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 A039678 Smallest number m > 1 such that m^(p-1)-1 is divisible by p^2, where p = n-th prime. 34
 5, 8, 7, 18, 3, 19, 38, 28, 28, 14, 115, 18, 51, 19, 53, 338, 53, 264, 143, 11, 306, 31, 99, 184, 53, 181, 43, 164, 96, 68, 38, 58, 19, 328, 313, 78, 226, 65, 253, 259, 532, 78, 176, 276, 143, 174, 165, 69, 330, 44, 33, 332, 94, 263, 48, 79, 171, 747, 731, 20, 147, 91, 40 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Using Fermat's little theorem twice, it is easy to see that m=p^2-1 solves this problem for all odd primes p. In fact, there appear to be exactly p-1 values of m with 1 <= m <= p^2 for which m^(p-1) == 1 (mod p^2). See A096082 for the related open problem. - T. D. Noe, Aug 24 2008 That there are exactly p-1 values of 1 <= m <= p^2 for which m^(p-1) == 1 (mod p^2) follows immediately from Hensel's lifting lemma and Fermat's little theorem - every solution mod p corresponds to a unique solution mod p^2. - Phil Carmody, Jan 10 2011 For n > 2, prime(n) does not divide a(n)^2 - 1, so a(n) is the smallest m > 1 such that (m^(prime(n)-1) - 1)/(m^2 - 1) == 0 (mod prime(n)^2). - Thomas Ordowski, Nov 24 2018 REFERENCES P. Ribenboim, The New Book of Prime Number Records, Springer, 1996, 345-349. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A185103(A000040(n)). EXAMPLE For n=3, p=5 is the third prime and 5^2 = 25 divides 7^4 - 1 = 2400. MATHEMATICA dpa[n_]:=Module[{p=Prime[n], a=2}, While[PowerMod[a, p-1, p^2]!=1, a++]; a]; Array[dpa, 70] (* Harvey P. Dale, Sep 05 2012 *) PROG (PARI) a(n) = my(p=prime(n)); for(a=2, oo, if(Mod(a, p^2)^(p-1)==1, return(a))) \\ Felix Fröhlich, Nov 24 2018 (Python) from sympy import prime from sympy.ntheory.residue_ntheory import nthroot_mod def A039678(n): return 2**2+1 if n == 1 else int(nthroot_mod(1, (p:= prime(n))-1, p**2, True)) # Chai Wah Wu, May 18 2022 CROSSREFS Cf. A185103. Sequence in context: A053787 A314569 A314570 * A259234 A131040 A231786 Adjacent sequences: A039675 A039676 A039677 * A039679 A039680 A039681 KEYWORD nonn,nice AUTHOR Felice Russo EXTENSIONS More terms from David W. Wilson Definition adjusted by Felix Fröhlich, Jun 24 2014 Edited by Felix Fröhlich, Nov 24 2018 STATUS approved

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Last modified September 22 11:22 EDT 2023. Contains 365521 sequences. (Running on oeis4.)