A004810 -id:A004810 - cp's OEIS frontend

A003384 Numbers that are the sum of 6 nonzero 8th powers.

6, 261, 516, 771, 1026, 1281, 1536, 6566, 6821, 7076, 7331, 7586, 7841, 13126, 13381, 13636, 13891, 14146, 19686, 19941, 20196, 20451, 26246, 26501, 26756, 32806, 33061, 39366, 65541, 65796, 66051, 66306, 66561, 66816, 72101, 72356, 72611, 72866, 73121

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
23063715 is in the sequence as 23063715 = 1^8 + 4^8 + 4^8 + 5^8 + 7^8 + 8^8.
93544421 is in the sequence as 93544421 = 1^8 + 3^8 + 6^8 + 7^8 + 9^8 + 9^8.
120267520 is in the sequence as 120267520 = 4^8 + 4^8 + 6^8 + 6^8 + 8^8 + 10^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A133093.
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Row sums of A133093(n+1). - Gary W. Adamson, Sep 09 2007

Removed incorrect program. - David A. Corneth, Aug 01 2020

A003385 Numbers that are the sum of 7 nonzero 8th powers.

Original entry on oeis.org

7, 262, 517, 772, 1027, 1282, 1537, 1792, 6567, 6822, 7077, 7332, 7587, 7842, 8097, 13127, 13382, 13637, 13892, 14147, 14402, 19687, 19942, 20197, 20452, 20707, 26247, 26502, 26757, 27012, 32807, 33062, 33317, 39367, 39622, 45927, 65542, 65797, 66052

Offset: 1

N. J. A. Sloane

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 6751 terms from R. J. Mathar, replacing an earlier file that missed terms)

Cf. A001016 (8th powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

A003385 := proc(nmax::integer)
    local a, x,x8,y,y8,z,z8,u,u8,v,v8,w,w8,t,t8 ;
    a := {} ;
    for x from 1 do
        x8 := x^8 ;
        if 7*x8 > nmax then
            break;
        end if;
        for y from x do
            y8 := y^8 ;
            if x8+6*y8 > nmax then
                break;
            end if;
            for z from y do
                z8 := z^8 ;
                if x8+y8+5*z8 > nmax then
                    break;
                end if;
                for u from z do
                    u8 := u^8 ;
                    if x8+y8+z8+4*u8 > nmax then
                        break;
                    end if;
                    for v from u do
                        v8 := v^8 ;
                        if x8+y8+z8+u8+3*v8 > nmax then
                            break;
                        end if;
                        for w from v do
                            w8 := w^8 ;
                            if x8+y8+z8+u8+v8+2*w8 > nmax then
                                break;
                            end if;
                            for t from w do
                                t8 := t^8 ;
                                if x8+y8+z8+u8+v8+w8+t8 > nmax then
                                    break;
                                end if;
                                if x8+y8+z8+u8+v8+w8+t8 <= nmax then
                                    a := a  union {x8+y8+z8+u8+v8+w8+t8} ;
                                end if;
                            end do:
                        end do:
                    end do:
                end do:
            end do:
        end do:
    end do:
    sort(convert(a,list)) ;
end proc:
nmax := 117440512 ;
L:= A003385(nmax) ;
LISTTOBFILE(L,"b003385.txt",1) ; # R. J. Mathar, Aug 01 2020

M = 217168099;
m = M^(1/8) // Ceiling;
Table[s = a^8+b^8+c^8+d^8+e^8+f^8+g^8; If[s>M, Nothing, s], {a, m}, {b, m}, {c, m}, {d, m}, {e, m}, {f, m}, {g, m}] // Flatten // Union (* Jean-François Alcover, Dec 01 2020 *)

\\ also works for nmax=117440512 producing 6751 terms
nmax=67000;v=vectorsmall(nmax);L=ceil(#v^(1/8));for(k1=1,L, for(k2=k1,L, for(k3=k2,L, for(k4=k3,L, for(k5=k4,L, for(k6=k5,L, for(k7=k6,L, my(s=k1^8+k2^8+k3^8+k4^8+k5^8+k6^8+k7^8); if(s<=#v,v[s]++))))))));for(k=1,#v,if(v[k],print1(k,", "))) \\ Hugo Pfoertner, Aug 01 2020

Incorrect program removed by David A. Corneth, Aug 04 2020

A003387 Numbers that are the sum of 9 nonzero 8th powers.

