A338899 -id:A338899 - cp's OEIS frontend

A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

1, 0, 0, 1, 1, 2, 3, 4, 6, 9, 11, 16, 23, 29, 39, 53, 69, 90, 118, 150, 195, 249, 315, 398, 506, 629, 789, 982, 1219, 1504, 1860, 2277, 2798, 3413, 4161, 5051, 6137, 7406, 8948, 10765, 12943, 15503, 18571, 22153, 26432, 31432, 37352, 44268, 52444, 61944, 73141

Offset: 0

Gus Wiseman, Dec 09 2020

nonn

These are also integer partitions that can be partitioned into not necessarily distinct edges (pairs of distinct parts). For example, (3,3,2,2) can be partitioned as {{2,3},{2,3}}, so is counted under a(10), but (4,2,2,2) and (4,2,1,1,1,1) cannot be partitioned into edges. The multiplicities of such a partition form a multigraphical partition (A209816, A320924).

			The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
              (41)  (51)    (52)    (62)    (63)      (73)
                    (2211)  (61)    (71)    (72)      (82)
                            (3211)  (3221)  (81)      (91)
                                    (3311)  (3321)    (3322)
                                    (4211)  (4221)    (4321)
                                            (4311)    (4411)
                                            (5211)    (5221)
                                            (222111)  (5311)
                                                      (6211)
                                                      (322111)

Eric Weisstein's World of Mathematics, Graphical partition.

A096373 counts the complement in even-length partitions.
A320911 gives the Heinz numbers of these partitions.
A339560 is the strict case.
A339562 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.
A320656 counts factorizations into squarefree semiprimes.
A320921 counts connected graphical partitions, ranked by A320923.
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).
- 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, A005117, A007717, A030229, A320655, A322353, A338899, A338903.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]],{n,0,30}]

A027187(n) = a(n) + A096373(n).

A339617 Number of non-graphical integer partitions of 2n.

Original entry on oeis.org

0, 1, 3, 6, 13, 25, 46, 81, 141, 234, 383, 615, 968, 1503, 2298, 3468, 5176, 7653, 11178, 16212, 23290, 33218, 46996, 66091, 92277, 128122, 176787, 242674, 331338, 450279, 608832, 819748, 1098907, 1467122, 1951020, 2584796, 3411998

Offset: 0

Gus Wiseman, Dec 13 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. See A209816 for multigraphical partitions, A000070 for non-multigraphical partitions. Graphical partitions are counted by A000569.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The a(1) = 1 through a(4) = 13 partitions:
  (2)  (4)    (6)      (8)
       (2,2)  (3,3)    (4,4)
       (3,1)  (4,2)    (5,3)
              (5,1)    (6,2)
              (3,2,1)  (7,1)
              (4,1,1)  (3,3,2)
                       (4,2,2)
                       (4,3,1)
                       (5,2,1)
                       (6,1,1)
                       (3,3,1,1)
                       (4,2,1,1)
                       (5,1,1,1)
For example, the partition (2,2,2,2) is not counted under a(4) because there are three possible graphs with the prescribed degrees:
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Graphical partition.

A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
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).
- A339656 counts loop-graphical partitions (A339658).
- A339617 [this sequence] 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, A007717, A025065, A320921, A320922, A338899, A339564, A339619, A339660, A339661.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[prptns[#],UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) + A000569(n) = A000041(2*n).

A339618 Heinz numbers of non-graphical integer partitions of even numbers.

