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A072995
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.
6
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
OFFSET
1,2
COMMENTS
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.]
Conjecture: a(n)=0 iff n/2 is in A061346. - Robert G. Wilson v, Feb 23 2014
[n=420 seems to be a counterexample to the above conjecture. - M. F. Hasler, Feb 24 2014]
From Robert G. Wilson v, Mar 05 2014: (Start)
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
When a(n) > 1, then 2n <= a(n) <= n^2. - Robert G. Wilson v, Dec 10 2014
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..1013 (first 200 terms from Don Reble)
FORMULA
First occurrence of n in A072994.
MATHEMATICA
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 *)
PROG
(PARI) A072995(n)=(n%2||n%2^(omega(n)-1)==0)&&for(k=1, 9e9, A072994(k)==n&&return(k)) \\ M. F. Hasler, Feb 23 2014
CROSSREFS
Cf. A072994.
Sequence in context: A373319 A253560 A050399 * A354165 A073395 A064549
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Aug 21 2002
EXTENSIONS
More terms from Don Reble, Feb 23 2014
Edited, at the suggestion of Don Reble, by M. F. Hasler, Feb 23 2014
STATUS
approved