Original entry on oeis.org

9, 264, 519, 774, 1029, 1284, 1539, 1794, 2049, 2304, 6569, 6824, 7079, 7334, 7589, 7844, 8099, 8354, 8609, 13129, 13384, 13639, 13894, 14149, 14404, 14659, 14914, 19689, 19944, 20199, 20454, 20709, 20964, 21219, 26249, 26504, 26759, 27014, 27269

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
5820102 is in the sequence as 5820102 = 1^8 + 1^8 + 1^8 + 1^8 + 5^8 + 5^8 + 6^8 + 6^8 + 6^8.
9960580 is in the sequence as 9960580 = 5^8 + 5^8 + 5^8 + 5^8 + 6^8 + 6^8 + 6^8 + 6^8 + 6^8.
11260068 is in the sequence as 11260068 = 1^8 + 1^8 + 2^8 + 4^8 + 5^8 + 6^8 + 6^8 + 6^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3854 terms from R. J. Mathar, replacing an earlier file that was missing terms)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

A003387 := proc(nmax::integer)
    local a, x,x8,y,y8,z,z8,u,u8,v,v8,w,w8,t,t8,s,s8,r,r8 ;
    a := {} ;
    for x from 1 do
        x8 := x^8 ;
        if 9*x8 > nmax then
            break;
        end if;
        for y from x do
            y8 := y^8 ;
            if x8+8*y8 > nmax then
                break;
            end if;
            for z from y do
                z8 := z^8 ;
                if x8+y8+7*z8 > nmax then
                    break;
                end if;
                for u from z do
                    u8 := u^8 ;
                    if x8+y8+z8+6*u8 > nmax then
                        break;
                    end if;
                    for v from u do
                        v8 := v^8 ;
                        if x8+y8+z8+u8+5*v8 > nmax then
                            break;
                        end if;
                        for w from v do
                            w8 := w^8 ;
                            if x8+y8+z8+u8+v8+4*w8 > nmax then
                                break;
                            end if;
                            for t from w do
                                t8 := t^8 ;
                                if x8+y8+z8+u8+v8+w8+3*t8 > nmax then
                                    break;
                                end if;
                                for s from t do
                                    s8 := s^8 ;
                                    if x8+y8+z8+u8+v8+w8+t8+2*s8 > nmax then
                                        break;
                                    end if;
                                    for r from s do
                                        r8 := r^8 ;
                                        if x8+y8+z8+u8+v8+w8+t8+s8+r8 > nmax then
                                            break ;
                                        end if;
                                        if x8+y8+z8+u8+v8+w8+t8+s8+r8 <= nmax then
                                            a := a  union {x8+y8+z8+u8+v8+w8+t8+s8+r8} ;
                                        end if;
                                    end do:
                                end do:
                            end do:
                        end do:
                    end do:
                end do:
            end do:
        end do:
    end do:
    sort(convert(a,list)) ;
end proc:
nmax := 15116544 ;
L:= A003387(nmax) ;
LISTTOBFILE(L,"b003387.txt",1) ; # R. J. Mathar, Aug 01 2020

M = 45711012; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++, For[i = h, i <= m, i++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8 + i^8;
If[s <= M, Sow[s]]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

Incorrect program removed by David A. Corneth, Aug 01 2020

A003388 Sum of 10 nonzero 8th powers.

