A385325 -id:A385325 - cp's OEIS frontend

A385356 Numbers x such that there exist two integers 00 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.

2, 40, 164, 196, 224, 1120, 3040, 13440, 22932, 44200, 76160, 90848, 91720, 174592, 530200, 619840, 687184, 872960, 1686400, 1767040, 1807120, 1927680, 1990912, 2154880, 3653760, 4286880, 5637632, 5759680, 6442128, 8225280, 8943800, 9264320, 9465600, 9694080

Offset: 1

S. I. Dimitrov, Jun 26 2025

nonn

The numbers x, y and z form a sigma-quadratic triple. See Dimitrov link.

			(40, 58, 56) is such a triple because sigma(40)^2 = sigma(58)^2 = 90^2 = 40^2 + 58^2 + 56^2.

Chai Wah Wu, Table of n, a(n) for n = 1..161
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A066784, A096907, A096908, A096909, A096910, A385325.

from itertools import count, islice
from sympy import divisor_sigma
from sympy.ntheory.primetest import is_square
def A385356_gen(startvalue=1): # generator of terms >= startvalue
    for x in count(max(startvalue,1)):
        sx, x2 = int(divisor_sigma(x)), x**2
        sx2 = sx**2
        if sx2>x2:
            for y in count(x):
                if (k:=sx2-x2-y**2)<=0:
                    break
                if is_square(k) and sx==divisor_sigma(y):
                    yield x
                    break
A385356_list = list(islice(A385356_gen(),8)) # Chai Wah Wu, Jul 02 2025

Data corrected by David A. Corneth, Jun 27 2025

a(18)-a(34) from Chai Wah Wu, Jul 02 2025

A385397 Numbers x such that there exist three integers 00 and w>0 such that sigma(x)^3 = sigma(y)^3 = x^3 + y^3 + z^3 + w^3.

Original entry on oeis.org

153, 216, 255, 324, 672, 735, 1074, 1170, 1218, 2430, 2655, 2736, 3482, 4148, 4605, 4935, 5220, 5446, 5916, 6048, 7140, 9340, 11000, 11160, 12768, 14090, 14098, 14980, 17220, 17696, 18984, 21068, 21948, 22128, 23022, 23205, 24297, 24570, 25284, 25740, 29058, 29640, 30240, 30690, 31008, 31190, 32760, 37140, 39840

Offset: 1

S. I. Dimitrov, Jun 27 2025

The numbers x, y, z and w form a sigma-cubic quadruple. See Dimitrov link.

			(255, 321, 84, 312) is such a quadruple because sigma(255)^3 = sigma(321)^3 = 432^3 = 255^3 + 321^3 + 84^3 + 312^3.

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A385325, A385356.

issc(n) = if (n>0, for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))); \\ A003325
isok(x) = my(s=sigma(x)); for (y=1, x, if (s == sigma(y), if (issc(s^3-x^3-y^3), return(1)););); \\ Michel Marcus, Jun 27 2025

More terms from David A. Corneth, Jun 27 2025

A386378 Integers x such that there exist four integers 00 and w>0 such that sigma(x)^3 = sigma(y)^3 = sigma(z)^3 = x^3 + y^3 + z^3 + t^3 + w^3.

Original entry on oeis.org

30, 62, 90, 174, 238, 357, 390, 440, 495, 552, 762, 870, 894, 924, 1056, 1146, 1248, 1386, 1560, 1740, 1770, 1782, 1824, 1880, 1938, 1992, 2046, 2208, 2262, 2472, 2568, 2625, 2670, 2686, 2730, 2840, 2856, 3000, 3190, 3382, 3630, 3666, 3720, 3738, 3828, 3885, 3960, 3984

Offset: 1

S. I. Dimitrov, Jul 20 2025

nonn

The numbers x, y, z, t and w form a sigma-cubic quintuple. See Dimitrov link.

			(174, 190, 323, 5, 94) is such a quintuple because sigma(174)^3 = sigma(190)^3 = sigma(323)^3 = 360^3 = 174^3 + 190^3 + 323^3 + 5^3 + 94^3.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A003325, A003328, A385325, A385397.

is23(n) = my(z); for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3, &z) && return([k,z]));
isok3(x) = my(s=sigma(x), v=select(z->(z>=x), invsigma(s))); if (#v >= 1, for (i=1, #v, for (j=1, #v, my(k=s^3 - x^3 - v[i]^3-v[j]^3); if (k>0, my(tw = is23(k)); if (tw, return([x, v[i], v[j], tw[1], tw[2]])););););); \\ Michel Marcus, Jul 22 2025

Corrected and extended by Michel Marcus, Jul 22 2025

cp's OEIS Frontend

A385356 Numbers x such that there exist two integers 00 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.

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A385397 Numbers x such that there exist three integers 00 and w>0 such that sigma(x)^3 = sigma(y)^3 = x^3 + y^3 + z^3 + w^3.

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A386378 Integers x such that there exist four integers 00 and w>0 such that sigma(x)^3 = sigma(y)^3 = sigma(z)^3 = x^3 + y^3 + z^3 + t^3 + w^3.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions