A392745 The number of partitions of n into parts, so that there exists a part i with multiplicity i.

0, 1, 0, 1, 2, 2, 3, 6, 6, 11, 13, 20, 24, 36, 43, 61, 77, 102, 128, 172, 208, 276, 339, 434, 536, 681, 830, 1047, 1277, 1587, 1929, 2388, 2880, 3540, 4267, 5195, 6245, 7571, 9052, 10925, 13028, 15630, 18588, 22214, 26318, 31332, 37030, 43892, 51739, 61137

Offset

0,5

Comments

This sequence was inspired by a puzzle, "IQ Tests," from the book Mathematical Puzzles and Curiosities.

Consider a multiple-choice question: "How many correct answer choices are there: (a) 1, (b) 1, (c) 2, (d) 2?" Answer 2 can be considered correct as it appears twice. This sequence counts the number of multiple-choice questions with a total n that contain a 'correct' answer.

References

I. David, T. Khovanova, and Y. Shpilman, Mathematical Puzzles and Curiosities, World Scientific, 2026, p.5.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k>0} 1/(1-x^k) - Product_{k>0} (1/(1-x^k) - x^(k^2)).

a(n) = A000041(n) - A276429(n).

Example

Consider 5 partitions of 4: (1,1,1,1), (2,1,1), (2,2), (3,1), 4. Only partitions (2,2) and (3,1) satisfy the requirement. In the first case, value 2 appears twice; in the second case, value 1 appears once. Thus, a(5) = 2.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> combinat[numbpart](n)-b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2026

Mathematica

Table[Total@ Map[Boole@ AnyTrue[Tally[#], SameQ @@ # &] &, IntegerPartitions[n]], {n, 0, 49}] (* Michael De Vlieger, Jan 29 2026 *)

Crossrefs

Cf. A000041, A276429, A336273.

Sequence in context: A078921 A011961 A182713 * A229626 A358088 A091770

Adjacent sequences: A392742 A392743 A392744 * A392746 A392747 A392748

Keyword

nonn

Author

Tanya Khovanova and PRIMES STEP senior group, Jan 21 2026

Status

approved