A353321 a(n) = floor(1/erfc(n/sqrt(2))).

1, 3, 21, 370, 15787, 1744277, 506797345, 390682215445, 803734397655347, 4430313100526836692, 65618063552490194383194, 2616897361902846669558232537, 281455127862349591601857362987343, 81737217988908649002650313009555641846, 64155724364921456082725604130103414484969173

Offset

0,2

Comments

a(n) is the residual (inverted) of the standard normal distribution in two-sided range of n sigma.

The 1/sqrt(2) factor appears in the argument of the erfc function because the standard normal cumulative distribution function has integrand exp(-(1/2)(x/sigma)^2).

The n=5 value, a(5)=1744277, appeared in discussions of the 5-sigma (two-sided probability) discovery of the Higgs boson at CERN (see articles in Links).

Links

Table of n, a(n) for n=0..14.

E. Lamb, Five sigma what's that?

C. Pease, The Higgs Boson and 5 Sigma

D. Spiegelhalter, Higgs: is it one-sided or two-sided?

Formula

Straightforward asymptotics of erfc (e.g., see Abramowitz and Stegun) gives leading order as a(n) ~ sqrt(Pi/2)*n*exp((1/2)*n^2).

Example

For n=2, a(2)=21 corresponds roughly to the statement that 95.5% of normally distributed measurements fall into the range of two sigma (plus and minus), since 1/21 = 1-0.955 (approximately). Nearest-integer version (A275366) is always more accurate (e.g., a(2)=22).

Mathematica

Table[Floor[1/Erfc[n/Sqrt[2]]], {n, 1, 16}]

Crossrefs

Cf. A275366 (rounded).

Sequence in context: A320949 A361056 A101389 * A108716 A358950 A271570

Adjacent sequences: A353318 A353319 A353320 * A353322 A353323 A353324

Keyword

nonn

Author

Greg Huber, Jul 04 2022

Status

approved