Original entry on oeis.org

3, 7, 9, 10, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 147

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime indices of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the unordered prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         43: {14}        79: {22}
      7: {4}         46: {1,9}       82: {1,13}
      9: {2,2}       49: {4,4}       84: {1,1,2,4}
     10: {1,3}       52: {1,1,6}     85: {3,7}
     13: {6}         53: {16}        87: {2,10}
     19: {8}         55: {3,5}       88: {1,1,1,5}
     21: {2,4}       57: {2,8}       89: {24}
     22: {1,5}       61: {18}        91: {4,6}
     25: {3,3}       62: {1,11}      94: {1,15}
     28: {1,1,4}     63: {2,2,4}    100: {1,1,3,3}
     29: {10}        66: {1,2,5}    101: {26}
     30: {1,2,3}     70: {1,3,4}    102: {1,2,7}
     34: {1,7}       71: {20}       107: {28}
     37: {12}        75: {2,3,3}    111: {2,12}
     39: {2,6}       76: {1,1,8}    113: {30}
For example, there are three possible multigraphs with degrees (1,1,3,3):
  {{1,2},{1,2},{1,2},{3,4}}
  {{1,2},{1,2},{1,3},{2,4}}
  {{1,2},{1,2},{1,4},{2,3}}.
Since none of these is a graph, the Heinz number 100 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A181819 applied to A320894 gives this sequence.
A300061 is a superset.
A339617 counts these partitions.
A320922 ranks the complement, counted by A000569.
A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
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).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618 [this sequence]).
- 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. A001358, A007717, A050326, A320923, A338899.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&&strs[Times@@Prime/@nrmptn[#]]=={}&]

Equals A300061 \ A320922.

For all n, A181821(a(n)) and A304660(a(n)) belong to A320894.

A112141 Product of the first n semiprimes.

Original entry on oeis.org

4, 24, 216, 2160, 30240, 453600, 9525600, 209563200, 5239080000, 136216080000, 4495130640000, 152834441760000, 5349205461600000, 203269807540800000, 7927522494091200000, 364666034728195200000, 17868635701681564800000, 911300420785759804800000

Offset: 1

Jonathan Vos Post, Nov 28 2005

Semiprime analog of primorial (A002110). Equivalent for product of what A062198 is for sum.

			a(10) = 4*6*9*10*14*15*21*22*25*26 = 136216080000, the product of the first 10 semiprimes.
From _Gus Wiseman_, Dec 06 2020: (Start)
The sequence of terms together with their prime signatures begins:
                        4: (2)
                       24: (3,1)
                      216: (3,3)
                     2160: (4,3,1)
                    30240: (5,3,1,1)
                   453600: (5,4,2,1)
                  9525600: (5,5,2,2)
                209563200: (6,5,2,2,1)
               5239080000: (6,5,4,2,1)
             136216080000: (7,5,4,2,1,1)
            4495130640000: (7,6,4,2,2,1)
          152834441760000: (8,6,4,2,2,1,1)
         5349205461600000: (8,6,5,3,2,1,1)
       203269807540800000: (9,6,5,3,2,1,1,1)
      7927522494091200000: (9,7,5,3,2,2,1,1)
    364666034728195200000: (10,7,5,3,2,2,1,1,1)
  17868635701681564800000: (10,7,5,5,2,2,1,1,1)
(End)

T. D. Noe, Table of n, a(n) for n = 1..100

Partial sums of semiprimes are A062198.
First differences of semiprimes are A065516.
A000040 lists primes, with partial products A002110 (primorials).
A000142 lists factorials, with partial products A000178 (superfactorials).
A001358 lists semiprimes, with partial products A112141 (this sequence).
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial products A339191.
A101048 counts partitions into semiprimes (restricted: A338902).
A320655 counts factorizations into 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.
Cf. A001222, A084126, A084127, A115392, A168472, A320732, A320892.

A112141 := proc(n)
    mul(A001358(i),i=1..n) ;
end proc:
seq(A112141(n),n=1..10) ; # R. J. Mathar, Jun 30 2020

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Times @@ NestList[ NextSemiPrime@# &, 2^2, n - 1]; Array[f, 18] (* Robert G. Wilson v, Jun 13 2013 *)
FoldList[Times,Select[Range[30],PrimeOmega[#]==2&]] (* Gus Wiseman, Dec 06 2020 *)

a(n)=my(v=vector(n),i,k=3);while(iCharles R Greathouse IV, Apr 04 2013

from sympy import factorint
def aupton(terms):
    alst, k, p = [], 1, 1
    while len(alst) < terms:
        if sum(factorint(k).values()) == 2:
            p *= k
            alst.append(p)
        k += 1
    return alst
print(aupton(18)) # Michael S. Branicky, Aug 31 2021

a(n) = Product_{i=1..n} A001358(i).

A001222(a(n)) = 2*n.

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).

A339113 Products of primes of squarefree semiprime index (A322551).

