A320892 -id:A320892 - cp's OEIS frontend

A339658 Heinz numbers of loop-graphical partitions (of even numbers).

1, 3, 4, 9, 10, 12, 16, 25, 27, 28, 30, 36, 40, 48, 63, 64, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 147, 160, 175, 189, 192, 196, 198, 208, 210, 220, 225, 243, 250, 252, 256, 264, 270, 280, 300, 324, 336, 343, 352, 360, 400, 432, 441, 448, 462, 468, 480

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

Equals the image of A181819 applied to the set of terms of A320912.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A 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.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The sequence of terms > 1 together with their prime indices begins:
      3: {2}               70: {1,3,4}          192: {1,1,1,1,1,1,2}
      4: {1,1}             75: {2,3,3}          196: {1,1,4,4}
      9: {2,2}             81: {2,2,2,2}        198: {1,2,2,5}
     10: {1,3}             84: {1,1,2,4}        208: {1,1,1,1,6}
     12: {1,1,2}           88: {1,1,1,5}        210: {1,2,3,4}
     16: {1,1,1,1}         90: {1,2,2,3}        220: {1,1,3,5}
     25: {3,3}            100: {1,1,3,3}        225: {2,2,3,3}
     27: {2,2,2}          108: {1,1,2,2,2}      243: {2,2,2,2,2}
     28: {1,1,4}          112: {1,1,1,1,4}      250: {1,3,3,3}
     30: {1,2,3}          120: {1,1,1,2,3}      252: {1,1,2,2,4}
     36: {1,1,2,2}        144: {1,1,1,1,2,2}    256: {1,1,1,1,1,1,1,1}
     40: {1,1,1,3}        147: {2,4,4}          264: {1,1,1,2,5}
     48: {1,1,1,1,2}      160: {1,1,1,1,1,3}    270: {1,2,2,2,3}
     63: {2,2,4}          175: {3,3,4}          280: {1,1,1,3,4}
     64: {1,1,1,1,1,1}    189: {2,2,2,4}        300: {1,1,2,3,3}
For example, the four loop-graphs with degrees y = (3,1,1,1) are:
  {{1,1},{1,2},{3,4}}
  {{1,1},{1,3},{2,4}}
  {{1,1},{1,4},{2,3}}
  {{1,2},{1,3},{1,4}},
so the Heinz number 40 is in the sequence. On the other hand, the three loop-multigraphs with degrees y = (4,4) are
  {{1,1},{1,1},{2,2},{2,2}}
  {{1,1},{1,2},{1,2},{2,2}}
  {{1,2},{1,2},{1,2},{1,2}},
but none of these is a loop-graph, so the Heinz number 49 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A320912 has these prime shadows (see A181819).
A339656 counts these partitions.
A339657 ranks the complement, counted by A339655.
A001358 lists semiprimes, with squarefree case A006881.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A338914 can be partitioned into strict pairs (A320911).
- A338916 can be partitioned into distinct pairs (A320912).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A320892, A338898, A338912, A338913, A339112.

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[25],Select[mpsbin[nrmptn[#]],UnsameQ@@#&]!={}&]

A300061 = A339657 \/ A339658.

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.

A339620 Heinz numbers of non-multigraphical partitions of even numbers.

Original entry on oeis.org

3, 7, 10, 13, 19, 21, 22, 28, 29, 34, 37, 39, 43, 46, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 155, 156, 159, 163, 166, 171, 172, 173

