Displaying 1-10 of 12 results found.
Generalized repunits in base 14.
+10
38
1, 15, 211, 2955, 41371, 579195, 8108731, 113522235, 1589311291, 22250358075, 311505013051, 4361070182715, 61054982558011, 854769755812155, 11966776581370171, 167534872139182395, 2345488209948553531
COMMENTS
Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=14, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
FORMULA
a(n) = (14^n - 1)/13.
a(n) = Sum_{i=0..n-1} 13^i*binomial(n,n-1-i). - Bruno Berselli, Nov 12 2015 G.f.: x/((1-x)*(1-14*x)).
E.g.f.: (1/13)*(exp(14*x) - exp(x)). (End)
EXAMPLE
a(4) = 2955 because (14^4-1)/13 = 38416/13 = 2955.
For n=6, a(6) = 1*6 + 13*15 + 169*20 + 2197*15 + 28561*6 + 371293*1 = 579195. - Bruno Berselli, Nov 12 2015
MATHEMATICA
Table[FromDigits[PadRight[{}, n, 1], 14], {n, 20}] (* or *) LinearRecurrence[{15, -14}, {1, 15}, 20] (* Harvey P. Dale, Aug 29 2016 *)
PROG
(Sage) [gaussian_binomial(n, 1, 14) for n in range(1, 15)] # Zerinvary Lajos, May 28 2009
CROSSREFS
Cf. A000225, A001022, A002450, A002452, A003462, A003463, A003464, A016123, A016125, A023000, A023001, A135278.
AUTHOR
Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008
Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.
+10
9
3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3
COMMENTS
a(n) = A084740(n) for all n except n = p-1, where p is an odd prime, for which A084740(n) = 2. All nonzero terms are odd primes.
a(n) = 0 for n = {4,9,16,25,32,36,49,64,81,100,121,125,144,...}, which are the perfect powers with exceptions of the form n^(p^m) where p>2 and (n^(p^(m+1))-1)/(n^(p^m)-1) are prime and m>=1 (in which case a(n^(p^m))=p). - Max Alekseyev, Jan 24 2009 a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime. a(n) is the least number k such that (n^k - 1)/(n-1) is a Brazilian prime, or 0 if no such Brazilian prime exists. - Bernard Schott, Apr 23 2017 a(152) = 270217, see the top PRP link. - Eric Chen, Jun 04 2018 a(184) = 16703, a(200) = 17807, a(210) = 19819, a(306) = 26407, a(311) = 36497, a(326) = 26713, a(331) = 25033; a(185) > 66337, a(269) > 63659, a(281) > 63421, and there are 48 unknown a(n) for n <= 1024. - Eric Chen, Jun 04 2018 Six more terms found: a(522)=20183, a(570)=12907, a(684)=22573, a(731)=15427, a(820)=12043, a(996)=14629. - Michael Stocker, Apr 09 2020
LINKS
Eric Weisstein's World of Mathematics, Repunit
EXAMPLE
a(7) = 5 because (7^5 - 1)/6 = 2801 = 11111_7 is prime and (7^k - 1)/6 = 1, 8, 57, 400 for k = 1, 2, 3, 4. - Bernard Schott, Apr 23 2017
MATHEMATICA
Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[(m^k - 1)/(m - 1)], k++]; k, 0]]@ If[Set[e, GCD @@ #[[All, -1]]] > 1, {#, IntegerExponent[n, #]} &@ Power[n, 1/e] /. {{k_, m_} /; Or[Not[PrimePowerQ@ m], Prime@ m, FactorInteger[m][[1, 1]] == 2] :> 0, {k_, m_} /; m > 1 :> n}, n] &@ FactorInteger@ n, {n, 2, 17}] (* Michael De Vlieger, Apr 24 2017 *)
PROG
(PARI) a052409(n) = my(k=ispower(n)); if(k, k, n>1)
a052410(n) = if (ispower(n, , &r), r, n)
is(n) = issquare(n) || (ispower(n) && !ispseudoprime((n^a052410(a052409(n))-1)/(n-1)))
CROSSREFS
Cf. A000043, A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A133857, A006035, A127995, A127996, A127997, A204940, A127998, A127999, A128000, A181979, A098438, A128002, A209120, A185073, A128003, A128004, A181987, A128005, A239637, A240765, A294722, A242797, A243279, A267375, A245237, A245442, A173767. (numbers n such that (b^n-1)/(b-1) is prime for b = 2 to 53)
Numbers n such that (43^n - 1)/42 is prime.
