A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

0, 0, 1, 1, 3, 1, 7, 1, 14, 10, 26, 1, 55, 1, 90, 68, 167, 1, 292, 1, 482, 345, 761, 1, 1291, 266, 1949, 1518, 3091, 1, 4793, 1, 7177, 5612, 10566, 2623, 16007, 1, 22912, 18992, 33619, 1, 48529, 1, 68758, 59187, 96571, 1, 137489, 11418, 189979, 167502, 264299

Offset

0,5

Comments

a(n) = 1 if and only if n is prime. - Chai Wah Wu, Nov 12 2018

Links

Table of n, a(n) for n=0..52.

Formula

a(n) = A000041(n) - A321451(n).

Example

The a(2) = 1 through a(9) = 10 partitions:
  (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)        (333)
               (211)            (222)                (422)       (3321)
               (1111)           (321)                (431)       (32211)
                                (2211)               (2222)      (33111)
                                (3111)               (3221)      (222111)
                                (21111)              (3311)      (321111)
                                (111111)             (4211)      (2211111)
                                                     (22211)     (3111111)
                                                     (32111)     (21111111)
                                                     (41111)     (111111111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The partition (32111) can be partitioned as ((13)(112)), and the blocks both sum to 4, so (32111) is counted under a(8).

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[facs[Times@@Prime/@#], SameQ@@hwt/@#&]]>1&]], {n, 10}]

Crossrefs

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321451, A321453, A321454, A321455.

Sequence in context: A098093 A160627 A114712 * A089741 A089736 A205479

Adjacent sequences: A321449 A321450 A321451 * A321453 A321454 A321455

Keyword

nonn

Author

Gus Wiseman, Nov 10 2018

Extensions

a(26)-a(52) from Alois P. Heinz, Nov 11 2018

Status

approved