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.
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 strict pairs (a multiset of edges);
(2) n can be factored into squarefree semiprimes;
(3) the unordered prime signature of n is multigraphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         53: {16}          94: {1,15}
      7: {4}         55: {3,5}        101: {26}
     10: {1,3}       57: {2,8}        102: {1,2,7}
     13: {6}         61: {18}         107: {28}
     19: {8}         62: {1,11}       111: {2,12}
     21: {2,4}       66: {1,2,5}      113: {30}
     22: {1,5}       71: {20}         115: {3,9}
     28: {1,1,4}     76: {1,1,8}      116: {1,1,10}
     29: {10}        79: {22}         117: {2,2,6}
     34: {1,7}       82: {1,13}       118: {1,17}
     37: {12}        85: {3,7}        129: {2,14}
     39: {2,6}       87: {2,10}       130: {1,3,6}
     43: {14}        88: {1,1,1,5}    131: {32}
     46: {1,9}       89: {24}         133: {4,8}
     52: {1,1,6}     91: {4,6}        134: {1,19}
For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
  {{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 multigraph (they have loops), the Heinz number 66 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A000070 counts these partitions.
A300061 is a superset.
A320891 has image under A181819 equal to this set of terms.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320656 counts factorizations into 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 [this sequence]).
- A209816 counts multigraphical partitions (A320924).
- A147878 counts connected multigraphical partitions (A320925).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- 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, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.

prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Select[Subsets[Union[m],{2}],MemberQ[#,m[[1]]]&]}]];
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[#]]]&&prpts[nrmptn[#]]=={}&]

Equals A300061 \ A320924.

For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.

A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 06 2018

nonn

A semiprime (A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - Gus Wiseman, Dec 31 2020

Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...

			a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
  4*15   6*35    4*6*35    4*9*35    4*15*77    4*6*15*35    4*6*10*77
  6*10   10*21   4*10*21   4*15*21   4*21*55    4*6*21*25    4*6*14*55
         14*15   4*14*15   6*10*21   4*33*35    4*9*10*35    4*6*22*35
                 6*10*14   6*14*15   6*10*77    4*9*14*25    4*10*14*33
                           9*10*14   6*14*55    4*10*15*21   4*10*21*22
                                     6*22*35    6*10*14*15   4*14*15*22
                                     10*14*33                6*10*14*22
                                     10*21*22
                                     14*15*22
(End)

Unlabeled multiset partitions of this type are counted by A007717.
The version for partitions is A112020, or A101048 without distinctness.
The non-strict version is A320655.
Positions of zeros include A320892.
Positions of nonzero terms are A320912.
The case of squarefree factors is A339661, or A320656 without distinctness.
Allowing prime factors gives A339839, or A320732 without distinctness.
A322661 counts loop-graphs, ranked by A320461.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A037143 lists primes and semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes.
A339846 counts even-length factorizations, with ordered version A174725.
Cf. A001221, A006125, A006129, A028260, A320893, A338915, A339841.

Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], ?(Times @@ # == n &)], {n, 105}] (* _Michael De Vlieger, Dec 11 2020 *)

A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ Antti Karttunen, Dec 10 2020

a(n) = Sum_{d|n} (-1)^A001222(d) * A339839(n/d). - Gus Wiseman, Dec 31 2020

A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.

Original entry on oeis.org

4, 10, 22, 25, 34, 46, 55, 62, 82, 85, 94, 115, 118, 121, 134, 146, 155, 166, 187, 194, 205, 206, 218, 235, 253, 254, 274, 289, 295, 298, 314, 334, 335, 341, 358, 365, 382, 391, 394, 415, 422, 451, 454, 466, 482, 485, 514, 515, 517, 527, 529, 538, 545, 554

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      4: {1,1}     146: {1,21}    314: {1,37}
     10: {1,3}     155: {3,11}    334: {1,39}
     22: {1,5}     166: {1,23}    335: {3,19}
     25: {3,3}     187: {5,7}     341: {5,11}
     34: {1,7}     194: {1,25}    358: {1,41}
     46: {1,9}     205: {3,13}    365: {3,21}
     55: {3,5}     206: {1,27}    382: {1,43}
     62: {1,11}    218: {1,29}    391: {7,9}
     82: {1,13}    235: {3,15}    394: {1,45}
     85: {3,7}     253: {5,9}     415: {3,23}
     94: {1,15}    254: {1,31}    422: {1,47}
    115: {3,9}     274: {1,33}    451: {5,13}
    118: {1,17}    289: {7,7}     454: {1,49}
    121: {5,5}     295: {3,17}    466: {1,51}
    134: {1,19}    298: {1,35}    482: {1,53}

