Search: a128346 -id:a128346
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A128352
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Numbers k such that (17^k - 5^k)/12 is prime.
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+10
19
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=17; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128353, A128354.
Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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A128353
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Numbers k such that (18^k - 5^k)/13 is prime.
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+10
19
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2, 3, 19, 23, 31, 37, 251, 283, 977, 28687, 32993
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=18; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128354.
Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A128354
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Numbers k such that (19^k - 5^k)/14 is prime.
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+10
19
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=19; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353.
Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A128349
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Numbers k such that (13^k - 5^k)/8 is prime.
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+10
18
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=13; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128350, A128351, A128352, A128353, A128354.
Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A128350
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Numbers k such that (14^k - 5^k)/9 is prime.
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+10
18
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=14; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n, 1, 200}]
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PROG
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CROSSREFS
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Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128351, A128352, A128353, A128354. Cf. A004061, A082182, A121877, A059802. Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
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STATUS
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approved
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A128351
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Numbers k such that (16^k - 5^k)/11 is prime.
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+10
18
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=16; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128352, A128353, A128354, A004061, A082182, A121877, A059802, A057171, A082387, A122853, A128335-A128342.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A128338
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Numbers k such that (8^k + 5^k)/13 is prime.
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+10
17
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=8; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128339, A128340, A128341, A128342, A128343. Cf. A004061, A082182, A121877, A059802. Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A128343
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Numbers k such that (14^k + 5^k)/19 is prime.
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+10
11
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OFFSET
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1,1
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COMMENTS
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All terms are primes.
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LINKS
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MATHEMATICA
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k=14; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n, 1, 100}]
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PROG
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CROSSREFS
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Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A247093
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Triangle read by rows: T(m,n) = smallest odd prime p such that (m^p-n^p)/(m-n) is prime (0<n<m), or 0 if no such p exists.
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+10
1
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3, 3, 3, 0, 0, 3, 3, 5, 13, 3, 3, 0, 0, 0, 5, 5, 3, 3, 5, 3, 3, 3, 0, 3, 0, 19, 0, 7, 0, 3, 0, 0, 3, 0, 3, 7, 19, 0, 3, 0, 0, 0, 31, 0, 3, 17, 5, 3, 3, 5, 3, 5, 7, 5, 3, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 5, 3, 7, 5, 5, 3, 7, 3, 3, 251, 3, 17, 3, 0, 5, 0, 151, 0, 0, 0, 59, 0, 5, 0, 3, 3, 5, 0, 1097, 0, 0, 3, 3, 0, 0, 7, 0, 17, 3
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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T(m,n) is 0 if and only if m and n are not coprime or A052409(m) and A052409(n) are not coprime. (The latter has some exceptions, like T(8,1) = 3. In fact, if p is a prime and does not equal to A052410(gcd(A052409(m),A052409(n))), then (m^p-n^p)/(m-n) is composite, so if it is not 0, then it is A052410(gcd(A052409(m),A052409(n))).) - Eric Chen, Nov 26 2014
a(i) = T(m,n) corresponds only to probable primes for (m,n) = {(15,4), (18,1), (19,18), (31,6), (37,22), (37,25), ...} (i={95, 137, 171, 441, 652, 655, ...}). With the exception of these six (m,n), all corresponding primes up to a(663) are definite primes. - Eric Chen, Nov 26 2014
a(n) is currently known up to n = 663, a(664) = T(37, 34) > 10000. - Eric Chen, Jun 01 2015
For n up to 1000, a(n) is currently unknown only for n = 664, 760, and 868. - Eric Chen, Jun 01 2015
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LINKS
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EXAMPLE
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Read by rows:
m\n 1 2 3 4 5 6 7 8 9 10 11
2 3
3 3 3
4 0 0 3
5 3 5 13 3
6 3 0 0 0 5
7 5 3 3 5 3 3
8 3 0 3 0 19 0 7
9 0 3 0 0 3 0 3 7
10 19 0 3 0 0 0 31 0 3
11 17 5 3 3 5 3 5 7 5 3
12 3 0 0 0 3 0 3 0 0 0 3
etc.
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MATHEMATICA
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t1[n_] := Floor[3/2 + Sqrt[2*n]]
m[n_] := Floor[(-1 + Sqrt[8*n-7])/2]
t2[n_] := n-m[n]*(m[n]+1)/2
b[n_] := GCD @@ Last /@ FactorInteger[n]
is[m_, n_] := GCD[m, n] == 1 && GCD[b[m], b[n]] == 1
Do[k=2, If[is[t1[n], t2[n]], While[ !PrimeQ[t1[n]^Prime[k] - t2[n]^Prime[k]], k++]; Print[Prime[k]], Print[0]], {n, 1, 663}] (* Eric Chen, Jun 01 2015 *)
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PROG
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(PARI) a052409(n) = my(k=ispower(n)); if(k, k, n>1);
a(m, n) = {if (gcd(m, n) != 1, return (0)); if (gcd(a052409(m), a052409(n)) != 1, return (0)); forprime(p=3, , if (isprime((m^p-n^p)/(m-n)), return (p)); ); }
tabl(nn) = {for (m=2, nn, for(n=1, m-1, print1(a(m, n), ", "); ); print(); ); } \\ Michel Marcus, Nov 19 2014
(PARI) t1(n)=floor(3/2+sqrt(2*n))
t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2)
b(n)=my(k=ispower(n)); if(k, k, n>1)
a(n)=if(gcd(t1(n), t2(n)) !=1 || gcd(b(t1(n)), b(t2(n))) !=1, 0, forprime(p=3, 2^24, if(ispseudoprime((t1(n)^p-t2(n)^p)/(t1(n)-t2(n))), return(p)))) \\ Eric Chen, Jun 01 2015
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CROSSREFS
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Cf. A000043 (2,1), A028491 (3,1), A057468 (3,2), A059801 (4,3), A004061 (5,1), A082182 (5,2), A121877 (5,3), A059802 (5,4), A004062 (6,1), A062572 (6,5), A004063 (7,1), A215487 (7,2), A128024 (7,3), A213073 (7,4), A128344 (7,5), A062573 (7,6), A128025 (8,3), A128345 (8,5), A062574 (8,7), A173718 (9,2), A128346 (9,5), A059803 (9,8), A004023 (10,1), A128026 (10,3), A062576 (10,9), A005808 (11,1), A210506 (11,2), A128027 (11,3), A216181 (11,4), A128347 (11,5), A062577 (11,10), A004064 (12,1), A128348 (12,5), A062578 (12,11).
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KEYWORD
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AUTHOR
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STATUS
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approved
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A273010
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Numbers n such that (9^n - 7^n)/2 is prime.
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+10
0
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OFFSET
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1,1
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COMMENTS
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All terms are prime.
The corresponding primes: 193, 21121, 1979713, ...
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LINKS
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MATHEMATICA
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Select[Range[1, 10000], PrimeQ[(9^# - 7^#)/2] &]
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PROG
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(PARI) for(n=1, 10000, if(isprime((9^n - 7^n)/2), print1(n, ", ")))
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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