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 A247093 Triangle read by rows: T(m,n) = smallest odd prime p such that (m^p-n^p)/(m-n) is prime (0
 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)
 OFFSET 1,1 COMMENTS 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 LINKS Eric Chen, Table of n, a(n) for n = 1..663 Eric Chen, Table of n, a(n) for n = 1..1000 status EXAMPLE 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. MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A128164 (n,1), A125713 (n+1,n), A125954 (2n+1,2), A122478 (2n+1,2n-1). 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). Sequence in context: A289893 A265466 A002073 * A329273 A130719 A143606 Adjacent sequences: A247090 A247091 A247092 * A247094 A247095 A247096 KEYWORD nonn,tabl AUTHOR Eric Chen, Nov 18 2014 STATUS approved

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Last modified September 7 23:15 EDT 2024. Contains 375749 sequences. (Running on oeis4.)