login
A307699
Numbers k such that there is no integer partition of k with exactly k-1 submultisets.
4
0, 1, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 38, 42, 44, 48, 50, 54, 60, 62, 66, 68, 72, 74, 80, 84, 86, 90, 92, 98, 102, 104, 108, 110, 114, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 158, 164, 168, 170, 174, 180, 182, 186, 192, 194, 198, 200, 206
OFFSET
1,3
COMMENTS
After a(1) = 0, first differs from A229488 in lacking 56.
The number of submultisets of a partition is the product of its multiplicities, each plus one.
{a(n)-1} contains all odd numbers m = p*q*... such that gcd(p-1, q-1, ...) > 2. In particular, {a(n)-1} contains all powers of all primes > 3. Proof: If g is the greatest common divisor, then all factors of k are congruent to 1 modulo g, and thus all multiplicities of any valid multiset are divisible by g. However, the required sum is congruent to 2 modulo g, and so no such multiset can exist. - Charlie Neder, Jun 06 2019
EXAMPLE
The sequence of positive terms together with their prime indices begins:
1: {}
2: {1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
14: {1,4}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
26: {1,6}
30: {1,2,3}
32: {1,1,1,1,1}
38: {1,8}
42: {1,2,4}
44: {1,1,5}
48: {1,1,1,1,2}
50: {1,3,3}
54: {1,2,2,2}
60: {1,1,2,3}
Partitions realizing the desired number of submultisets for each non-term are:
3: (3)
4: (22)
5: (41)
7: (511)
9: (621)
10: (4411)
11: (71111)
13: (9211)
15: (9111111)
16: (661111)
17: (9521)
19: (94411)
21: (981111)
22: (88111111)
23: (32222222222)
25: (99421)
27: (3222222222222)
28: (994411)
29: (98222222)
MATHEMATICA
Select[Range[50], Function[n, Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==n-1&]=={}]]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 30 2019
EXTENSIONS
More terms from Alois P. Heinz, May 30 2019
STATUS
approved