+10
7
5, 13, 6277, 26777, 27299, 40031, 44773, 194119
MATHEMATICA
Select[Prime[Range[100000]], PrimeQ[(43^#-1)/42]&]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005.
Least k such that Phi(k,n), the k-th cyclotomic polynomial evaluated at n, is prime.
+10
5
2, 2, 1, 1, 3, 1, 5, 1, 6, 2, 9, 1, 5, 1, 3, 2, 3, 1, 19, 1, 3, 2, 5, 1, 6, 4, 3, 2, 5, 1, 7, 1, 3, 6, 21, 2, 10, 1, 6, 2, 3, 1, 5, 1, 19, 2, 10, 1, 14, 3, 6, 2, 11, 1, 6, 4, 3, 2, 3, 1, 7, 1, 5, 204, 12, 2, 6, 1, 3, 2, 3, 1, 5, 1, 3, 6, 3, 2, 5, 1, 6, 2, 5, 1, 5, 11, 7, 2, 3, 1, 6, 12, 7, 4, 7, 2, 17, 1, 3
COMMENTS
Note that a(n)=1 iff n-1 is prime because Phi(1,x)=x-1. For n<2048, we have the bound a(n)<251. However, a(2048) is greater than 10000. Is a(n) defined for all n? For fixed n, there are many sequences listing the k that make Phi(k,n) prime: A000043, A028491, A004061, A004062, A004063, A004023, A005808, A016054, A006032, A006033, A006034, A006035.
MATHEMATICA
Table[k=1; While[ !PrimeQ[Cyclotomic[k, n]], k++ ]; k, {n, 100}]
CROSSREFS
Cf. A117544 (least k such that Phi(n, k) is prime).
Numbers n such that (45^n - 1)/44 is prime.
+10
5
19, 53, 167, 3319, 11257, 34351, 216551
COMMENTS
a(7) > 10^5.
Numbers corresponding to a(4)-a(6) are probable primes.
All terms are prime.
MATHEMATICA
Select[Prime[Range[100000]], PrimeQ[(45^# - 1)/44] &]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765.
EXTENSIONS
a(7)=216551 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020
Numbers n such that (46^n - 1)/45 is prime.
+10
5
2, 7, 19, 67, 211, 433, 2437, 2719, 19531
COMMENTS
a(10) > 10^5.
Numbers corresponding to a(7)-a(9) are probable primes.
All terms are prime.
MATHEMATICA
Select[Prime[Range[100000]], PrimeQ[(46^# - 1)/45] &]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765, A242797.
Numbers n such that (23^n - 1)/22 is prime.
+10
4
5, 3181, 61441, 91943, 121949, 221411
MATHEMATICA
Select[Prime[Range[100]], PrimeQ[(23^#-1)/22]&]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A128000, A098438, A128002, A128003, A128004, A128005.
EXTENSIONS
a(5)=121949 corresponds to a probable prime discovered by Paul Bourdelais, Oct 19 2017 a(6)=221411 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020
Numbers n such that (48^n - 1)/47 is prime.
+10
3
19, 269, 349, 383, 1303, 15031, 200443
COMMENTS
a(7) > 10^5.
All terms are prime.
MATHEMATICA
Select[Prime[Range[100000]], PrimeQ[(48^# - 1)/47] &]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765, A242797, A243279.
EXTENSIONS
a(7) corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020
Numbers n such that (50^n - 1)/49 is prime.
+10
2
3, 5, 127, 139, 347, 661, 2203, 6521, 210319
COMMENTS
a(9) > 10^5.
All terms are prime.
MATHEMATICA
Select[Prime[Range[100000]], PrimeQ[(50^# - 1)/49] &]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765, A242797, A243279, A245237.
EXTENSIONS
a(9)=210319 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020
Numbers n such that (39^n - 1)/38 is prime.
+10
1
349, 631, 4493, 16633, 36341
MATHEMATICA
Select[Prime[Range[100000]], PrimeQ[(39^#-1)/38]&]
CROSSREFS
Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005.
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