A338911 is the even instead of odd version.
A339003 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists semiprimes with relatively prime indices.
A318990 lists semiprimes with divisible indices.
A338904 groups semiprimes by weight.
A338906/A338907 are semiprimes of even/odd weight.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give prime indices of squarefree semiprimes.
A338909 lists semiprimes with non-relatively prime indices.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A195017, A320655, A320732, A320892, A339004.

q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
    numtheory[pi](i[1])::odd, l))(ifactors(n)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, Nov 23 2020

Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

from math import isqrt
from sympy import primepi, primerange
def A338910(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),-1) if a&1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001222(m) = A195017(m) = 2. - Peter Munn, Jan 17 2021

A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

Original entry on oeis.org

9, 21, 39, 49, 57, 87, 91, 111, 129, 133, 159, 169, 183, 203, 213, 237, 247, 259, 267, 301, 303, 321, 339, 361, 371, 377, 393, 417, 427, 453, 481, 489, 497, 519, 543, 551, 553, 559, 579, 597, 623, 669, 687, 689, 703, 707, 717, 749, 753, 789, 791, 793, 813, 817

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      9: {2,2}     237: {2,22}    481: {6,12}
     21: {2,4}     247: {6,8}     489: {2,38}
     39: {2,6}     259: {4,12}    497: {4,20}
     49: {4,4}     267: {2,24}    519: {2,40}
     57: {2,8}     301: {4,14}    543: {2,42}
     87: {2,10}    303: {2,26}    551: {8,10}
     91: {4,6}     321: {2,28}    553: {4,22}
    111: {2,12}    339: {2,30}    559: {6,14}
    129: {2,14}    361: {8,8}     579: {2,44}
    133: {4,8}     371: {4,16}    597: {2,46}
    159: {2,16}    377: {6,10}    623: {4,24}
    169: {6,6}     393: {2,32}    669: {2,48}
    183: {2,18}    417: {2,34}    687: {2,50}
    203: {4,10}    427: {4,18}    689: {6,16}
    213: {2,20}    453: {2,36}    703: {8,12}

A338910 is the odd instead of even version.
A339004 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A338899, A270650, A270652 list prime indices of squarefree semiprimes.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists semiprimes with relatively prime indices.
A318990 lists semiprimes with divisible indices.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
A338909 lists semiprimes with non-relatively prime indices.
A338912 and A338913 list prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A128301, A195017, A320655, A320732, A320892, A338898, A339002, A339003.

q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
    numtheory[pi](i[1])::even, l))(ifactors(n)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, Nov 23 2020

Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

from math import isqrt
from sympy import primerange, primepi
def A338911(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),-1) if a&1^1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001222(m) = 2 and A195017(m) = -2. - Peter Munn, Jan 17 2021

A339112 Products of primes of semiprime index (A106349).

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A semiprime (A001358) is a product of any two prime numbers.

Also MM-numbers of labeled multigraphs with loops (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 (A..F = 10..15):
     1:            149:   (34)     313:     (36)
     7:   (11)     161: (11)(22)   329:   (11)(23)
    13:   (12)     163:   (18)     343: (11)(11)(11)
    23:   (22)     167:   (26)     347:     (29)
    29:   (13)     169: (12)(12)   373:     (1C)
    43:   (14)     199:   (19)     377:   (12)(13)
    47:   (23)     203: (11)(13)   389:     (45)
    49: (11)(11)   227:   (44)     421:     (1D)
    73:   (24)     233:   (27)     439:     (37)
    79:   (15)     257:   (35)     443:     (1E)
    91: (11)(12)   269:   (28)     449:     (2A)
    97:   (33)     271:   (1A)     467:     (46)
   101:   (16)     293:   (1B)     487:     (2B)
   137:   (25)     299: (12)(22)   491:     (1F)
   139:   (17)     301: (11)(14)   499:     (38)

Robert Israel, Table of n, a(n) for n = 1..10000

These primes (of semiprime index) are listed by A106349.
The strict (squarefree) case is A340020.
The prime instead of semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The squarefree semiprime instead of semiprime version:
  strict: A309356
  primes: A322551
  products: A339113
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
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, A005117, A007097, A289509, A320461.

