A386703 The residue of p(n) modulo q(n) in the interval (-q(n)/2, q(n)/2], where p(n) = A000041(n) and q(n) = A000009(n).

0, 0, 1, 1, 1, -1, 0, -2, -2, 2, -4, 2, -7, 3, -13, 7, -7, 17, 4, -13, 32, 23, 7, -11, -30, -39, -62, -56, -43, -20, 42, 159, -161, 22, 258, -59, 357, 95, -239, -504, 483, 412, 471, 719, -978, -426, 434, -1137, 533, -622, -1780, 2087, 2251, -2669, -1562, 831, -3372, 1772

Offset: 1

Author

Zhi-Wei Sun, Jul 30 2025

Keywords

• sign

Comments

Conjecture: |a(n)| > 1 for all n > 7.

This has been verified for all n = 8..10^5.

Verified for all n <= 2000000. - Vaclav Kotesovec, Jul 30 2025

Examples

			a(6) = -1  since p(6) = 11 is congruent to -1 modulo q(6) = 4.
a(7) = 0 since p(7) = 15 is congruent to 0 modulo q(7) = 5.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, A conjecture involving the partition function and the strict partition function, Question 498447 at MathOverflow, July 30, 2025.

Crossrefs

Cf. A000009, A000041.

Programs

Mathematica

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,(1-n)/2];
a[n_]=rMod[PartitionsP[n],PartitionsQ[n]];Table[a[n],{n,1,70}]