Original entry on oeis.org

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 841, 907, 929, 937

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also MM-numbers of labeled multigraphs (without uncovered vertices). 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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with the corresponding multigraphs begins:
      1: {}               233: {{2,7}}          487: {{2,11}}
     13: {{1,2}}          257: {{3,5}}          491: {{1,15}}
     29: {{1,3}}          269: {{2,8}}          499: {{3,8}}
     43: {{1,4}}          271: {{1,10}}         559: {{1,2},{1,4}}
     47: {{2,3}}          293: {{1,11}}         577: {{1,16}}
     73: {{2,4}}          313: {{3,6}}          607: {{2,12}}
     79: {{1,5}}          347: {{2,9}}          611: {{1,2},{2,3}}
    101: {{1,6}}          373: {{1,12}}         631: {{3,9}}
    137: {{2,5}}          377: {{1,2},{1,3}}    647: {{1,17}}
    139: {{1,7}}          389: {{4,5}}          653: {{4,7}}
    149: {{3,4}}          421: {{1,13}}         673: {{1,18}}
    163: {{1,8}}          439: {{3,7}}          677: {{2,13}}
    167: {{2,6}}          443: {{1,14}}         727: {{2,14}}
    169: {{1,2},{1,2}}    449: {{2,10}}         751: {{4,8}}
    199: {{1,9}}          467: {{4,6}}          757: {{1,19}}

These primes (of squarefree semiprime index) are listed by A322551.
The strict (squarefree) case is A309356.
The prime instead of squarefree semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of squarefree semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The semiprime instead of squarefree semiprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A002100 counts partitions into squarefree semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices, which are listed by A112798.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339561 lists products of distinct squarefree semiprimes (ranking: A339560).
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A289509, A320461.

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]

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

A339114 Least semiprime whose prime indices sum to n.

Original entry on oeis.org

4, 6, 9, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 2

Gus Wiseman, Nov 28 2020

nonn

Converges to A100484.
After a(4) = 9, also the least squarefree semiprime whose prime indices sum to n.
A semiprime is a product of any two 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.

			The sequence of terms together with their prime indices begins:
      4: {1,1}     106: {1,16}    254: {1,31}
      6: {1,2}     118: {1,17}    262: {1,32}
      9: {2,2}     122: {1,18}    274: {1,33}
     14: {1,4}     134: {1,19}    278: {1,34}
     21: {2,4}     142: {1,20}    298: {1,35}
     26: {1,6}     146: {1,21}    302: {1,36}
     34: {1,7}     158: {1,22}    314: {1,37}
     38: {1,8}     166: {1,23}    326: {1,38}
     46: {1,9}     178: {1,24}    334: {1,39}
     58: {1,10}    194: {1,25}    346: {1,40}
     62: {1,11}    202: {1,26}    358: {1,41}
     74: {1,12}    206: {1,27}    362: {1,42}
     82: {1,13}    214: {1,28}    382: {1,43}
     86: {1,14}    218: {1,29}    386: {1,44}
     94: {1,15}    226: {1,30}    394: {1,45}

A024697 is the sum of the same semiprimes.
A098350 has this sequence as antidiagonal minima.
A338904 has this sequence as row minima.
A339114 (this sequence) is the squarefree case for n > 4.
A339115 is the greatest among the same semiprimes.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A037143 lists primes and semiprimes.
A056239 gives the sum of prime indices of n.
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A320655 counts factorizations into semiprimes.
A332765/A332877 is the greatest squarefree semiprime of weight n.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338907/A338906 list semiprimes of odd/even weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A001222, A014342, A025129, A062198, A112798, A338905, A339116.

Table[Min@@Table[Prime[k]*Prime[n-k],{k,n-1}],{n,2,30}]
Take[DeleteDuplicates[SortBy[{Times@@#,Total[PrimePi[#]]}&/@Tuples[ Prime[ Range[ 200]],2],{Last,First}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]],60] (* Harvey P. Dale, Sep 06 2022 *)

a(n) = vecmin(vector(n-1, k, prime(k)*prime(n-k))); \\ Michel Marcus, Dec 03 2020

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

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

cp's OEIS Frontend

A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A339617 Number of non-graphical integer partitions of 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A339618 Heinz numbers of non-graphical integer partitions of even numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A112141 Product of the first n semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

A339113 Products of primes of squarefree semiprime index (A322551).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

A339114 Least semiprime whose prime indices sum to n.