A248044 Least positive integer m such that m + n divides pi(m)^2 + pi(n)^2, where pi(x) denotes the number of primes not exceeding x.

1, 3, 1, 4, 12, 11, 1, 8, 7, 16, 2, 5, 26, 25, 24, 4, 228, 227, 46, 45, 44, 43, 16, 6, 5, 1, 27, 26, 45, 44, 12526, 12525, 12524, 12523, 2970, 502, 351, 350, 46, 45, 236, 235, 10, 9, 8, 4, 1078, 1077, 576, 575, 574, 198, 63, 62, 61, 176, 16, 10, 362, 70

Offset

1,2

Comments

Conjecture: a(n) exists for any n > 0.

a(n) has been computed for n = 1..100000, with 99998 values resolved within the search bound n <= 10^13. The two unresolved cases, n = 19623 and n = 19624, satisfy a(n) > 10^13, giving a(n)/n > 5.096*10^8, the largest lower bounds in the dataset. Both belong to a cluster sharing pi(n) = 2225 = 5^2 * 89, whose prime factorization consists entirely of primes == 1 (mod 4), maximizing the modular obstruction. - Carlo Corti, Mar 23 2026

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1387 from Zhi-Wei Sun)

Carlo Corti, Table of n, a(n) for n = 1..100000

Carlo Corti, Source code, data and verification scripts.

Carlo Corti, Zenodo archive.

Carlo Corti, Logarithmic scatterplot of A248044 (n=1..100000).

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685 [math.NT], 2014-2017.

Example

a(5) = 12 since 12 + 5 = 17 divides pi(12)^2 + pi(5)^2 = 5^2 + 3^2 = 34.

Mathematica

Do[m=1; Label[aa]; If[Mod[PrimePi[m]^2+PrimePi[n]^2, m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000720, A247824, A247975, A248035, A248036.

Sequence in context: A065253 A010756 A191857 * A153278 A010284 A095328

Adjacent sequences: A248041 A248042 A248043 * A248045 A248046 A248047

Keyword

nonn,look

Author

Zhi-Wei Sun, Sep 30 2014

Status

approved