OFFSET

1,5

COMMENTS

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 123 terms from Robert G. Wilson v)

FORMULA

log(a(n)) ~ log(A258268) * sqrt(n) - log(n). - Vaclav Kotesovec, May 30 2021

EXAMPLE

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).

MAPLE

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))):

seq(a(n), n=1..100); # Alois P. Heinz, Aug 02 2018

MATHEMATICA

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]]]];

Array[a, 100] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

PROG

(PARI) a(n) = my(nb = 0); forpart(p=n, nb++, sqrtint(n)-issquare(n)); nb; \\ Michel Marcus, Jul 15 2018

CROSSREFS

KEYWORD

nonn

AUTHOR

Richard Locke Peterson, Jun 29 2018

EXTENSIONS

More terms from Michel Marcus, Jul 15 2018

STATUS

approved