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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 (list; graph; refs; listen; history; text; internal format)
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)

Robert G. Wilson v, Graph of n, a(n) for n = 1..1000

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: A268597 A253560 A050399 * A073395 A064549 A304203

Adjacent sequences:  A072992 A072993 A072994 * A072996 A072997 A072998

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

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Last modified April 13 07:25 EDT 2021. Contains 342935 sequences. (Running on oeis4.)