A161874 Bases with smallest unhappy number (in that base) > 2.

16, 18, 20, 30, 130, 256, 1042, 4710, 7202, 10082, 47274, 65536, 65600, 351634, 426530, 431730, 764930, 5921514, 26639560, 32435910, 88605010, 97025190, 99562110

Offset: 1

Author

Jud McCranie, Jun 20 2009

Keywords

• nonn
• base
• more

Comments

a(12) > 50000, if it exists. - Amiram Eldar, Jun 16 2020

a(18) > 1.5*10^6, if it exists. The smallest unhappy numbers corresponding to bases a(1)-a(17) are 3, 7, 3, 5, 20, 3, 12, 3, 3, 14, 3, 3, 3, 3, 3, 3, 23. - Giovanni Resta, Jun 25 2020

The smallest unhappy numbers in bases (a(18), ..., a(23)) are (3, 23, 3, 261, 6, 12). - Lucas A. Brown, Apr 25 2023

a(24) > 10^8, if it exists. - Lucas A. Brown, Apr 25 2023

Examples

			In base 16, 2 is happy because the sequence it generates is 2 -> 4 -> (1,0) -> 1, while 3 is unhappy because the sequence it generates is 3 -> 9 -> (5,1) -> (1,10) -> (6,5) -> (3,13) -> (11,2) -> (7,13) -> (13,10) -> (1,0,13) -> (10,10) -> (12,8) -> (13,0) -> (10,9) -> (11,5) -> (9,2) -> (5,5) -> (3,2) -> (0,13) -> (10,9) -> ..., which repeats with period 6.
		

Links

• Lucas A. Brown, A161874.c

Crossrefs

Cf. A031177 (unhappy numbers in base 10), A161872, A362026.

Programs

Mathematica

happyQ[n_, b_] := NestWhile[Total[IntegerDigits[#, b]^2] &, n, UnsameQ, All] == 1; Select[Range[2, 256], !MemberQ[{2, 4}, #] && happyQ[2, #] &] (* Amiram Eldar, Jun 16 2020 *)

Extensions

a(11) from Amiram Eldar, Jun 16 2020

a(12)-a(17) from Giovanni Resta, Jun 25 2020

a(18) from Lucas A. Brown, Aug 17 2022

a(19) from Lucas A. Brown, Aug 26 2022

a(20)-a(23) from Lucas A. Brown, Apr 25 2023