (42)    (52)    (53)    (63)      (64)
                    (2111)  (51)    (61)    (62)    (72)      (73)
                            (2211)  (2221)  (71)    (81)      (82)
                            (3111)  (3211)  (3221)  (3222)    (91)
                                    (4111)  (3311)  (3321)    (3322)
                                            (4211)  (4221)    (3331)
                                            (5111)  (4311)    (4222)
                                                    (5211)    (4321)
                                                    (6111)    (4411)
                                                    (222111)  (5221)
                                                    (321111)  (5311)
                                                              (6211)
                                                              (7111)
                                                              (322111)
                                                              (421111)
For example, the partition (4,2,1,1,1,1) can be partitioned into {{1,1},{1,2},{1,4}}, and thus is counted under a(10).

Eric Weisstein's World of Mathematics, Graphical partition.

A320912 gives the Heinz numbers of these partitions.
A338915 counts the complement in even-length partitions.
A339563 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A007717, A025065, A320656, A320732, A320893, A320921, A338898, A338902, A339564.

stfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],stfs[Times@@Prime/@#]!={}&]],{n,0,20}]

A027187(n) = a(n) + A338915(n).

More terms from Jinyuan Wang, Feb 14 2025

A339656 Number of loop-graphical integer partitions of 2n.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 49, 84, 140, 229, 367, 577, 895, 1368, 2064, 3080, 4547, 6642, 9627, 13825, 19704, 27868, 39164, 54656, 75832, 104584

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the unordered prime signature of n is loop-graphical.

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)    (2,2)      (3,3)          (3,3,2)
      (1,1)  (3,1)      (2,2,2)        (4,2,2)
             (2,1,1)    (3,2,1)        (4,3,1)
             (1,1,1,1)  (4,1,1)        (2,2,2,2)
                        (2,2,1,1)      (3,2,2,1)
                        (3,1,1,1)      (3,3,1,1)
                        (2,1,1,1,1)    (4,2,1,1)
                        (1,1,1,1,1,1)  (5,1,1,1)
                                       (2,2,2,1,1)
                                       (3,2,1,1,1)
                                       (4,1,1,1,1)
                                       (2,2,1,1,1,1)
                                       (3,1,1,1,1,1)
                                       (2,1,1,1,1,1,1)
                                       (1,1,1,1,1,1,1,1)
For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely
  {{1,1},{2,2},{3,4}}
  {{1,1},{2,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,4},{2,3}}
  {{1,3},{1,4},{2,2}},
so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely
  {{1,1},{1,1},{2,2}}
  {{1,1},{1,2},{1,2}},
but neither of these is a loop-graph, so z is not counted under a(3).

Eric Weisstein's World of Mathematics, Graphical partition.

A339658 ranks these partitions.
A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A062740 counts labeled connected loop-graphs.
A320461 ranks normal loop-graphs.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A322661 counts covering loop-graphs.
A339845 counts the same partitions by length, or A339844 with zeros.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A000569 counts graphical partitions (A320922).
- A058696 counts partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n (A339618).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 [this sequence] counts loop-graphical partitions (A339658).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]!={}&]],{n,0,5}]

A058696(n) = a(n) + A339655(n).

a(8)-a(25) from Andrew Howroyd, Jan 10 2024

A002135 Number of terms in a symmetrical determinant: a(n) = na(n-1) - (n-1)(n-2)*a(n-3)/2.

