A377587 a(n) is the smallest odd integer m with m-2^k not squarefree for all 1<=k<=n

11, 29, 533, 849, 434977, 10329791, 28819433, 129747557, 6915752957, 2569472629649, 23373845739407, 60690478781437

Offset

1,1

Comments

After he had proved that infinitly many positive odd integers cannot be expressed as the sum of a prime and a power of two, Paul Erdős conjectured that every odd integer 1<m can be expressed as the sum of a squarefree number s and a power of two (with positive integer exponent): m=s+2^k. This sequence gives the smallest odd integers 1<m with m-2^k not being squarefree for all 1<=k<=n. Thus, if Erdős' Conjecture is true the sequence is monotonically increasing; and if the conjecture is false it becomes stationary at the first m without a representation as m=s+2^k.

There are no other terms < 2^50.

References

Paul Erdős, Problems in number theory, New Zealand J. Math. (1997), 155-160.

Links

Table of n, a(n) for n=1..12.

T. Bloom, Is every odd n the sum of a squarefree number and a power of 2?, Erdős Problems.

Paul Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math., Vol. 2 (1950), p. 113-125.

Andrew Granville and K. Soundararajan, A binary additive problem of Erdős and the order of 2 mod p^2, Raman. J., Vol. 2 (1998) pp. 283-298.

Christian Hercher, On the Sum of Squarefree Integers and a Power of Two, arXiv:2411.01964 [math.NT], 2024.

Example

a(3)=533 since 533-2^1=3^2*59, 533-2^2=23^2, 533-2^3=21*5^2, and there is no smaller odd integer 1<m<533 with m-2^1, m-2^2, and m-2^3 not being squarefree.

Crossrefs

Cf. A005117, A013929.

Sequence in context: A027758 A351353 A285992 * A355696 A271348 A057739

Adjacent sequences: A377584 A377585 A377586 * A377588 A377589 A377590

Keyword

nonn,more

Author

Christian Hercher, Nov 02 2024

Status

approved