A086748 -id:A086748 - cp's OEIS frontend

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

M. F. Hasler, Jul 12 2024

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

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

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.

/* 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])

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

A374221 Indices of 1's in A374200, where A374200 is the minimum value of A347381 that it obtains among unitary divisors of n larger than 1.

Original entry on oeis.org

3, 4, 5, 12, 15, 20, 21, 24, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 57, 60, 65, 68, 69, 75, 76, 80, 85, 87, 92, 93, 95, 96, 100, 105, 108, 111, 115, 116, 120, 123, 124, 129, 132, 135, 141, 145, 147, 148, 155, 156, 159, 160, 164, 165, 168, 172, 177, 180, 183, 185, 188, 189, 192, 195, 196, 201, 204, 205, 212, 213

Offset: 1

Antti Karttunen, Jul 09 2024

nonn

It seems that for all odd terms x in this sequence, A374204(x) = 1, and they form a subsequence of all terms k for which A374204(k) = 1, that might be sequence A086748.

Antti Karttunen, Table of n, a(n) for n = 1..20001

Cf. A000203, A005940, A347381, A374200, A374204, A374220 (characteristic function).

Cf. also A086748 and A347391 (very likely a subsequence).

PARI
```
isA374221 = A374220;
```

cp's OEIS Frontend

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

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