A387291 Integers x such that there exist two numbers y,z with x <= y <= z such that psi(x) = psi(y) = psi(z) = (x + y + z)/2.

2, 4, 6, 8, 16, 18, 28, 32, 44, 54, 64, 70, 105, 110, 128, 150, 162, 165, 182, 200, 238, 240, 256, 280, 310, 315, 364, 382, 468, 486, 512, 520, 585, 590, 644, 735, 750, 780, 790, 795, 800, 1000, 1024, 1034, 1162, 1246, 1260, 1274, 1410, 1434, 1456, 1458, 1472, 1540, 1575

Offset: 1

Author

S. I. Dimitrov, Aug 25 2025

Keywords

• nonn
• new

Comments

The numbers x, y and z form a psi-amicable triple.

Examples

			2 is in the sequence since psi(2) = 3 = (2 + 2 + 2)/2.
28 is in the sequence since psi(28) = psi(33) = psi(35) = 48 = (28 + 33 + 35)/2.
		

Links

• S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Crossrefs

Cf. A001615, A385852, A386726, A386901, A386933, A387292, A387486.

Programs

PARI

mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(k) = my(vk=invpsi(mypsi(k))); vk = select(x->x>=k, vk); if (#vk, my(x = 1); for (y=x, #vk, for (z=y, #vk, if (apsi(k) == (vk[x]+vk[y]+vk[z])/2, return(1););););); \\ Michel Marcus, Sep 04 2025