A340785 -id:A340785 - cp's OEIS frontend

A349906 Number of factorizations of n into even factors > 1 (a(1) = 1 by convention).

1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 11, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 12, 0, 1, 0, 3, 0, 1, 0, 4, 0

Offset: 1

Antti Karttunen, Dec 13 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001055, A340785 (even bisection), A349907.

A349906(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A349906(n/d, d))); (s));

A341449 Heinz numbers of integer partitions into odd parts > 1.

Original entry on oeis.org

1, 5, 11, 17, 23, 25, 31, 41, 47, 55, 59, 67, 73, 83, 85, 97, 103, 109, 115, 121, 125, 127, 137, 149, 155, 157, 167, 179, 187, 191, 197, 205, 211, 227, 233, 235, 241, 253, 257, 269, 275, 277, 283, 289, 295, 307, 313, 331, 335, 341, 347, 353, 365, 367, 379, 389

Offset: 1

Gus Wiseman, Feb 15 2021

nonn

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.

			The sequence of partitions together with their Heinz numbers begins:
      1: ()        97: (25)       197: (45)       307: (63)
      5: (3)      103: (27)       205: (13,3)     313: (65)
     11: (5)      109: (29)       211: (47)       331: (67)
     17: (7)      115: (9,3)      227: (49)       335: (19,3)
     23: (9)      121: (5,5)      233: (51)       341: (11,5)
     25: (3,3)    125: (3,3,3)    235: (15,3)     347: (69)
     31: (11)     127: (31)       241: (53)       353: (71)
     41: (13)     137: (33)       253: (9,5)      365: (21,3)
     47: (15)     149: (35)       257: (55)       367: (73)
     55: (5,3)    155: (11,3)     269: (57)       379: (75)
     59: (17)     157: (37)       275: (5,3,3)    389: (77)
     67: (19)     167: (39)       277: (59)       391: (9,7)
     73: (21)     179: (41)       283: (61)       401: (79)
     83: (23)     187: (7,5)      289: (7,7)      415: (23,3)
     85: (7,3)    191: (43)       295: (17,3)     419: (81)

Note: A-numbers of ranking sequences are in parentheses below.
Partitions with no ones are A002865 (A005408).
The case of even parts is A035363 (A066207).
These partitions are counted by A087897.
The version for factorizations is A340101.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A056239 adds up prime indices.
A078408 counts partitions with odd parts, length, and sum (A300272).
A112798 lists the prime indices of each positive integer.
A257991/A257992 count odd/even prime indices.
Cf. A026804 (A340932), A160786 (A340931), A340386, A340604, A340607, A340785, A340854/A340855, A341446.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&OddQ[Times@@primeMS[#]]&]

A357859 Number of integer factorizations of 2n into distinct even factors.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 17 2022

nonn

			The a(n) factorizations for n = 2, 4, 12, 24, 32, 48, 60, 96:
  (4)  (8)    (24)    (48)     (64)     (96)      (120)     (192)
       (2*4)  (4*6)   (6*8)    (2*32)   (2*48)    (2*60)    (2*96)
              (2*12)  (2*24)   (4*16)   (4*24)    (4*30)    (4*48)
                      (4*12)   (2*4*8)  (6*16)    (6*20)    (6*32)
                      (2*4*6)           (8*12)    (10*12)   (8*24)
                                        (2*6*8)   (2*6*10)  (12*16)
                                        (2*4*12)            (4*6*8)
                                                            (2*4*24)
                                                            (2*6*16)
                                                            (2*8*12)

The version for partitions instead of factorizations is A000009.
Positions of 1's are A004280.
The non-strict version is A340785.
Including odd n gives A357860.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime-power divisors.
A050361 counts strict factorizations into prime powers.
Cf. A000688, A000961, A023894, A295935, A318721.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[2*n],UnsameQ@@#&&OddQ[Times@@(#+1)]&]],{n,100}]

A357860 Number of integer factorizations of n into distinct even factors.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 3, 0, 1, 0

Offset: 1

Gus Wiseman, Oct 17 2022

nonn

			The factorizations of 36..48 are (empty columns indicated by dots):
  36    .  38  .  40    .  42  .  44    .  46  .  48
  2*18            2*20            2*22            6*8
                  4*10                            2*24
                                                  4*12
                                                  2*4*6

The non-strict version is A349906.
Same as A357859 with zeros every other term.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime-power divisors.
A050361 counts strict factorizations into prime powers.
Cf. A000009, A000688, A000961, A023894, A295935, A318721, A340785.

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],UnsameQ@@#&&OddQ[Times@@(#+1)]&]],{n,100}]

cp's OEIS Frontend

A349906 Number of factorizations of n into even factors > 1 (a(1) = 1 by convention).

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A341449 Heinz numbers of integer partitions into odd parts > 1.

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Mathematica

A357859 Number of integer factorizations of 2n into distinct even factors.

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Mathematica

A357860 Number of integer factorizations of n into distinct even factors.

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Mathematica