A373469 Least odd k such that C(2k, k) == 1 (mod A007775(n)), or 0 if no such k exists.

1, 17, 13, 2383, 37, 3, 3391, 185, 129, 419, 95, 139, 7, 7373, 497, 21, 89, 27, 319, 7, 23, 191, 277, 25, 33635, 137, 1957, 347, 879, 889, 47, 57, 411, 263, 63, 57, 63, 143, 62561, 363, 1679, 861, 285735, 1017, 545, 2605, 913, 1873, 735, 206349, 817, 407, 485, 49, 7605, 179817

Offset: 1

Author

M. F. Hasler, Jul 12 2024

Keywords

• nonn
• hard

Comments

A007775 lists the odd numbers not divisible by 3 or 5. It seemed that these are exactly the odd numbers not in A086748 (= odd m such that C(2k,k) == 1 (mod m) has no odd solution k), i.e., the numbers in A086748 would exactly be the odd multiples of 3 and 5, but so far there was no proof or disproof for that. The present sequence gives an explicit proof, if it exists, for each x in A007775, that x is not in A086748.

It is highly possible that a(n) = 0 for n with m = A007775(n) divisible by three or more distinct primes, in which case values of k such that C(2k,k) coprime to m, let alone C(2k,k) == 1 (mod m), are very sparse and possibly finite. See A030979 for a similar problem. - Max Alekseyev, Jul 14 2024

Examples for moduli that have 3 distinct prime factors >5: a(603) = 57 associated with modulus A007775(603) = 2261 = 7*17*19. a(4333) = 23 associated with modulus A007775(4333) = 16247 = 7*11*211. a(6621) = 1709 associated with 11*37*61. a(6797)=19999 assocated with 7*11*331. - R. J. Mathar, Aug 09 2024

Links

• M. F. Hasler and Max Alekseyev, Table of n, a(n) for n = 1..266

Crossrefs

Cf. A007775 (odd numbers not divisible by 3 or 5), A086748 (odd m such that C(2k,k)==1 (mod m) has no odd solution k).

Cf. A030979.

Programs

PARI

/* helper function: compute C(n,k) mod prime p */
LucasT(n,k,p)={if(n>=k, my(kp = digits(k,p), np = digits(n,p)[-#kp..-1]); prod(i=1, #kp, binomial(np[i], kp[i]), Mod(1,p)))}
is1(k,f)={for(i=1,matsize(f)[1], LucasT(2*k, k, f[i,1])==1||return); vecmax(f[,2])==1 || binomial(2*k,k)%factorback(f)==1}
apply( {A373469(n, m=A007775(n), f=factor(m))=!f || forstep(k=3, oo, 2, is1(k,f) && return(k))}, [1..50])

Extensions

a(43)-a(56) from Max Alekseyev, Jul 12 2024