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A167860
Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).
4
7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, 4127, 5087, 5431, 6911, 8887, 9127, 9791, 9887, 12391, 13151, 14407, 15551, 16607, 19543, 20399, 21031, 21319, 21839, 23039, 25391, 26399, 28087, 28463, 28711, 29287, 33223, 39551, 43103, 44879, 46271
OFFSET
1,1
COMMENTS
Apparently A167860 is a subset of primes of the form 8*k + 7 (A007522).
Every A167859(m) from m=(p-1)/2 to m=(p-1) is divisible by prime p belonging to A167860.
7^3 divides A167859(13) and 7^2 divides A167859(10)-A167859(13).
Every A167859(m) from m=(kp-1 - (p-1)/2) to m=(kp-1) is divisible by prime p from A167860.
Every A167859(m) from m=((p^2-1)/2) to m=(p^2-1) is divisible by prime p from A167860. For p=7 every A167859(m) from m=((p^3-1)/2) to m=(p^3-1) and from m=((p^4-1)/2) to m(p^4-1)is divisible by p^2.
LINKS
MAPLE
A167859 := proc(n)
option remember;
if n <= 1 then
add( (binomial(2*k, k)/2^k)^2, k=0..n) ;
4^n*% ;
else
4*(5*n^2 - 4*n + 1)*procname(n-1) - 16*(2*n - 1)^2*procname(n-2) ;
%/n^2 ;
end if;
end proc:
isA167860 := proc(p)
local m ;
for m from (p-1)/2 to p-1 do
if modp(A167859(m), p) > 0 then
return false;
end if;
end do:
true ;
end proc:
A167860 := proc(n)
option remember ;
if n = 0 then
2;
else
p := nextprime(procname(n-1)) ;
while not isA167860(p) do
p := nextprime(p) ;
end do ;
return p;
end if;
end proc:
seq(A167860(n), n=1..10) ; # R. J. Mathar, Jan 22 2025
PROG
(PARI) is(p) = if(isprime(p)&&p%2, my(m=Mod(1, p), s=m); for(k=1, p\2, s+=(m*=(2*k-1)/k)^2); !s, 0); \\ Jinyuan Wang, Jul 24 2022
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Nov 13 2009
EXTENSIONS
More terms from Jinyuan Wang, Jul 24 2022
STATUS
approved