A066663 -id:A066663 - cp's OEIS frontend

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

Joseph L. Pe, Feb 19 2002

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)

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

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

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

Cf. A093945, A091439.

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

(*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[ # ]] &]

More terms from Farideh Firoozbakht, Dec 30 2006

A248857 Composite numbers n such that n - phi(n) is a power of 10.

Original entry on oeis.org

1320, 1640, 1768, 1996, 13200, 16400, 19984, 19996, 132000, 164000, 199996, 1320000, 1640000, 1999936, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996, 132000000, 164000000, 164296960, 166662400, 199999936, 199999984, 1320000000

Offset: 1

Farideh Firoozbakht, Dec 31 2014

nonn

Numbers n such that n - phi(n) is a positive power of 10.
Numbers n such that phi(n) = n - 10^floor(log(10,n)).
The most significant digit of all terms is equal to 1, since all terms are even and for even numbers n, phi(n) <= n/2.
If p = 5^(2n-1)*10^m-1 is prime then s = 4^n*p is in the sequence, because s - phi(s) = 10^(2n+m-1).
For n=1,2, ..., 6, ... smallest such term of the sequence respectively are 1996, 19984, 1999936, 1999999744, 19999999998976,19999999995904, ... .
Sequence A248858 gives number of digits of these terms.

			1320 is in the sequence because 1320 - phi(1320) = 10^3.

Giovanni Resta, Table of n, a(n) for n = 1..47 (terms < 10^12, first 41 terms from Robert G. Wilson v)

Cf. A000010, A067206, A066663, A244440, A248854, A248858.

a248857[n_] := Select[Select[Range@n, CompositeQ[#] &], IntegerQ[Log10[# - EulerPhi[#]]] &]; a248857[10^7] (* Michael De Vlieger, Jan 07 2015 *)

lista(nn) = forcomposite(n=2, nn, if (ispower(n-eulerphi(n),,&d) && (d==10), print1(n, ", "))); \\ Michel Marcus, Jan 06 2015

a(22)-a(28) from Giovanni Resta, Apr 17 2017

cp's OEIS Frontend

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

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