A161874
Bases with smallest unhappy number (in that base) > 2.
Original entry on oeis.org
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
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.
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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 *)
A217705
Smallest number greater than 1 that is happy under bases 2 through n.
Original entry on oeis.org
2, 3, 3, 23, 79, 2207, 58775, 569669, 11814485, 210511543, 73748383237
Offset: 2
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.
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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
A362026
Smallest unhappy number in base A161874(n).
Original entry on oeis.org
3, 7, 3, 5, 20, 3, 12, 3, 3, 14, 3, 3, 3, 3, 3, 3, 23, 3, 23, 3, 261, 6, 12
Offset: 1
The first term in this sequence corresponds to base 16. 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.
Showing 1-3 of 3 results.
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