A267495 Autobiographical numbers in base 7: numbers which are fixed or belong to a cycle under the operator T.

22, 10213223, 10311233, 10313314, 10313315, 10313316, 21322314, 21322315, 21322316, 31123314, 31123315, 31123316, 31331415, 31331416, 31331516, 1031223314, 1031223315, 1031223316, 3122331415, 3122331416, 3122331516, 103142132415, 104122232415, 103142132416, 104122232416, 314213241516, 412223241516, 1011112131415, 1011112131416, 1011112131516, 1011112141516, 1011113141516, 1111213141516, 10414213142516, 10413223241516, 10512223142516, 10512213341516, 101112213141516

Offset: 1

Author

Antonia Münchenbach, Jan 16 2016

Keywords

• nonn
• base
• fini
• full

Comments

The T operator numerically summarizes the frequency of digits 0 through 6 in that order when they occur in a number. The numbers and the frequency are written in base 7.

These are all autobiographical numbers in base 7 which lead to a fixed-point or belong to a cycle.

There are three cycles with length 2 (103142132415 /104122232415, 103142132416/104122232416, 314213241516/412223241516), one cycle with length 3 (10512213341516/10512223142516/10414213142516). 29 numbers are fixed-points.

Examples

			10213223 contains one 0, two 1's, three 2's and two 3's, so T(10213223) = 1 0 2 1 3 2 2 3, and this is fixed under T.
103142132415 and 104122232415 belong to the cycle of length 2, so
T(T(103142132415)) = T(1 0 4 1 2 2 2 3 2 4 1 5) = 1 0 3 1 4 2 1 3 2 4 1 5.
		

References

• Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Links

• Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Crossrefs

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.