cp's OEIS Frontend

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the sequence A000079, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.

If the ABC conjecture is true this sequence is finite.

The associated numbers B for this case are A147638, the associated A are A147640.

The standard way to search for records in the ABC conjecture is to run with the C parameter through all the integers A000027. If this search space is diluted by admitting only powers of 2 as in A147639, the sequence of records changes. This sequence here lists the A such that the triples (A=a(n), B=A147638(n), C=A147639(n)) locate records for this search when C is restricted to powers of 2.

The minima are reached for m values given in A147803.

This is related to the abc conjecture.

All terms of this sequence are even, so one could also consider A147800/2 = 1, 3, 11, 21, 111, 183, 1023, 6981, 5313, 39963, 146631, ...

The product of distinct prime divisors of (2^n)*m*(2^n-m) is also called the radical of that number: rad((2^n)*m*(2^n-m)).

For numbers m see A143700.

If the ABC conjecture is true this sequence is finite.

For numbers A for this case see A147643.

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the form 23^x, see A009967, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.

For associated B for this case see A147641, for associated A see A147643.

Related to the abc conjecture. The minima are reached for m values given in A147802.

All terms of this sequence are even, so one could also consider A147801/2 = 1, 1, 5, 5, 11, 55, 139, 119, 527, 671, 5533, 3059, 9367, 53515, 278635, 81515, ...

The minima are given in A147800.

This is related to the abc conjecture: Since m is coprime to 5, it is also coprime to 5^n and thus to 5^n-m. Thus the squarefree kernel A007947(m*(5^n-m)*5^n) = 5*A007947(m*(5^n-m)).

Original title was: "Numbers c such that (c^2-1)c is square free and gcd(c-1,c,c+1)=1", but (c^2-1)c is never squarefree for odd c, and gcd(n,n+1) is always = 1. - M. F. Hasler, Nov 03 2013

These numbers c with distribution a+b=c such that a=(c-1)/2 (see A172186) and b=(c+1)/2 (see A179019) have minimal possible values with function L(a,b,c) = log(c)/log(N(a,b,c)) = log(c)/log((c^2-1)c/4).

This function is minimal orbital in hypothesis (a,b,c).

There are no numbers or distributions which have value L less than log(c)/log((c^2-1)*c/4).

Equivalently, odd squarefree numbers c such that (c^2 - 1)/4 is also squarefree. - Jon E. Schoenfield, Feb 13 2023

The asymptotic density of this sequence is Product_{p prime} (1 - 3/p^2) = A206256 = 0.125486980905... (Tsang, 1985). - Amiram Eldar, Feb 26 2024

The minima are reached for m values given in A147804.

This is related to the abc conjecture.

All terms of this sequence are even, so one could also consider A147799/2 = 3, 3, 15, 15, 141, 1131, 8517, 18003, 35535, ... So far these terms are also multiples of 3, but this might be a coincidence.