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A342701
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a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.
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3
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3, 7, 5, 14, 9, 34, 7, 16, 15, 26, 11, 68, 39, 28, 15, 32, 33, 72, 25, 40, 35, 56, 17, 101, 45, 37, 45, 56, 29, 152, 31, 61, 39, 56, 35, 144, 37, 61, 39, 74, 41, 128, 35, 88, 45, 161, 47, 192, 49, 82, 51, 74, 95, 216, 43, 97, 75, 203, 59, 304, 91, 88, 63, 122
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OFFSET
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1,1
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COMMENTS
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Sierpiński (1956) proved that there is at least one solution for all n>=1.
Schinzel (1958) proved that there are at least two solutions k to phi(n+k) = phi(k) for all n <= 8*10^47. Schinzel and Wakulicz (1959) increased this bound to 2*10^58.
Schinzel (1958) observed that under the prime k-tuple conjecture there is a second solution for all even n.
Holt (2003) proved that there is a second solution for all even n <= 1.38 * 10^26595411.
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, section B36, page 138-142.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 217-219.
Wacław Sierpiński, Sur une propriété de la fonction phi(n), Publ. Math. Debrecen, Vol. 4 (1956), pp. 184-185.
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LINKS
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EXAMPLE
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a(1) = 3 since the solutions to the equation phi(1+k) = phi(k) are k = 1, 3, 15, 104, 164, ... (A001274), and 3 is the second solution.
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MATHEMATICA
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f[n_, 0] = 0; f[n_, k0_] := Module[{k = f[n, k0 - 1] + 1}, While[EulerPhi[n + k] != EulerPhi[k], k++]; k]; Array[f[#, 2] &, 100]
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PROG
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(PARI) a(n) = my(k=1, nb=0); while ((nb += (eulerphi(n+k)==eulerphi(k))) != 2, k++); k; \\ Michel Marcus, Mar 19 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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