A068571 Number of happy numbers <= 10^n.

1, 3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294

Offset

0,2

Links

Bryan Wolf, Table of n, a(n) for n = 0..1000 (first 122 terms from Lambert Klasen)

Justin Gilmer, On the density of happy numbers, arXiv:1110.3836 [math.NT], 2011-2015.

Lambert Klasen, Xeen3d's happy.html.

Walter Schneider, Happy Numbers.

Eric Weisstein's World of Mathematics, Happy Number.

Formula

From Bryan Wolf, Nov 06 2012: (Start)

h(n,x) = h(n-1,x-0^2) + h(n-1,x-1^2) + ... + h(n-1,x-9^2) where h() is the number of numbers of length n whose sum of the squares of their digits is x.

a(n) is the sum of all h(n, 0 < x <= 81*n), where x is a happy number, plus 1 for 10^n. (End)

Example

For n=0, h(0,0) = 1 and h(0,x >0) = 0.

Prog

(PARI) ssd(n)=n=digits(n); sum(i=1, #n, n[i]^2)
happy(n)=while(n>6, n=ssd(n)); n==1
a(n)=my(f=n!, s, d); forvec(v=vector(9, i, [0, n]), d=vector(9, i, if(i>8, n, v[i+1])-v[i]); if(happy(sum(i=1, 9, d[i]*i^2)), s+=f/prod(i=1, 9, d[i]!)/v[1]!), 1); s+1 \\ Charles R Greathouse IV, Nov 01 2016

Crossrefs

Cf. A007770, A124095, A219667.

Sequence in context: A371411 A009156 A074573 * A074569 A026303 A154627

Adjacent sequences: A068568 A068569 A068570 * A068572 A068573 A068574

Keyword

nonn,base

Author

Sascha Kurz, Mar 26 2002

Extensions

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.de), Nov 03 2004

Status

approved