Original entry on oeis.org

1, 1, 2, 5, 17, 73, 388, 2461, 18155, 152531, 1436714, 14986879, 171453343, 2134070335, 28708008128, 415017867707, 6416208498137, 105630583492969, 1844908072865290, 34071573484225549, 663368639907213281, 13580208904207073801

Offset: 0

N. J. A. Sloane

a(n) is the number of collections of necklaces created by using exactly n different colored beads (to make the entire collection). - Geoffrey Critzer, Apr 19 2009
a(n) is the number of ways that a deck with 2 cards of each of n types may be dealt into n hands of 2 cards each, assuming that the order of the hands and the order of the cards in each hand are irrelevant. See the Art of Problem Solving link for proof. - Joel B. Lewis, Sep 30 2012
From Bruce Westbury, Jan 22 2013: (Start)
It follows from the respective exponential generating functions that A002135 is the binomial transform of A002137:
A002135(n) = Sum_{k=0..n} binomial(n,k)*A002137(k),
2 = 1.1 + 2.0 + 1.1,
5 = 1.1 + 3.0 + 3.1 + 1.1,
17 = 1.1 + 4.0 + 6.1 + 4.1 + 1.6, ...
A002137 arises from looking at the dimension of the space of invariant tensors of the r-th tensor power of the adjoint representation of the symplectic group Sp(2n) (for n large compared to r).
(End)
a(n) is the number of representations required for the symbolic central moments of order 2 for the multivariate normal distribution, that is,   E[X1^2 X2^2 .. Xn^2|mu=0, Sigma] (Phillips 2010). These representations are the upper-triangular, positive integer matrices for which for each i, the sum of the i-th row and i-th column equals 2, the power of each component. This can be shown starting from the formulation by Joel B Lewis. See "Proof for multivariate normal moments" link below for a proof. - Kem Phillips, Aug 20 2014
Equivalent to Critzer's comment, a(n) is the number of ways to cover n labeled vertices by disjoint undirected cycles, hence the exponential transform of A001710(n - 1). - Gus Wiseman, Oct 20 2018

			For n = 3, the a(3) = 5 ways to deal out the deck {1, 1, 2, 2, 3, 3} into three two-card hands are {11, 22, 33}, {12, 12, 33}, {13, 13, 22}, {11, 23, 23}, {12, 13, 23}. - _Joel B. Lewis_, Sep 30 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a_n.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.9 and Problem 5.22.

T. D. Noe, Table of n, a(n) for n=0..100
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5. [Annotated scanned copy]
Art of Problem Solving, Partitioning a deck with 2 cards in n types into pairs
A. Cayley, On the number of distinct terms in a symmetrical or partially symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, pp. 185-190. [Annotated scanned copy]
Tomislav Došlic and Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - _N. J. A. Sloane_, May 01 2012
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.
Toufik Mansour, The length of the initial longest increasing sequence in a permutation, Art Disc. Appl. Math. (2023).
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]
K. Phillips, R functions to symbolically compute the central moments of the multivariate normal distribution, Journal of Statistical Software, Feb 2010.
Kem Phillips, Proof for multivariate normal moments
Richard Stanley and J. Riordan, Problem E2297, Amer. Math. Monthly, 79 (1972), 519-520.

A diagonal of A260338.
Row sums of A215771.
Column k=2 of A257463 and A333467.
Cf. A002137, A059422, A059423, A059424.
Cf. A001147, A007717, A037143, A001710, A215771, A319225, A319226, A320655, A320656.

G:=proc(n) option remember; if n <= 1 then 1 elif n=2 then
2 else n*G(n-1)-binomial(n-1,2)*G(n-3); fi; end;

a[x_]:=Log[1/(1-x)^(1/2)]+x/2+x^2/4;Range[0, 20]! CoefficientList[Series[Exp[a[x]], {x, 0, 20}], x]
RecurrenceTable[{a[0]==a[1]==1,a[2]==2,a[n]==n*a[n-1]-(n-1)(n-2)* a[n-3]/2}, a,{n,30}] (* Harvey P. Dale, Dec 16 2011 *)
Table[Sum[Binomial[k, i] Binomial[i - 1/2, n - k] (3^(k - i) n!)/(4^k k!) (-1)^(n - k - i), {k, 0, n}, {i, 0, k}], {n, 0, 12}] (* Emanuele Munarini, Aug 25 2017 *)

a(n):=sum(sum(binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-k-i),i,0,k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 25 2017 */

a(n) = if(n<3, [1,1,2][n+1], n*a(n-1) - (n-1)*(n-2)*a(n-3)/2 ); /* Joerg Arndt, Apr 07 2013 */

