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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

Table of n, a(n) for n=1..59.

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.)