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A143548 Irregular triangle of numbers k < p^2 such that p^2 divides k^(p-1)-1, with p=prime(n). 11
1, 1, 8, 1, 7, 18, 24, 1, 18, 19, 30, 31, 48, 1, 3, 9, 27, 40, 81, 94, 112, 118, 120, 1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288, 1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n begins with 1 and has prime(n)-1 terms. The first differences of each row are symmetric. For k > p^2, the solutions are just shifted by m*p^2 for m > 0. An open question is whether every integer appears in this sequence. For instance, 2 does not appear until the prime 1093 and 5 does not appear until the prime 20771.

For row n > 1, the sum of the terms in row n is (p-1)*p^2*(p+1)/2, which is A138416. - T. D. Noe, Aug 24 2008, corrected by Robert Israel, Sep 27 2016

Max Alekseyev points out that there is a much faster method of computing these numbers. Let p=prime(n) and let r be a primitive root of p (see A001918 and A060749). Then the terms in row n are r^(k*p) (mod p^2) for k=0..p-2. - T. D. Noe, Aug 26 2008

The numbers in this sequence are the bases to Euler pseudoprimes q, which are squares of prime numbers, such that n^((q-1)/2) == +-1 mod q. An exception is the first number 9 = 3*3, which is, following the strict definition in Crandall and Pomerance, no Fermat pseudoprime and hence no Euler pseudoprime. - Karsten Meyer, Jan 08 2011

For row n > 1, the sum is zero modulo p^2 (rows are antisymmetric due to Binomial Theorem). - Peter A. Lawrence, Sep 11 2016

REFERENCES

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2005

LINKS

T. D. Noe, Rows n=1..50 of triangle, flattened

EXAMPLE

(2)   1,

(3)   1, 8,

(5)   1, 7, 18, 24,

(7)   1, 18, 19, 30, 31, 48,

(11)  1, 3, 9, 27, 40, 81, 94, 112, 118, 120,

(13)  1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168,

(17)  1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288,

MAPLE

f:= proc(n) local p, j, x;

  p:= ithprime(n);

  x:= numtheory:-primroot(p);

  op(sort([seq(x^(i*p) mod p^2, i=0..p-2)]))

end proc:

map(f, [$1..20]); # Robert Israel, Sep 27 2016

MATHEMATICA

Flatten[Table[p=Prime[n]; Select[Range[p^2], PowerMod[ #, p-1, p^2]==1&], {n, 50}]] (* T. D. Noe, Aug 24 2008 *)

Flatten[Table[p=Prime[n]; r=PrimitiveRoot[p]; b=PowerMod[r, p, p^2]; Sort[NestList[Mod[b*#, p^2]&, 1, p-2]], {n, 50}]] (* Faster version from T. D. Noe, Aug 26 2008 *)

CROSSREFS

Cf. A039678, A056020, A056021, A056022, A056024, A056025, A056027, A056028, A056031, A056034, A056035, A096082, A138416.

Sequence in context: A234614 A246750 A199872 * A231929 A154460 A021554

Adjacent sequences:  A143545 A143546 A143547 * A143549 A143550 A143551

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, Aug 24 2008

STATUS

approved

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Last modified August 7 17:34 EDT 2020. Contains 336278 sequences. (Running on oeis4.)