Original entry on oeis.org

10, 265, 520, 775, 1030, 1285, 1540, 1795, 2050, 2305, 2560, 6570, 6825, 7080, 7335, 7590, 7845, 8100, 8355, 8610, 8865, 13130, 13385, 13640, 13895, 14150, 14405, 14660, 14915, 15170, 19690, 19945, 20200, 20455, 20710, 20965, 21220, 21475, 26250, 26505

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
3431590 is in the sequence as 3431590 = 1^8 + 1^8 + 1^8 + 1^8 + 1^8 + 2^8 + 3^8 + 4^8 + 6^8 + 6^8.
6276517 is in the sequence as 6276517 = 1^8 + 1^8 + 2^8 + 4^8 + 5^8 + 5^8 + 5^8 + 6^8 + 6^8 + 6^8.
8045029 is in the sequence as 8045029 = 1^8 + 2^8 + 3^8 + 3^8 + 4^8 + 4^8 + 4^8 + 5^8 + 6^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

offset corrected by David A. Corneth, Aug 01 2020

A003393 Numbers that are the sum of 4 positive 9th powers.

Original entry on oeis.org

4, 515, 1026, 1537, 2048, 19686, 20197, 20708, 21219, 39368, 39879, 40390, 59050, 59561, 78732, 262147, 262658, 263169, 263680, 281829, 282340, 282851, 301511, 302022, 321193, 524290, 524801, 525312, 543972, 544483, 563654, 786433, 786944, 806115, 1048576, 1953128, 1953639

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
12964419872 is in the sequence as 12964419872 = 3^9 + 5^9 + 11^9 + 13^9.
59116436980 is in the sequence as 59116436980 = 5^9 + 6^9 + 14^9 + 15^9.
79254744137 is in the sequence as 79254744137 = 6^9 + 11^9 + 15^9 + 15^9. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Removed incorrect program. - David A. Corneth, Aug 01 2020

A004803 Numbers that are the sum of 3 nonzero 10th powers.

Original entry on oeis.org

3, 1026, 2049, 3072, 59051, 60074, 61097, 118099, 119122, 177147, 1048578, 1049601, 1050624, 1107626, 1108649, 1166674, 2097153, 2098176, 2156201, 3145728, 9765627, 9766650, 9767673, 9824675, 9825698, 9883723, 10814202, 10815225, 10873250

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
17258390288153 is in the sequence as 17258390288153 = 14^10 + 14^10 + 21^10.
42930989049225 is in the sequence as 42930989049225 = 19^10 + 20^10 + 22^10.
323760702520401 is in the sequence as 323760702520401 = 23^10 + 26^10 + 26^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

kmax = 9*10^15; (* max term *)
m = kmax^(1/10) // Ceiling;
Table[k = x^10 + y^10 + z^10; If[k <= kmax, k, Nothing], {x, 1, m}, {y, x, m}, {z, y, m}] // Flatten // Union (* Jean-François Alcover, Jul 19 2017, updated May 02 2023 *)

Removed incorrect program. - David A. Corneth, Aug 01 2020

A004804 Numbers that are the sum of 4 nonzero 10th powers.

Original entry on oeis.org

4, 1027, 2050, 3073, 4096, 59052, 60075, 61098, 62121, 118100, 119123, 120146, 177148, 178171, 236196, 1048579, 1049602, 1050625, 1051648, 1107627, 1108650, 1109673, 1166675, 1167698, 1225723, 2097154, 2098177, 2099200, 2156202, 2157225

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
65969099123 is in the sequence as 65969099123 = 7^10 + 7^10 + 9^10 + 12^10.
1099804917226 is in the sequence as 1099804917226 = 4^10 + 5^10 + 7^10 + 16^10.
1164925542026 is in the sequence as 1164925542026 = 5^10 + 9^10 + 12^10 + 16^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

k = 4; p = 10; amax = 3*10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j-1], bmax}, {j, 1, k}]]] // Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

Removed incorrect program. - David A. Corneth, Aug 01 2020

A004805 Numbers that are the sum of 5 positive 10th powers.

