A060626 -id:A060626 - cp's OEIS frontend

A175101 The number of bases b for which the odd squarefree semiprime A046388(n) is a Fermat pseudoprime.

2, 2, 2, 2, 2, 2, 2, 2, 14, 2, 2, 14, 2, 34, 2, 2, 2, 2, 2, 2, 2, 34, 2, 2, 14, 2, 2, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 34, 2, 14, 2, 2, 34, 2, 2, 34, 14, 2, 2, 2, 2, 2, 34, 2, 14, 2, 2, 2, 2, 2, 2, 2, 2, 98, 2, 14, 2, 14, 2, 2, 2, 2, 34, 2, 2, 2, 2, 2, 34, 2, 14, 2, 98, 2, 34, 2, 2, 142, 14, 2, 14, 2

Offset: 1

T. D. Noe, Dec 02 2010

nonn

A number x is a Fermat pseudoprime for base b if b^(x-1) = 1 (mod x).
Comment from Karsten Meyer: (Start) Each term pq of the sequence A046388 is at least a Fermat pseudoprime to the two bases which have the property that |l*p - m*q| = 2 and b is the number between l*p and m*q. There are no more bases of this form below pq.
There may exist other bases smaller than pq, but just two bases have the property that they are direct neighbors of a multiple of p and a multiple of q. For example, 39=3*13 is a Fermat pseudoprime to the bases 14 and 25 because 14 is the number between 13 and 3*5 and 25 is the number between 3*8 and 2*13.
91=7*13 is a Fermat pseudoprime to the bases 27 and 64 because 27 is the number between 2*13 and 4*7 and 64 is the number between 9*7 and 5*13. For 91, the bases 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88 also exist, but neither of them lies between a multiple of 7 and a multiple of 13. (End)
Looking at odd squarefree semiprimes less than 10000, it appears that the number of bases is always of the form 2(2k^2-1), which is A060626 and twice A056220. Using the formula in A063994, the number of bases for pq (including bases 1 and pq-1) is gcd(p-1,pq-1) * gcd(q-1,pq-1).

			For A046388(1) = 15, the bases b in the range [2,13] are 4 and 11. So a(1) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046388, A063994 (number of bases b for which b^(n-1) = 1 (mod n)).

a(n) = A063994(A046388(n)) - 2.

A349766 Numbers of the form 2*t^2-4 when t > 1 is a term in A001541.

Original entry on oeis.org

14, 574, 19598, 665854, 22619534, 768398398, 26102926094, 886731088894, 30122754096398, 1023286908188734, 34761632124320654, 1180872205318713598, 40114893348711941774, 1362725501650887306814, 46292552162781456489998, 1572584048032918633353214, 53421565080956452077519374

Offset: 1

Bernard Schott, Dec 04 2021

nonn

Equivalently: integers k such that k$ / (k/2+1)! and k$ / (k/2+2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information).

The 3 subsequences of A349081 are A035008, A139098 and this one.

			A001541(1) = 3, then for t = 3, 2*t^2-4 = 14; also for k = 14, 14$ / 8! = 1309248519599593818685440000000^2 and 14$ / 9! = 436416173199864606228480000000^2. Hence, 14 is a term.

Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (35, -35, 1).

Cf. A000178, A035008, A139098, A348692, A349079, A349080, A349081.

Subsequence of A060626.

with(orthopoly):
sequence = (2*T(n,3)^2-4, n=1..20);

(2*#^2 - 4) & /@ LinearRecurrence[{6, -1}, {3, 17}, 17] (* Amiram Eldar, Dec 04 2021 *)
LinearRecurrence[{35, -35, 1},{14, 574, 19598},17] (* Ray Chandler, Mar 01 2024 *)

a(n) = my(t=subst(polchebyshev(n), 'x, 3)); 2*t^2-4; \\ Michel Marcus, Dec 04 2021

a(n) = 2*(cosh(2*n*arcsinh(1)))^2 - 4.

a(n) = 16*A001110(n) - 2. - Hugo Pfoertner, Dec 04 2021

A292116 Numbers for which there exists a nontrivial bisection of binomial coefficients as given by Theorem 12 of Ionascu et al. (2016).

Original entry on oeis.org

13, 14, 33, 34, 61, 62, 97, 98, 103, 141, 142, 193, 194, 253, 254, 321, 322, 397, 398, 481, 482, 573, 574, 673, 674, 713, 781, 782, 897, 898, 1021, 1022, 1153, 1154, 1293, 1294, 1441, 1442, 1597, 1598, 1761, 1762, 1933, 1934, 2113, 2114, 2301, 2302, 2497, 2498, 2701, 2702, 2913, 2914, 3133, 3134

Offset: 1

N. J. A. Sloane, Sep 10 2017

nonn

It would be nice to have a more precise definition.
From Ray Chandler, Sep 11 2017: (Start)
The sequence is the union of three types of numbers:
(1) A060626 beginning with the 2nd term.
(2) A089508 beginning with the 3rd term and omitting even values (every third term).
(3) A082109 beginning with the 2nd term.
Note that there appear to be other solutions that are not covered by Theorem 12.
(End)

Ray Chandler, Table of n, a(n) for n = 1..10000
Eugen J. Ionascu, Thor Martinsen, and Pantelimon Stanica, Bisecting binomial coefficients, arXiv:1610.02063 [math.CO], 2016. See p. 18.

Cf. A060626, A082109, A089508.

lim=3000; a0={};
k=4; While[c=k^2-3;c<=lim,a0=Join[a0,{c,c+1}];k+=2];
k=2; While[c=Fibonacci[2k]*Fibonacci[2k+1]-1;c<=lim,If[OddQ[c],AppendTo[a0,c]];k++];
a0=Sort[a0] (* Ray Chandler, Sep 11 2017 *)

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A175101 The number of bases b for which the odd squarefree semiprime A046388(n) is a Fermat pseudoprime.

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A349766 Numbers of the form 2*t^2-4 when t > 1 is a term in A001541.

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A292116 Numbers for which there exists a nontrivial bisection of binomial coefficients as given by Theorem 12 of Ionascu et al. (2016).

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