A168532 -id:A168532 - cp's OEIS frontend

A319299 Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.

1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 7, 2, 1, 1, 14, 1, 17, 3, 1, 1, 27, 2, 1, 34, 6, 1, 1, 55, 1, 63, 7, 3, 2, 1, 1, 100, 1, 119, 14, 1, 1, 167, 6, 2, 1, 209, 17, 3, 1, 1, 296, 1, 347, 27, 7, 2, 1, 1, 489, 1, 582, 34, 6, 3, 1, 1, 775, 14, 2, 1, 945, 55, 1, 1, 1254

Offset: 1

Gus Wiseman, Sep 16 2018

			Triangle begins:
    1
    1   1
    2   1
    3   1   1
    6   1
    7   2   1   1
   14   1
   17   3   1   1
   27   2   1
   34   6   1   1
   55   1
   63   7   3   2   1   1
  100   1
  119  14   1   1
  167   6   2   1
  209  17   3   1   1
  296   1
  347  27   7   2   1   1
  489   1
  582  34   6   3   1   1

Robert Israel, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened)

A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.

Cf. A018783, A027750, A078374, A078392, A289508, A289509, A303139, A305563, A319300.

# with table A000837 obtained from that sequence
f:= proc(n) local D,d;
  D:= sort(convert(numtheory:-divisors(n),list),`>`);
  seq(A000837[d],d=D)
end proc:
map(f, [$1..60]); # Robert Israel, Jul 09 2020

Table[Length[Select[IntegerPartitions[n],GCD@@#==k&]],{n,20},{k,Divisors[n]}]

T(n,k) = A000837(n/A027750(n,k)).

A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 3, 1, 0, 2, 0, 1, 3, 1, 0, 3, 2, 1, 4, 1, 1, 5, 0, 1, 5, 1, 2, 5, 1, 1, 5, 2, 2, 6, 0, 1, 9, 1, 0, 9, 0, 3, 9, 1, 1, 9, 5, 1, 11, 1, 0, 15, 1, 2, 13, 1, 5, 14, 0, 1, 18

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
  (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)
       (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)
               (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)
                       (27,15)  (25,15,5)  (33,15,9)  (33,27)
                                (27,15,3)  (33,21,3)  (35,25)
                                           (39,15,3)  (39,21)
                                                      (45,15)
                                                      (27,21,9,3)
                                                      (33,15,9,3)

This is the case of A000700 with a common divisor.
Including evens gives A303280.
The complement is counted by A366844, non-strict version A366843.
The non-strict version is A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, A366845, A366848.

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366750(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

More terms from Chai Wah Wu, Nov 02 2023

cp's OEIS Frontend

A319299 Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.

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