A257795 Bihappy numbers: numbers that reach 1 under iteration of the sum-of-squares-of-two-digits map s_2.

1, 10, 100, 103, 301, 367, 608, 806, 1000, 1030, 1826, 2363, 2618, 2896, 3010, 3056, 3640, 4036, 4498, 4596, 5294, 5630, 6080, 6323, 6703, 6791, 8060, 8484, 9167, 9452, 9628, 9645, 9844, 10000, 10003, 10275, 10300, 10451, 10979, 11241, 11540, 12336, 12770, 12939, 13623, 13929, 14015, 14112, 15104, 15161, 16151, 16286, 17027

Offset: 1

Author

Pieter Post, May 09 2015

Keywords

• nonn
• base

Comments

If n has an even number of digits, say n = abcdef, the map is n -> s_2(n) := (ab)^2+(cd)^2+(ef)^2. If n has an odd number of digits, say n = abcde, the map is n -> s_2(n) = a^2+(bc)^2+(de)^2. The sequence {s_2(n), n >= 0} does not have its own entry in the OEIS because it begins {0, 1, ..., 9801, 1, 2, 5, ...} and agrees with the squares (A000290) for the first 100 terms. - N. J. A. Sloane, May 10 2015

This sequence is infinite because it contains several infinite subsequences (powers of 10, for example).

Examples

			367 is in the sequence since 3^2+67^2 = 4498 => 44^2+98^2= 11540 => 1^2+15^2+40^2 = 1826 => 18^2+26^2 = 1000 => 10^2+0^2 = 100 => 1^2+0^2 = 1, so in 6 iterations 367 reaches 1.
		

Links

• Pieter Post and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 244 terms from Pieter Post)

Crossrefs

Cf. A257810 (minimum of the limiting cycle), A007770 (happy numbers), A055616, A000290 (the squares).

Programs

PARI

select( {is_A257795(n,S=[])=!while(1M. F. Hasler, Dec 20 2024

Formula

All 10^k are members of this sequence.

If n is a member each permutation of a set of pairs of digits gives another member.

Placing two zeros between the sets of two digits gives another member.

All other numbers have loops of lengths 1, 2, 4, 5, 6, 10, 14, 35 or 56.

The first number with a loop of length 2 is 51, which reaches the loop (5965, 7706) after 3 iterations.

The first number with a loop of length 4 is 342, loop of 5 is 57, loop of 6 is 389, loop of 10 is 21, loop of 14 is 28, loop of 35 is 2 and the first number with a loop of 56 is 5.

And there are some numbers which end up in a loop of length 1. The first such number is 1233 (= 12^2 + 33^2)

All numbers appear to end up in one of these loops.