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%I #24 Feb 24 2024 06:47:53
%S 1,1320,1640,1768,1996,2640,3960,13200,16400,19984,19996,26400,39600,
%T 132000,164000,199996,264000,396000,1320000,1640000,1999936,2640000,
%U 3960000,13200000,16400000,16666240,17999488,18515584,19999984,19999996
%N Numbers n such that the digits of n end in phi(n).
%C Comments from _Farideh Firoozbakht_, Dec 30 2006: (Start)
%C "(1). If n is in the sequence and 10 divides n then for each natural number k, n*10^k is in the sequence. So since 1320, 1640, 2640, 3960 & 16666240 are in the sequence, for each natural number k, 132*10^k, 164*10^k, 264*10^k, 396*10^k & 1666624*10^k are in the sequence. Hence the sequence is infinite.
%C "(2). If 5*10^k-1 is prime then 4*(5*10^k-1) is in the sequence. So 4*A093945 is a subsequence of this sequence.
%C "(3). If p=125*10^k-1 is prime then 16*p is in the sequence. For k = 1, 4, 5, 8, 13, 19, 25, 26, 76, 88, 167, 290, 389, ... p is prime.
%C "(4). If p=3125*10^k-1 is prime then 64*p is in the sequence. For k = 1, 3, 9, 33, 121, 223, 357, 363, 447, ... p is prime." (End)
%D Pickover, C. "Wonders of Numbers". Oxford Univ. Press, 2001.
%H Giovanni Resta, <a href="/A067206/b067206.txt">Table of n, a(n) for n = 1..66</a> (terms < 10^12, first 33 terms from Farideh Firoozbakht)
%H C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&format=complete">Zentralblatt review</a>
%e The digits of 1768 end in phi(1768) = 768, so 1768 is a term of the sequence.
%t (*returns true if a ends in b, false o.w.*) f[a_, b_] := Module[{c, d, e, g, h, i, r}, r = False; c = ToString[a]; d = ToString[b]; e = StringLength[c]; g = StringPosition[c, d]; h = Length[g]; If[h > 0, i = g[[h]]; If[i[[2]] == e, r = True]]; r]; Select[Range[10^5], f[ #, EulerPhi[ # ]] &]
%Y Cf. A093945, A091439.
%Y Cf. A066663. - _R. J. Mathar_, Sep 30 2008
%K base,nonn
%O 1,2
%A _Joseph L. Pe_, Feb 19 2002
%E More terms from _Farideh Firoozbakht_, Dec 30 2006
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