A066220
Least k > 0 such that t^k = 1 mod (prime(n) - t) for 0 < t < prime(n).
Original entry on oeis.org
1, 1, 2, 4, 6, 60, 60, 120, 144, 7920, 55440, 18480, 7920, 27720, 2520, 637560, 8288280, 480720240, 480720240, 480720240, 480720240, 480720240, 1442160720, 9854764920, 59128589520, 59128589520, 147821473800, 670124014560
Offset: 1
Michael Ulm (taga(AT)hades.math.uni-rostock.de), Dec 18 2001
a(5) = 6 because 2^6 = 1 mod 9, 3^6 = 1 mod 8, 4^6 = 1 mod 7, 5^6 = 1 mod 6, 6^6 = 1 mod 5, 7^6 = 1 mod 4, 8^6 = 1 mod 3, 9^6 = 1 mod 2 and 6 is the minimal exponent that satisfies this.
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a[p_?PrimeQ] := Module[{e = 1}, While[! And @@ Table[Mod[PowerMod[i, e, p - i] - 1, p - i] == 0, {i, p - 1}], e++]; e]; a /@ Prime[Range[10]]
A120284
Numerator of absolute value of Sum_{k=1..n} (-1)^(k+1)*(2*k+1)*(Sum_{i=1..k} 1/i).
Original entry on oeis.org
3, 9, 25, 125, 147, 343, 761, 6849, 7381, 81191, 86021, 1118273, 1171733, 1171733, 2436559, 41421503, 42822903, 271211719, 279175675, 55835135, 19093197, 439143531, 1347822955, 33695573875, 34395742267, 309561680403, 315404588903
Offset: 1
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Numerator[Abs[Table[Sum[(-1)^(k+1)*(2k+1)*Sum[1/i,{i,1,k}],{k,1,n}],{n,1,30}]]]
A127063
Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube and denominator Sum_{k=1..p-1} 1/k^4 is a fourth power and denominator Sum_{k=1..p-1} 1/k^5 is a fifth power.
Original entry on oeis.org
2, 3, 5, 17, 439, 443, 16400183, 16400191, 16400201, 16400203, 16400221, 16400231, 16400233, 16400269, 16400273, 16400299, 16400309, 16400317, 16400347, 16400383, 16400387, 16400389, 16400411, 16400413, 16400429, 16400431
Offset: 1
Cf.
A061002,
A034602,
A127029,
A127042,
A127043,
A127044,
A127046,
A127047,
A127048,
A127049,
A127051,
A127061,
A127062.
A330014
When prime(n) is an odd prime (n >= 2) and N(n) / D(n) = Sum_{k=1..prime(n)-1} 1/k^3, then prime(n) divides N(n) and a(n) = N(n) / prime(n).
Original entry on oeis.org
3, 407, 4081, 1742192177, 1964289620189, 26430927136768997, 12913609418092462447, 14639800647032731764901, 21461951639001843544904995612963, 489697309796854053100609288112563213, 97796057728171000155497946604711651753457
Offset: 2
For prime(4) = 7 then 1 + 1/2^3 + 1/3^3 + 1/4^3 + 1/5^3 + 1/6^3 = 28567/24000 and 28567/7 = 4081, a(4) = 4081.
- Guy Alarcon and Yves Duval, TS: Préparation au Concours Général, RMS, Collection Excellence, Paris, 2010, chapitre 10, Exercices de sélection de la délégation française en Octobre 2005 pour OIM 2006, Exercice 1, p. 169, p. 179.
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[(Numerator(&+ [1/(k-1)^3:k in [2..NthPrime(n)]])) / NthPrime(n):n in [2..12]]; // Marius A. Burtea, Nov 27 2019
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a[n_] := Numerator[Sum[1/(i- 1)^3, {i, 2,(p = Prime[n])}]]/p; Array[a, 11, 2] (* Amiram Eldar, Nov 27 2019 *)
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