A217705 Smallest number greater than 1 that is happy under bases 2 through n.

2, 3, 3, 23, 79, 2207, 58775, 569669, 11814485, 210511543, 73748383237

Offset

2,1

Comments

A happy number is a number that after iteration of sum of squares of digits eventually reaches 1 (A007770). The happy property is base-dependent. This sequence lists the smallest number that is happy in bases 2, 3, ..., n.

All numbers are happy in binary and base 4.

Links

Table of n, a(n) for n=2..12.

Example

a(8) = 58775 because:
Base 2: 1110010110010111 - 1010 - 10 - 1,
Base 3: 2222121212 - 1011 - 10 - 1,
Base 4: 321121113 - 132 - 32 - 31 - 22 - 20 - 10 - 1,
Base 5: 3340100 - 120 - 10 - 1,
Base 6: 1132035 - 121 - 10 - 1,
Base 7: 333233 - 100 - 1,
Base 8: 162627 - 202 - 10 - 1,
Base 9 fails since the end is the 58 - 108 - 72 cycle and fails to reach 1.

Prog

(PARI) ssd(n, b)=my(s); while(n, s+=(n%b)^2; n\=b); s
happy(k, b)=my(t=ssd(k, b)); k=ssd(t, b); while(t!=k&&k>1, t=ssd(t, b); k=ssd(ssd(k, b), b)); k==1
h3(k)=while(k>8, k=ssd(k, 3)); k==1 || k==3
a(n)=if(n<4, return(n)); my(k=2); while(k++, if(!h3(k), next); for(b=5, n, if(!happy(k, b), next(2))); return(k)) \\ Charles R Greathouse IV, Mar 22 2013

Crossrefs

Cf. A007770, A001273, A068571, A161871, A161873, A055629, A161872, A094406.

Sequence in context: A296836 A019229 A135353 * A084956 A216724 A281786

Adjacent sequences: A217702 A217703 A217704 * A217706 A217707 A217708

Keyword

base,hard,nonn

Author

Sergio Pimentel, Mar 20 2013

Extensions

a(9)-a(12) from Giovanni Resta, Mar 21 2013

Status

approved