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A316353
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Number of partitions of positive integer n such that all parts are less than the square root of n.
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1
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0, 1, 1, 1, 3, 4, 4, 5, 5, 14, 16, 19, 21, 24, 27, 30, 72, 84, 94, 108, 120, 136, 150, 169, 185, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 18138, 19928, 21873, 23961, 26226, 28652
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OFFSET
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1,5
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COMMENTS
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This sequence itself is not a semigroup, but the set of all the partitions enumerated by this sequence does form a semigroup (actually a subsemigroup of the set of all partitions) with the following binary operation: let alpha = the partition (a,b,c,... [this is of course a finite list]) be the partition of the number N1 [that is, a + b + c + ... = N1] and let ALPHA = (A,B,C,...) be the partition of N2. Then the binary operation given by alpha*ALPHA = (a,b,c,...)*(A,B,C,...) = (aA,aB,aC,...,bA,bB,bC,...,cA,cB,cC,...) is a partition of the integer N1*N2. Furthermore, since any part x of alpha is less than the square root of N1, and likewise for any part Y of ALPHA, then the part xY is less than the square root of N1*N2, so the set is a subsemigroup of the semigroup of all partitions under the given operation. If the sole partition (1) of 1 is adjoined, the semigroup becomes a monoid.
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LINKS
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FORMULA
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EXAMPLE
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a(3)=1, since the partition (1,1,1) is the only partition of 3 with all parts less than the square root of 3 ~ 1.73.
a(6)=4, since there are only 4 allowable partitions: (1,1,1,1,1,1,1), (1,1,1,1,2), (1,1,2,2), and (2,2,2).
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> b(n, (r-> `if`(r*r>=n, r-1, r))(isqrt(n))):
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MATHEMATICA
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Table[With[{s = Sqrt@ n}, Count[IntegerPartitions[n], _?(AllTrue[#, # < s &] &)]], {n, 53}] (* Michael De Vlieger, Jul 22 2018 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt[n - 1]]; Array[f, 50] (* Robert G. Wilson v, Jul 24 2018 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := b[n, Function[r, If[r*r >= n, r - 1, r]][Floor[Sqrt[n]]]];
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PROG
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(PARI) a(n) = my(nb = 0); forpart(p=n, nb++, sqrtint(n)-issquare(n)); nb; \\ Michel Marcus, Jul 15 2018
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CROSSREFS
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Cf. A000041 (the partition numbers), A097356 (with 'no greater' rather than less).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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