A000227 Nearest integer to e^n.

1, 3, 7, 20, 55, 148, 403, 1097, 2981, 8103, 22026, 59874, 162755, 442413, 1202604, 3269017, 8886111, 24154953, 65659969, 178482301, 485165195, 1318815734, 3584912846, 9744803446, 26489122130, 72004899337, 195729609429, 532048240602, 1446257064291, 3931334297144, 10686474581524

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

x = e^n is the location of the maximum of x^(1/x^(1/n)). One can define another sequence, c(n) as the value of the natural number k that maximizes k^(1/k^(1/n)). Empirically, despite the rounding, c(n) and a(n) match each other until at least n>24500 (see the link). - Stanislav Sykora, Jun 06 2014

References

• Federal Works Agency, Work Projects Administration for the City of NY, Tables of the Exponential Function. National Bureau of Standards, Washington, DC, 1939.
• A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 230.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=0..200
• Stanislav Sykora, Comments on A000227

Crossrefs

Cf. A000149 (floor e^n), A001671 (e^n rounded up), A002160 (nearest integer to Pi^n).

Programs

Maple

Digits := 40: [seq(round(exp(n)), n=0..30)];

Mathematica

Table[ Round[ N[E^n] ], {n, 0, 30} ]
Round[E^Range[0,30]] (* Harvey P. Dale, Jul 28 2025 *)

PARI

apply( A000227(n)=exp(n)\/1, [0..50]) \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018