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A307699 Numbers k such that there is no integer partition of k with exactly k-1 submultisets.

%I A307699 #51 Jul 24 2021 01:14:00
%S A307699 0,1,2,6,8,12,14,18,20,24,26,30,32,38,42,44,48,50,54,60,62,66,68,72,
%T A307699 74,80,84,86,90,92,98,102,104,108,110,114,122,126,128,132,134,138,140,
%U A307699 146,150,152,158,164,168,170,174,180,182,186,192,194,198,200,206
%N A307699 Numbers k such that there is no integer partition of k with exactly k-1 submultisets.
%C A307699 After a(1) = 0, first differs from A229488 in lacking 56.
%C A307699 The number of submultisets of a partition is the product of its multiplicities, each plus one.
%C A307699 {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
%e A307699 The sequence of positive terms together with their prime indices begins:
%e A307699    1: {}
%e A307699    2: {1}
%e A307699    6: {1,2}
%e A307699    8: {1,1,1}
%e A307699   12: {1,1,2}
%e A307699   14: {1,4}
%e A307699   18: {1,2,2}
%e A307699   20: {1,1,3}
%e A307699   24: {1,1,1,2}
%e A307699   26: {1,6}
%e A307699   30: {1,2,3}
%e A307699   32: {1,1,1,1,1}
%e A307699   38: {1,8}
%e A307699   42: {1,2,4}
%e A307699   44: {1,1,5}
%e A307699   48: {1,1,1,1,2}
%e A307699   50: {1,3,3}
%e A307699   54: {1,2,2,2}
%e A307699   60: {1,1,2,3}
%e A307699 Partitions realizing the desired number of submultisets for each non-term are:
%e A307699    3: (3)
%e A307699    4: (22)
%e A307699    5: (41)
%e A307699    7: (511)
%e A307699    9: (621)
%e A307699   10: (4411)
%e A307699   11: (71111)
%e A307699   13: (9211)
%e A307699   15: (9111111)
%e A307699   16: (661111)
%e A307699   17: (9521)
%e A307699   19: (94411)
%e A307699   21: (981111)
%e A307699   22: (88111111)
%e A307699   23: (32222222222)
%e A307699   25: (99421)
%e A307699   27: (3222222222222)
%e A307699   28: (994411)
%e A307699   29: (98222222)
%t A307699 Select[Range[50],Function[n,Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==n-1&]=={}]]
%Y A307699 Positions of zeros in A325836.
%Y A307699 Cf. A002033, A088880, A088881, A098859, A108917, A126796, A276024, A325694, A325792, A325798, A325828, A325830, A325833, A325834, A325835.
%K A307699 nonn
%O A307699 1,3
%A A307699 _Gus Wiseman_, May 30 2019
%E A307699 More terms from _Alois P. Heinz_, May 30 2019