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 A307699 Numbers n such that there is no integer partition of n with exactly n - 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 (list; graph; refs; listen; history; text; internal format)
 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 k = 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 LINKS 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&]=={}]] CROSSREFS Positions of zeros in A325836. Cf. A002033, A088880, A088881, A098859, A108917, A126796, A276024, A325694, A325792, A325798, A325828, A325830, A325833, A325834, A325835. Sequence in context: A047238 A189933 A229488 * A226485 A213638 A191965 Adjacent sequences:  A307696 A307697 A307698 * A307700 A307701 A307702 KEYWORD nonn AUTHOR Gus Wiseman, May 30 2019 EXTENSIONS More terms from Alois P. Heinz, May 30 2019 STATUS approved

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Last modified April 7 04:20 EDT 2020. Contains 333292 sequences. (Running on oeis4.)