E.g.f.: (1-x)^(-1/2)*exp(x/2+x^2/4).
D-finite with recurrence a(n+1) = (n+1)*a(n) - binomial(n, 2)*a(n-2). See Comtet.
Asymptotics: a(n) ~ sqrt(2)*exp(3/4-n)*n^n*(1+O(1/n)). - Pietro Majer, Oct 27 2009
E.g.f.: G(0)/sqrt(1-x) where G(k) = 1 + x*(x+2)/(4*(2*k+1) - 4*x*(x+2)*(2*k+1)/(x*(x+2) + 8*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-k-i). - Emanuele Munarini, Aug 25 2017

A339559 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 3, 7, 6, 14, 14, 23, 27, 41, 47, 70, 84, 114, 141, 190, 225, 303, 370, 475, 578, 738, 890, 1131, 1368, 1698, 2058, 2549, 3048, 3759, 4505, 5495, 6574, 7966, 9483, 11450, 13606, 16307, 19351, 23116, 27297, 32470, 38293, 45346, 53342, 62939

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a non-graphical partition.

			The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
  11   .   22     2111   33       2221     44         3222       55
           1111          2211     4111     2222       6111       3322
                         3111     211111   3311       222111     3331
                         111111            5111       321111     4222
                                           221111     411111     4411
                                           311111     21111111   7111
                                           11111111              222211
                                                                 322111
                                                                 331111
                                                                 421111
                                                                 511111
                                                                 22111111
                                                                 31111111
                                                                 1111111111
For example, the partition y = (4,4,3,3,2,2,1,1,1,1) can be partitioned into a multiset of edges in just three ways:
  {{1,2},{1,2},{1,3},{1,4},{3,4}}
  {{1,2},{1,3},{1,3},{1,4},{2,4}}
  {{1,2},{1,3},{1,4},{1,4},{2,3}}
None of these are strict, so y is counted under a(22).

Eric Weisstein's World of Mathematics, Graphical partition.

A320894 ranks these partitions (using Heinz numbers).
A338915 allows equal pairs (x,x).
A339560 counts the complement in even-length partitions.
A339564 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A089259, A292432, A320893, A338899, A338903, A339619.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]],{n,0,15}]

A027187(n) = a(n) + A339560(n).

More terms from Jinyuan Wang, Feb 14 2025

A339116 Triangle of all squarefree semiprimes grouped by greater prime factor, read by rows.

Original entry on oeis.org

6, 10, 15, 14, 21, 35, 22, 33, 55, 77, 26, 39, 65, 91, 143, 34, 51, 85, 119, 187, 221, 38, 57, 95, 133, 209, 247, 323, 46, 69, 115, 161, 253, 299, 391, 437, 58, 87, 145, 203, 319, 377, 493, 551, 667, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899

Offset: 2

Gus Wiseman, Dec 01 2020

A squarefree semiprime is a product of any two distinct prime numbers.

			Triangle begins:
   6
  10  15
  14  21  35
  22  33  55  77
  26  39  65  91 143
  34  51  85 119 187 221
  38  57  95 133 209 247 323
  46  69 115 161 253 299 391 437
  58  87 145 203 319 377 493 551 667
  62  93 155 217 341 403 527 589 713 899

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)

A339194 gives row sums.
A100484 is column k = 1.
A001748 is column k = 2.
A001750 is column k = 3.
A006094 is column k = n - 1.
A090076 is column k = n - 2.
A319613 is the central column k = 2*n.
A087112 is the not necessarily squarefree version.
A338905 is a different triangle of squarefree semiprimes.
A339195 is the generalization to all squarefree numbers, row sums A339360.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd terms A046388.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
A332765 gives the greatest squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A014342, A098350, A112798, A168472, A320656, A338901, A339003, A339114/A339115.
Subsequence of A019565.

