A067206 Numbers n such that the digits of n end in phi(n).

1, 1320, 1640, 1768, 1996, 2640, 3960, 13200, 16400, 19984, 19996, 26400, 39600, 132000, 164000, 199996, 264000, 396000, 1320000, 1640000, 1999936, 2640000, 3960000, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996

Offset: 1

Author

Joseph L. Pe, Feb 19 2002

Keywords

• base
• nonn

Comments

Comments from Farideh Firoozbakht, Dec 30 2006: (Start)

"(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.

"(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.

"(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.

"(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)

Examples

			The digits of 1768 end in phi(1768) = 768, so 1768 is a term of the sequence.
		

References

• Pickover, C. "Wonders of Numbers". Oxford Univ. Press, 2001.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..66 (terms < 10^12, first 33 terms from Farideh Firoozbakht)
• C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Crossrefs

Cf. A093945, A091439.

Cf. A066663. - R. J. Mathar, Sep 30 2008

Programs

Mathematica

(*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[ # ]] &]

Extensions

More terms from Farideh Firoozbakht, Dec 30 2006