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A300285
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The number of solutions to phi(x) = phi(x+1) below 10^n, where phi(x) is the Euler totient function.
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1
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2, 3, 10, 17, 36, 68, 142, 306, 651, 1267, 2567, 5236, 10755
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OFFSET
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1,1
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COMMENTS
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The terms were calculated by:
a(1)-a(2) - R. Ratat (1917).
a(3) - Victor L. Klee, Jr. (1947).
a(4)-a(5) - Mohan Lal and Paul Gillard (1972).
a(6) - David Ballew, Janell Case and Robert N. Higgins (1975).
a(7)-a(8) - Robert Baillie (1976).
a(9)-a(10) - Sidney West Graham, Jeffrey J. Holt, and Carl Pomerance (1999).
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REFERENCES
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R. Ratat, L'Intermédiaire des Mathématiciens, Vol. 24, pp. 101-102, 1917.
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LINKS
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Sidney West Graham, Jeffrey J. Holt, and Carl Pomerance,On the solutions to phi(n)= phi(n+ k), Number Theory in Progress, Proceedings of the International Conference in Honor of the 60th Birthday of A. Schinzel, Poland, 1997, Walter de Gruyter, 1999, pp. 867-882.
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FORMULA
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According to Thomas Ordowski's conjecture in A001274, a(n) ~ 10^(C*n/3), where C = 9/Pi^2 = 0.911891... Numerically it seems that C ~ 0.93.
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EXAMPLE
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Below 10^2 there are 3 solutions x = 1, 3, 15, hence a(2) = 3.
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MATHEMATICA
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With[{s = Array[EulerPhi, 10^6]}, Array[Count[Range[10^# - 1], _?(s[[#]] == s[[# + 1]] &)] &, IntegerLength@ Length@ s - 1]] (* Michael De Vlieger, Mar 04 2018 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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