Table[Prime[i]*Prime[j],{i,2,10},{j,i-1}]

row(n) = {prime(n)*primes(n-1)}
{ for(n=2, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

T(n,k) = prime(n) * prime(k) for k < n.

Offset corrected by Andrew Howroyd, Jan 19 2023

A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

Original entry on oeis.org

7, 13, 19, 21, 22, 29, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 154, 155, 156, 159, 163, 165, 166, 169, 171

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

Equals the image of A181819 applied to the set of terms of A320892.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656, with Heinz numbers A339658.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The sequence of terms together with their prime indices begins:
      7: {4}         57: {2,8}      107: {28}
     13: {6}         61: {18}       111: {2,12}
     19: {8}         62: {1,11}     113: {30}
     21: {2,4}       66: {1,2,5}    115: {3,9}
     22: {1,5}       71: {20}       116: {1,1,10}
     29: {10}        76: {1,1,8}    117: {2,2,6}
     34: {1,7}       79: {22}       118: {1,17}
     37: {12}        82: {1,13}     121: {5,5}
     39: {2,6}       85: {3,7}      129: {2,14}
     43: {14}        87: {2,10}     130: {1,3,6}
     46: {1,9}       89: {24}       131: {32}
     49: {4,4}       91: {4,6}      133: {4,8}
     52: {1,1,6}     94: {1,15}     134: {1,19}
     53: {16}       101: {26}       136: {1,1,1,7}
     55: {3,5}      102: {1,2,7}    138: {1,2,9}
For example, the three loop-multigraphs with degrees y = (5,2,1) are:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a loop-graph (they have multiple edges), the Heinz number 66 is in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A320892 has these prime shadows (see A181819).
A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.
A339655 counts these partitions.
A339658 ranks the complement, counted by A339656.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339844 counts loop-graphical partitions by length.
factorizations of n into distinct primes or squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657 [this sequence]).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A338898, A338912, A338913, A339742, A339839.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[50],EvenQ[Length[nrmptn[#]]]&&Select[mpsbin[nrmptn[#]],UnsameQ@@#&]=={}&]

A300061 = A339657 \/ A339658.

A025129 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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

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

			From _Gus Wiseman_, Dec 05 2020: (Start)
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)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sum of prime(i) * prime(j) for i + j = n, i != j.

Cf. A000040, A258323.
The nonsquarefree version is A024697 (shifted right).
Row sums of A338905 (shifted right).
A332765 is the greatest among these squarefree semiprimes.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A014342 is the self-convolution of the primes.
A056239 is the sum of prime indices of n.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339194 sums squarefree semiprimes grouped by greater prime factor.
Cf. A001221, A005117, A062198, A098350, A168472, A320656, A338900, A338901, A338904, A339114, A339116.

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)
-- Reinhard Zumkeller, Apr 07 2014

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 *)

A025129=n->sum(k=1,n\2,prime(k)*prime(n-k+1)) \\ M. F. Hasler, Apr 06 2014

a(n) = A024697(n) for even n. - M. F. Hasler, Apr 06 2014

Following suggestions by Robert Israel and N. J. A. Sloane, initial 0=a(1) added by M. F. Hasler, Apr 06 2014

A332765 Consider all permutations p_i of the first n primes; a(n) is the minimum over p_i of the maximal product of two adjacent primes in the permutation.

Original entry on oeis.org

6, 10, 15, 22, 35, 55, 77, 91, 143, 187, 221, 253, 323, 391, 493, 551, 667, 713, 899, 1073, 1189, 1271, 1517, 1591, 1763, 1961, 2183, 2419, 2537, 2773, 3127, 3233, 3599, 3953, 4189, 4331, 4757, 4897, 5293, 5723, 5963, 6499, 6887, 7171, 7663, 8051, 8633, 8989, 9797, 9991, 10403, 10807

Offset: 2

Bobby Jacobs, Apr 23 2020

nonn

The optimal permutation of n primes is {p_n, p_1, p_n-1, p_2, …, p_ceiling(n/2)}. - Ivan N. Ianakiev, Apr 28 2020

