A386726 -id:A386726 - cp's OEIS frontend

A386727 Numbers x such that there exist three integers 0

3, 10, 24, 51, 78, 105, 114, 136, 186, 220, 224, 255, 322, 348, 357, 370, 435, 478, 506, 616, 642, 710, 748, 820, 861, 885, 957, 996, 1004, 1068, 1113, 1214, 1221, 1276, 1292, 1336, 1390, 1485, 1491, 1562, 1564, 1581, 1605, 1660, 1670, 1704, 1716, 1724, 1815, 1869, 1880, 1912, 1947

Offset: 1

S. I. Dimitrov, Jul 31 2025

nonn

The numbers x, y, z and t form an amicable quadruple according to Yanney’s definition.

			114 is in the sequence since sigma(114) = sigma(158) = sigma(209) = sigma(239) = 240 = (114 + 158 + 209 + 239)/3.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Benjamin Franklin Yanney, Another definition of amicable numbers and some of their relations to Dickson's amicables, Amer. Math. Monthly, Vol. 30, No. 6 (1923), 311-315.

Cf. A000203, A036471, A386726.

isok(x1) = my(s=sigma(x1), vx=select(x->(x>=x1), invsigma(s)), v=vector(4, i, vx[1])); for (i=1, #vx, v[2] = vx[i]; for (j=1, #vx, v[3] = vx[j]; for (k=1, #vx, v[4] = vx[k]; if (vecsum(v) == 3*s, return(1));););); \\ Michel Marcus, Aug 01 2025

More terms from Michel Marcus, Aug 01 2025

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.

Original entry on oeis.org

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

S. I. Dimitrov, Aug 25 2025

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

			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.

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

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

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

cp's OEIS Frontend

A386727 Numbers x such that there exist three integers 0

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