k-morphic numbers

[cloitre]

Relations between k - morphic numbers :

Definition : n is a k-morphic number if n^k ends with n

Observations : 

A033819=A074321 (3-morphic numbers = 7-morphic numbers)

and 

A074332=A068407 (5-morphic numbers = 13-morphic numbers)
 

More generally are some of the following statements true ?

(i)-----------------------------------------------------------------


Let k>1 be an odd integer not equal to 1 (mod10) :

If k == 3 (mod4) the k-morphic sequence = the 3-morphic sequence (A033819)

If k == 1 (mod4) the k-morphic sequence = the 5-morphic sequence (A068407)

(ii)-----------------------------------------------------------------

Let k' be an odd integer equal to 1 (mod10)

If (k-1)/10 is even and not divisible by 10 the k-morphic sequence is :

2,3,4,5,6,7,8,9,11,12,13,16,17,19,21,23,24,25,27,28,29,31,32,33,36,37,39,41,
43,44,47,48,49,51,52,53,56,57,59,61,63,64,67,68,69,71,72,73,75,76,77,79,81,8
3,84,87,88,89,91,92,93,96,97,99,101,107,125,143,149,151,157,168,176,193,199,
201,207,224,232,243,249,251,257,293,299 (not in the EIS)

If (k-1)/10 is odd and not divisible by 5 the k-morphic sequence is :

4,5,6,9,11,16,19,21,24,25,29,31,36,39,41,44,49,51,56,59,61,64,69,71,75,76,79
,81,84,89,91,96,99,101,125,149,151,176,199,201,224,249,251,299,301,349,351,3
75,376,399,401,424,449,451,499,501,549,551,576,599,601,624,625,649,651,699,7
01,749,751,776,799,801,824 (not in the EIS)



(iii)------------------------------------------------------------------

If k is even and does not equal 6 (mod10) the k - morphic sequence = A003226

If k is even and equals 6 (mod10) the k - morphic sequence = A068408


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Benoit Cloitre
abcloitre at wanaddo.fr