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