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A072995
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Least k > 0 such that the number of solutions to x^k == 1 (mod k) 1 <= x <= k is equal to n, or 0 if no such k exists.
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6
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1, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 225, 32, 289, 54, 361, 110, 147, 242, 529, 72, 125, 338, 81, 196, 841, 0, 961, 64, 1089, 578, 1225, 108, 1369, 722, 507, 100, 1681, 0, 1849, 484, 675, 1058, 2209, 144, 343, 250, 2601, 1378, 2809
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OFFSET
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1,2
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COMMENTS
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A072989 lists the indices for which a(n) differs from A050399(n), e.g., in n = 20, 40, 52, ... in addition to the zeros in this sequence (n = 30, 42, 66, 70, 78, 90, ...). See also A009195 vs. A072994. [Corrected and extended by M. F. Hasler, Feb 23 2014]
The sequence seems difficult to extend, as the next term a(30) is larger than 5100. However, a(32)=64, a(64)=128 and a(128)=256 can be easily calculated. It thus appears that a(2^k)=2^(k+1), for k=1,2,3,.... Is this known to be true? - John W. Layman, Aug 05 2003 -- Answer: It's true. One could have defined the sequence so that a(1)=2: then it would be true for 2^0 also. - Don Reble, Feb 23 2014
a(30), if it exists, is greater than 400000. - Ryan Propper, Sep 10 2005
a(30) doesn't exist: If N is even, and divisible by D different odd primes, but not divisible by 2^D, then a(N) doesn't exist. - Don Reble, Feb 23 2014 [This and the preceding comment refer to the former definition lacking the clause "0 if no such k exists". - Ed.]
[n=420 seems to be a counterexample to the above conjecture. - M. F. Hasler, Feb 24 2014]
Observation:
If n = 1 then a(n) = 1 by definition;
If, but not iff, n (an even number) is a member of A238367 then a(n) = 0;
If n (an even number not in A238367) is {684, 954, ...}, then a(n) = 0;
If n (an odd number) is {273, 399, 651, 741, 777, 903, ...}, then a(n) = 0;
If p is a prime [A000040] and e is its exponent, then a(p^e) = p^(e+1);
If p is a prime then a(2p^e) = 2p^(e+1);
If p is a prime then a(n) # p since the f(p)=1.
(End)
Often A072995(n) equals A050399(n). They differ at n: 20, 30, 40, 42, 52, 60, 66, 68, 70, 78, 80, 84, 90, 100, 102, 104, 110, 114, 116, 120, 126, 130, 132, ... - Robert G. Wilson v, Dec 06 2014
When A072995(n)>0 and does not equal A050399(n): 20, 40, 52, 60, 68, 80, 84, 100, 104, 116, 120, 132, 136, 140, 148, 156, 160, 164, 168, 171, 180, 200, ... - Robert G. Wilson v, Dec 06 2014
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LINKS
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FORMULA
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MATHEMATICA
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t = Table[0, {1000}]; f[n_] := (d = If[EvenQ@ n, 2, 1]; d*Length@ Select[ Range[ n/d], PowerMod[#, n, n] == 1 &]); f[1] = 1; k = 1; While[k < 520001, If[ PrimeQ@ k, k++]; a = f@ k; If[a < 1001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t (* Robert G. Wilson v, Dec 12 2014 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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