A068407 Automorphic numbers: numbers k such that k^5 ends with k. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 1 (mod 4).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99, 125, 193, 249, 251, 307, 375, 376, 432, 443, 499, 501, 557, 568, 624, 625, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501, 2943, 3125, 3307, 3568, 3749, 3751

Offset: 1

Author

Benoit Cloitre, Mar 08 2002

Keywords

• nonn
• base

Examples

			13568 is a term because 13568^5 = 459810807237016813568 ends with 13568.
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to automorphic numbers

Crossrefs

Cf. A003226, A033819.

Programs

Magma

[n : n in [0..3749] | Intseq(n^5)[1..#Intseq(n)] eq Intseq(n)]; // Arkadiusz Wesolowski, Nov 15 2013

Mathematica

Select[Range[0,100000],PowerMod[#,5,10^IntegerLength[#]]==#&] (* Harvey P. Dale, Nov 04 2011 *)

Sage

def automorphic(maxdigits, pow, base=10) :
    morphs = [[0]]
    for i in range(maxdigits):
        T=[d*base^i+x for x in morphs[-1] for d in range(base)]
        morphs.append([x for x in T if x^pow % base^(i+1) == x])
    return sorted(set(sum(morphs,[])))
# (call with pow=5 for this sequence), Eric M. Schmidt, Jul 29 2013