Also the greatest squarefree semiprime whose prime indices sum to n + 1. A squarefree semiprime (A006881) 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 06 2020

			Here are the ways (up to reversal) to order the first four primes:
  2, 3, 5, 7: Products: 6, 15, 35;  Largest product: 35
  2, 3, 7, 5: Products: 6, 21, 35;  Largest product: 35
  2, 5, 3, 7: Products: 10, 15, 21; Largest product: 21
  2, 5, 7, 3: Products: 10, 35, 21; Largest product: 35
  2, 7, 3, 5: Products: 14, 21, 15; Largest product: 21
  2, 7, 5, 3: Products: 14, 35, 15; Largest product: 35
  3, 2, 5, 7: Products: 6, 10, 35;  Largest product: 35
  3, 2, 7, 5: Products: 6, 14, 35;  Largest product: 35
  3, 5, 2, 7: Products: 15, 10, 14; Largest product: 15
  3, 7, 2, 5: Products: 21, 14, 10; Largest product: 21
  5, 2, 3, 7: Products: 10, 6, 21;  Largest product: 21
  5, 3, 2, 7: Products: 15, 6, 14;  Largest product: 15
The minimum largest product is 15, so a(4) = 15.
From _Gus Wiseman_, Dec 06 2020: (Start)
The sequence of terms together with their prime indices begins:
      6: {1,2}     551: {8,10}    3127: {16,17}
     10: {1,3}     667: {9,10}    3233: {16,18}
     15: {2,3}     713: {9,11}    3599: {17,18}
     22: {1,5}     899: {10,11}   3953: {17,19}
     35: {3,4}    1073: {10,12}   4189: {17,20}
     55: {3,5}    1189: {10,13}   4331: {18,20}
     77: {4,5}    1271: {11,13}   4757: {19,20}
     91: {4,6}    1517: {12,13}   4897: {17,23}
    143: {5,6}    1591: {12,14}   5293: {19,22}
    187: {5,7}    1763: {13,14}   5723: {17,25}
    221: {6,7}    1961: {12,16}   5963: {19,24}
    253: {5,9}    2183: {12,17}   6499: {19,25}
    323: {7,8}    2419: {13,17}   6887: {20,25}
    391: {7,9}    2537: {14,17}   7171: {20,26}
    493: {7,10}   2773: {15,17}   7663: {22,25}
(End)

Cf. A332877, A333747.
A338904 and A338905 have this sequence as row maxima.
A339115 is the not necessarily squarefree version.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A025129 gives the sum of squarefree semiprimes of weight n.
A056239 (weight) gives the sum of prime indices of n.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339114 is the least (squarefree) semiprime of weight n.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A014342, A024697, A046388, A062198, A098350, A112798, A168472, A320655, A338901.

primes[n_]:=Reverse[Prime/@Range[n]]; partition[n_]:=Partition[primes[n],UpTo[Ceiling[n/2]]];
riffle[n_]:=Riffle[partition[n][[1]],Reverse[partition[n][[2]]]];
a[n_]:=Max[Table[riffle[n][[i]]*riffle[n][[i+1]],{i,1,n-1}]];a/@Range[2,53]
(* Ivan N. Ianakiev, Apr 28 2020 *)

It appears that a(n) = A332877(n - 1) for n > 5.

a(12)-a(13) from Jinyuan Wang, Apr 24 2020

More terms from Ivan N. Ianakiev, Apr 28 2020

cp's OEIS Frontend

A340655 Number of twice-balanced factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A338901 Position of the first appearance of prime(n) as a factor in the list of squarefree semiprimes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A339656 Number of loop-graphical integer partitions of 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A002135 Number of terms in a symmetrical determinant: a(n) = n*a(n-1) - (n-1)*(n-2)*a(n-3)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A339559 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A339116 Triangle of all squarefree semiprimes grouped by greater prime factor, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A002135 Number of terms in a symmetrical determinant: a(n) = na(n-1) - (n-1)(n-2)*a(n-3)/2.