A295866 Number of decimal digits in the number of partitions of n.

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Author

José Hernández, Feb 13 2018

Keywords

• nonn
• base

Comments

In his book on analytic number theory, Don Newman tells this amusing story regarding the number of digits in p(n): "This is told of Major MacMahon who kept a list of these partition numbers arranged one under another up into the hundreds. It suddenly occurred to him that, viewed from a distance, the outline of the digits seemed to form a parabola! Thus the number of digits in p(n), the number of partitions of n, is around C*sqrt(n), or p(n) itself is very roughly e^(a*sqrt(n)). The first crude assessment of p(n)!"

References

• D. J. Newman, Analytic number theory, Springer Verlag, 1998, p. 17.

Crossrefs

Cf. A000041, A055642, A072212, A097985.

Programs

Mathematica

Join[{1}, IntegerLength[PartitionsP[#]] & /@ Range[99]]

PARI

a(n) = #digits(numbpart(n)); \\ Michel Marcus, Feb 17 2018

Formula

a(n) = A055642(A000041(n)).