N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
  p:= nextprime(p);
  if 2*p > N then break fi;
  Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
  SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
  Rp:= {}:
  for k from 1 while p^k <= N do
    Rpk:= select(`<=`,R, N/p^k);
    Rp:= Rp union map(`*`,Rpk, p^k);
  od;
  R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024

semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]

A339661 Number of factorizations of n into distinct squarefree semiprimes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Dec 19 2020

nonn

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

Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

			The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
  (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)
  (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)
  (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)
                      (10*14*33)  (10*39*77)   (6*14*15*55)
                      (10*21*22)  (14*33*65)   (6*15*22*35)
                      (14*15*22)  (14*39*55)   (10*14*15*33)
                                  (15*22*91)   (10*15*21*22)
                                  (15*26*77)
                                  (21*22*65)
                                  (21*26*55)
                                  (22*35*39)
                                  (26*33*35)
                                  (6*35*143)
                                  (10*21*143)
                                  (14*15*143)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008836 (Liouville's lambda) with A339742.
A050326 allows all squarefree numbers, non-strict case A050320.
A320656 is the not necessarily strict version.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A322794 counts uniform factorizations, such as these.
A339561 lists positions of nonzero terms.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A320655 counts factorizations into semiprimes, with strict case A322353.
The following count vertex-degree partitions and give their Heinz numbers:
- 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).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- 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. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.

bfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[bfacs[n]],{n,100}]

A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA280710(d), s += A339661(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

More terms and secondary offset added by Antti Karttunen, May 02 2022

A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

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

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of squarefree semiprimes.
A101048 counts partitions into semiprimes.
A338902 is the not necessarily squarefree version.
A339113 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.
Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

a(n) = A002100(A006881(n)).

A339740 Non-products of distinct primes or squarefree semiprimes.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.

			The sequence of terms together with their prime indices begins:
      4: {1,1}             80: {1,1,1,1,3}
      8: {1,1,1}           81: {2,2,2,2}
      9: {2,2}             88: {1,1,1,5}
     16: {1,1,1,1}         96: {1,1,1,1,1,2}
     24: {1,1,1,2}        104: {1,1,1,6}
     25: {3,3}            108: {1,1,2,2,2}
     27: {2,2,2}          112: {1,1,1,1,4}
     32: {1,1,1,1,1}      121: {5,5}
     40: {1,1,1,3}        125: {3,3,3}
     48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
     49: {4,4}            135: {2,2,2,3}
     54: {1,2,2,2}        136: {1,1,1,7}
     56: {1,1,1,4}        144: {1,1,1,1,2,2}
     64: {1,1,1,1,1,1}    152: {1,1,1,8}
     72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.

A013929 allows only primes.
A320894 does not allow primes (but omega is assumed even).
A339741 is the complement.
A339742 has zeros at these positions.
A339840 allows squares of primes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339839 into distinct primes or 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).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339617 counts non-graphical partitions of 2n (A339618).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 counts all of even length (A028260).
- A096373/A339737 cannot be partitioned into strict pairs (A320891).
- A338915/A339662 cannot be partitioned into distinct pairs (A320892).
- A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.

sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqps[#]=={}&]

cp's OEIS Frontend

A339658 Heinz numbers of loop-graphical partitions (of even numbers).

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

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A339620 Heinz numbers of non-multigraphical partitions of even numbers.

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A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

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A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.

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A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

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A339112 Products of primes of semiprime index (A106349).

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A339661 Number of factorizations of n into distinct squarefree semiprimes.