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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A257950 Numbers n which are both happy (A007770) and bihappy (A257795) numbers.

Original entry on oeis.org

1, 10, 100, 103, 301, 367, 608, 806, 1000, 1030, 3010, 3056, 5630, 6080, 6703, 6791, 8060, 9167, 10000, 10003, 10275, 10300, 11241, 12770, 12939, 13929, 14112, 17027, 17502, 20175, 21921, 22119, 27501, 30001, 30067, 30100, 30616, 31606, 36700
Offset: 1

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Author

Pieter Post, May 14 2015

Keywords

Comments

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

Examples

			367 is member of this sequence because 367 = 3^2+6^2+7^2= 94 => 9^2+4^2 = 97 => 9^2+7^2 = 130 => 1^2+3^2+0^2 = 10 => 1^2+0^2 = 1, so after five iterations 367 reaches 1. And 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

Crossrefs

Cf. A007770, A257795.

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 members (example 367 and 6703).
Placing two zeros between the sets of two digits gives another member.

A257810 Smallest number of the cycle in which n ends under iteration of sum-of-the-square-of-two-digits.

Original entry on oeis.org

1, 37, 37, 37, 41, 41, 41, 41, 37, 1, 41, 41, 41, 41, 37, 37, 41, 41, 37, 37, 1268, 41, 41, 41, 41, 41, 41, 1946, 37, 37, 41, 41, 37, 37, 41, 41, 37, 37, 41, 37, 41, 41, 1946, 41, 37, 41, 1946, 37, 41, 41, 5965, 41, 41, 41, 41, 37, 1781, 41, 37, 41, 41, 41, 1268, 41, 41, 37, 37, 41, 37, 41, 41, 41, 41, 5965, 37, 37, 41, 41, 41, 41, 37, 37, 41, 37, 41, 41, 37, 41, 41, 37, 41, 41, 41, 41, 37, 41, 41, 41, 41, 1
Offset: 1

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Author

Pieter Post, May 09 2015

Keywords

Comments

For the definition of the 'sum-of-the-square-of-two-digits' see the comment on A257795 where the map is called s_2.
The following statements, densities and conjectures are based on calculations for n = 1..10000.
The map s_2 has fixed points 1, 1233, 3388, ... These are cycles of length 1. For the two four digit numbers see A055616.
Because s_2(3312) = 1233 and s_2(8833) = 3388 one has a(3312) = 1233 and a(8833) = 3388.
The numbers that end under iteration of s_2 in the cycle (1) are called the bihappy numbers (A257795). They have a density of 0.33%.
It is conjectured that iterations of s_2 always end in cycles of finite period length and besides the 1-cycles for fixed points and the bihappy numbers there are ten different cycles of length > 1. The period lengths are: 2, 2, 4, 5, 5, 6, 10, 14, 35 or 56.
Two cycles with a period length of 2: 5965 => 7706 => 5965, first number that reaches this 2-cycle is 51, the second 2-cycle is: 3869 => 6205 => 3869, first number that reaches this 2-cycle is 562. Density of both 2-cycles together is 0.9%.
Cycle with a period length of 4: 3460 => 4756 => 5345 => 4834 => 3460. First number to reach this 4-cycle is 342. Density is 0.69%.
Two cycles with a period length of 5: (1781, 6850, 7124, 5617, 3425, 1781), first number to reach this 5-cycle is 57. And (3770, 6269, 8605, 7421, 5917), first number to reach this 5-cycle is 162. Density of both 5-cycles together is 1.78%.
Cycle with a period length of 6: (4973, 7730, 6829, 5465, 7141, 6722, 4973). First number to reach this 6-cycle is 389. Density exactly 1%.
Cycle with a period length of 10: (1268, 4768, 6833, 5713, 3418, 1480, 6596, 13441, 2838, 2228, 1268). First number to reach this 10-cycle is 21. Density 0.48%.
Cycle with a period length of 14: (1946, 2477, 6505, 4250, 4264, 5860, 6964, 8857, 10993, 8731, 8530, 8125, 7186, 12437, 1946). First number to reach this 14-cycle is 28. Density 5.5%.
Cycle with a period length of 35: (37, 1369, 4930, 3301, 1090, 8200, 6724, 5065, 6725, 5114, 2797, 10138, 1446, 2312, 673, 5365, 7034, 6056, 6736, 5785, 10474, 5493, 11565, 4451, 4537, 3394, 9925, 10426, 693, 8685, 14621, 2558, 3989, 9442, 10600, 37). First number to reach this 35-cycle is 2. Density is 27.89%.
Cycle with a period length of 56: (41, 1681, 6817, 4913, 2570, 5525, 3650, 3796, 10585, 7251, 7785, 13154, 3878, 7528, 6409, 4177, 7610, 5876, 9140, 9881, 16165, 7947, 8450, 9556, 12161, 4163, 5650, 5636, 4432, 2960, 4441, 3617, 1585, 7450, 7976, 12017, 690, 8136, 7857, 9333, 9738, 10853, 2874, 6260, 7444, 7412, 5620, 3536, 2521, 1066, 4456, 5072, 7684, 12832, 1809, 405, 41). First number to reach this 56-cycle is 5. Density 61.38%.

Examples

			s_2^[3](51)=5965, since 51^2 = 2601 => 26^2+1^2 = 677 => 6^2+77^2 = 5965 => 59^2+ 65^2 = 7706 => 77^2+6^2 = 5965. Three iterations are needed to reach the 2-cycle (5965, 7706).

