Displaying 1-10 of 13 results found.
Numbers k such that 13^k - 2 is a prime.
+10
14
1, 2, 4, 5, 12, 78, 80, 90, 117, 120, 813, 1502, 2306, 2946, 6308, 13320, 26369, 31868, 44265, 81008
COMMENTS
13320 is a term found by Lelio R Paula 11/2006.
Numbers corresponding to a(13)..a(16) are probable primes. If n is of the form 4k+3 then 13^n-2 is composite, because 13^n-2 == (3^4)^k*3^3 - 2 == 25 == 0 (mod 5). So there is no term of the form 4k+3. - Farideh Firoozbakht, Dec 07 2009
MATHEMATICA
Do[ f = 13^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 1000} ]
CROSSREFS
Cf. A084714 (smallest prime of the form (2n-1)^k - 2). Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Numbers k such that 17^k - 2 is a prime.
+10
14
6, 24, 30, 106, 184, 232, 460, 1258, 3480, 5458, 32886
MATHEMATICA
Do[ f = 17^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 1000} ]
CROSSREFS
Cf. A084714 (smallest prime of the form (2n-1)^k - 2). Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Numbers k such that 19^k - 2 is a prime.
+10
14
1, 2, 3, 13, 14, 19, 20, 23, 38, 1124, 7592, 11755, 12155, 12915, 14172, 15500, 20255, 28388, 184650
MATHEMATICA
Do[ f = 19^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 1000} ]
CROSSREFS
Cf. A084714 (smallest prime of the form (2n-1)^k - 2). Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Numbers k such that 15^k - 2 is a prime.
+10
13
1, 2, 3, 7, 12, 17, 19, 51, 65, 550, 1460, 1641, 7035, 18002, 20963, 21163, 42563, 94906, 148048
MATHEMATICA
Do[ f = 15^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 1000} ]
Do[If[PrimeQ[15^n - 2], Print[n]], {n, 10^4}] (* Ryan Propper, Jun 06 2007 *)
CROSSREFS
Cf. A084714 (smallest prime of the form (2n-1)^k - 2). Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
a(n) is the smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists.
+10
13
0, 79, 6103515623, 5764799, 31381059607
COMMENTS
a(6) = 11^22420 - 2 was found by Rick L. Shepherd on Sep 29 2007. It has 23349 decimal digits and it is too large to include. a(7) through a(12): {771936328432730777189183517369830159827426282764863750131729657829597399846468418688727, 98526125335693359373, 339448671314611904643504117119, 37589973457545958193355599, 1136272165922724266740722458520499, 480250763996501976790165756943039}.
CROSSREFS
Cf. A084714 (smallest prime of the form (2n-1)^k - 2). Cf. A133856 (least number k > (2n-1) such that (2n-1)^k - 2 is prime).
Numbers k such that 21^k - 2 is a prime.
+10
12
1, 2, 4, 10, 21, 25, 27, 32, 60, 88, 106, 120, 146, 264, 828, 965, 1944, 4822, 12089, 14427, 17354, 42335, 46395, 58348, 190632
MATHEMATICA
Do[ f = 21^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 1000} ]
CROSSREFS
Cf. A084714 (smallest prime of the form (2n-1)^k - 2). Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
9, 119, 1329, 14639, 161049, 1771559, 19487169, 214358879, 2357947689, 25937424599, 285311670609, 3138428376719, 34522712143929, 379749833583239, 4177248169415649, 45949729863572159, 505447028499293769
COMMENTS
There are only two known primes in a(n): a(4) = 14639 and a(6) = 1771559 (see A128472 = smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists). 3 divides a(2k-1). 7 divides a(3k-1). 13 divides a(12k-5). 17 divides a(16k-14). Final digit of a(n) is 9.
Final two digits of a(n) are periodic with period 10: a(n) mod 100 = {09, 19, 29, 39, 49, 59, 69, 79, 89, 99}.
Final three digits of a(n) are periodic with period 50: a(n) mod 1000 = {009, 119, 329, 639, 049, 559, 169, 879, 689, 599, 609, 719, 929, 239, 649, 159, 769, 479, 289, 199, 209, 319, 529, 839, 249, 759, 369, 079, 889, 799, 809, 919, 129, 439, 849, 359, 969, 679, 489, 399, 409, 519, 729, 039, 449, 959, 569, 279, 089, 999}.
MATHEMATICA
LinearRecurrence[{12, -11}, {9, 119}, 17] (* Ray Chandler, Aug 26 2015 *)
CROSSREFS
Cf. A001020, A024127, A034524. Cf. A104096 = Largest prime <= 11^n. Cf. A084714 = smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists. Cf. A128472 = smallest prime of the form (2n-1)^k - 2 for k>(2n-1), or 0 if no such number exists. Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.
Primes of the form 11^k - 2.
+10
3
COMMENTS
Last digit of all terms is 9.
The nest term (11^22420-2) is too large to be displayed; see A133982 for the corresponding k. - Joerg Arndt, Nov 28 2020
EXAMPLE
a(1) = 11^4 - 2 = 14639,
a(2) = 11^6 - 2 = 1771559.
Square matrix T(m,n)=1 if (2m+1)^(2n-1)-2 is prime, 0 otherwise; read by antidiagonals.
+10
2
0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
COMMENTS
In some sense the "minimal" possible generalization of the pattern of Mersenne primes (cf. A000043) is to consider powers of odd numbers minus 2. Here only odd powers are considered.
PROG
(PARI) T = matrix( 19, 19, m, n, isprime((2*m+1)^(2*n-1)-2)) ;
A155899 = concat( vector( vecmin( matsize(T)), i, vector( i, j, T[j, i-j+1])))
CROSSREFS
Cf. A084714, A128472, A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.
Least number k > (2n-1) such that (2n-1)^k - 2 is prime, or 0 if no such number exists.
+10
1
0, 4, 14, 8, 11, 22420, 78, 17, 24, 20, 25, 24, 63, 30, 42, 69, 128, 50, 119, 204, 2816, 76, 52, 288, 64, 66, 184, 153, 67, 268, 78, 210, 438, 295, 96, 74, 136, 128, 2900, 1898, 130, 92, 381, 106, 18626, 97, 98, 1650, 747, 109, 214, 113, 312, 354, 1702, 560, 2798, 123, 171, 554, 11210, 834, 208, 990, 9271
FORMULA
A128472(n) = (2n-1)^a(n) - 2 for n > 1.
CROSSREFS
Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists). Cf. A084714 (smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists).
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