Links

Crossrefs

Cf. A257795 (bi-happy numbers: indices of 1s in this sequence), A007770 (happy numbers), A055616 (fixed points of a similar map).

Programs

  • PARI
    apply( {A257810(n)=for(i=0,1, my(S=[n]); while(!setsearch(S, n=norml2(digits(n, 100))), S=setunion(S, [n])); i && n=S[1]); n}, [1..99]) \\ M. F. Hasler, Dec 20 2024

Extensions

Edited by Wolfdieter Lang, Jun 08 2015

A257948 Length of cycle in which n ends under iteration of sum-of-squares-of-two-digits map s_2.

Original entry on oeis.org

1, 35, 35, 35, 56, 56, 56, 56, 35, 1, 56, 56, 56, 56, 35, 35, 56, 56, 35, 35, 10, 56, 56, 56, 56, 56, 56, 14, 35, 35, 56, 56, 35, 35, 56, 56, 35, 35, 56, 35, 56, 56, 14, 56, 35, 56, 14, 35, 56, 56, 2, 56, 56, 56, 56, 35, 5, 56, 35, 56, 56, 56, 10, 56, 56, 35, 35, 56, 35, 56, 56, 56, 56, 2, 35, 35, 56, 56, 56, 56, 35, 35, 56, 35, 56, 56, 35, 56, 56, 35, 56, 56, 56, 56, 35, 56, 56, 56, 56, 1
Offset: 1

Views

Author

Pieter Post, May 14 2015

Keywords

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 following statements, densities and conjectures are based on calculations for n = 1...10000.
The map s_2 has fixed points 1, 1233, 3388. These are cycles of length 1. For the two 4-digit numbers see A055616. There are more numbers that end under iteration of s_2 in 1233 or 3388. Like a(3312) = 1233 or a(3388) = 8833.
The numbers that end under iteration of s_2 in 1 are called the bihappy numbers (A257795). They have a density of 0.33%.
It is conjectured that iterations of s_2 always end in cycles of finite period length and besides the 1-cycles there are ten different cycles of length > 1. The period lengths are 2, 2, 4, 5, 5, 6, 10, 14, 35 or 56.
Two cycles with a period length of 2: 5965 => 7706 => 5965, first number that reaches this 2-cycle is 51, the second 2-cycle is: 3869 => 6205 => 3869, first number that reaches this 2-cycle is 562. Density of both 2-cycles together is 0.9%.
Cycle with a period length of 4: 3460 => 4756 => 5345 => 4834 => 3460. First number to reach this 4-cycle is 342. Density is 0.69%.
Two cycles with a period length of 5: (1781, 6850, 7124, 5617, 3425, 1781), first number to reach this 5-cycle is 57. And (3770, 6269, 8605, 7421, 5917), first number to reach this 5-cycle is 162. Density of both 5-cycles together is 1.78%.
Cycle with a period length of 6: (4973, 7730, 6829, 5465, 7141, 6722, 4973). First number to reach this 6-cycle is 389. Density exactly 1%.
Cycle with a period length of 10: (1268, 4768, 6833, 5713, 3418, 1480, 6596, 13441, 2838, 2228, 1268). First number to reach this 10-cycle is 21. Density 0.48%.
Cycle with a period length of 14: (1946, 2477, 6505, 4250, 4264, 5860, 6964, 8857, 10993, 8731, 8530, 8125, 7186, 12437, 1946). First number to reach this 14-cycle is 28. Density 5.5%.
Cycle with a period length of 35: (37, 1369, 4930, 3301, 1090, 8200, 6724, 5065, 6725, 5114, 2797, 10138, 1446, 2312, 673, 5365, 7034, 6056, 6736, 5785, 10474, 5493, 11565, 4451, 4537, 3394, 9925, 10426, 693, 8685, 14621, 2558, 3989, 9442, 10600, 37). First number to reach this 35-cycle is 2. Density is 27.89%.
Cycle with a period length of 56: (41, 1681, 6817, 4913, 2570, 5525, 3650, 3796, 10585, 7251, 7785, 13154, 3878, 7528, 6409, 4177, 7610, 5876, 9140, 9881, 16165, 7947, 8450, 9556, 12161, 4163, 5650, 5636, 4432, 2960, 4441, 3617, 1585, 7450, 7976, 12017, 690, 8136, 7857, 9333, 9738, 10853, 2874, 6260, 7444, 7412, 5620, 3536, 2521, 1066, 4456, 5072, 7684, 12832, 1809, 405, 41). First number to reach this 56-cycle is 5. Density 61.38%.
Density is calculated over s_2(1) till s_2(10000).

Examples

			s_2^[9](2)= 35, because 2^2=4=>  4^2=16 => 16^2=256 =>   2^2+56^2=3140 =>   31^2+40^2=2561 =>  25^2+61^2=4346 => 43^2+46^2=3965 => 39^2+65^2=5746 => 57^2+46^2=5365=> 53^2+65^2= 7034. Nine iterations are needed to reach the 35-cycle.
s_2^[3](51)=2, since 51^2 = 2601 => 26^2+1^2 = 677 => 6^+77^2 = 5965 => 59^2+ 65^2 = 7706 => 77^2+6^2 = 5965. Three iterations are needed to reach the 2-cycle.

Links

Crossrefs

Cf. A257795, A055616, A007770.
Showing 1-3 of 3 results.

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