Original entry on oeis.org

5, 1028, 2051, 3074, 4097, 5120, 59053, 60076, 61099, 62122, 63145, 118101, 119124, 120147, 121170, 177149, 178172, 179195, 236197, 237220, 295245, 1048580, 1049603, 1050626, 1051649, 1052672, 1107628, 1108651, 1109674, 1110697, 1166676, 1167699, 1168722, 1225724, 1226747

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
10352707051 is in the sequence as 10352707051 = 1^10 + 5^10 + 6^10 + 7^10 + 10^10.
59130893253 is in the sequence as 59130893253 = 7^10 + 9^10 + 9^10 + 11^10 + 11^10.
69011865378 is in the sequence as 69011865378 = 6^10 + 6^10 + 9^10 + 9^10 + 12^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

k = 5; p = 10; amax = 2*10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j - 1], bmax}, {j, 1, k}]]] //Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

Removed incorrect program. - David A. Corneth, Aug 01 2020

A004814 Numbers that are the sum of 3 positive 11th powers.

Original entry on oeis.org

3, 2050, 4097, 6144, 177149, 179196, 181243, 354295, 356342, 531441, 4194306, 4196353, 4198400, 4371452, 4373499, 4548598, 8388609, 8390656, 8565755, 12582912, 48828127, 48830174, 48832221, 49005273, 49007320, 49182419, 53022430

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
204800049005272 is in the sequence as 204800049005272 = 3^11 + 5^11 + 20^11.
2518268235958260 is in the sequence as 2518268235958260 = 16^11 + 19^11 + 25^11.
3786934745885995 is in the sequence as 3786934745885995 = 10^11 + 19^11 + 26^11. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Removed incorrect program. - David A. Corneth, Aug 01 2020

A336725 A(n,k) is the n-th number that is a sum of k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 10, 8, 4, 5, 19, 17, 10, 5, 6, 36, 34, 24, 13, 6, 7, 69, 67, 49, 29, 17, 7, 8, 134, 132, 98, 64, 36, 18, 8, 9, 263, 261, 195, 129, 84, 43, 20, 9, 10, 520, 518, 388, 258, 160, 99, 55, 25, 10, 11, 1033, 1031, 773, 515, 321, 247, 114, 62, 26, 11, 12, 2058, 2056, 1542, 1028, 642, 384, 278, 129, 66, 29, 12

Offset: 1

Alois P. Heinz, Aug 01 2020

			Square array A(n,k) begins:
   1,  2,  3,   4,   5,   6,    7,    8,    9,   10, ...
   2,  5, 10,  19,  36,  69,  134,  263,  520, 1033, ...
   3,  8, 17,  34,  67, 132,  261,  518, 1031, 2056, ...
   4, 10, 24,  49,  98, 195,  388,  773, 1542, 3079, ...
   5, 13, 29,  64, 129, 258,  515, 1028, 2053, 4102, ...
   6, 17, 36,  84, 160, 321,  642, 1283, 2564, 5125, ...
   7, 18, 43,  99, 247, 384,  769, 1538, 3075, 6148, ...
   8, 20, 55, 114, 278, 734,  896, 1793, 3586, 7171, ...
   9, 25, 62, 129, 309, 797, 2193, 2048, 4097, 8194, ...
  10, 26, 66, 164, 340, 860, 2320, 6568, 4608, 9217, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-11 give: A000027, A000404, A003072, A003338, A003350, A003362, A003374, A003386, A003398, A004810, A004822.
Rows n=1-3 give: A000027, A052944, A145071.
Main diagonal gives A000337.
Cf. A336820.

A:= proc() local l, w, A; l, w, A:= proc() [] end, proc() [] end,
      proc(n, k) option remember; local b; b:=
        proc(x, y) option remember; `if`(x=0, {0}, `if`(y<1, {},
          {b(x, y-1)[], map(t-> t+l(k)[y], b(x-1, y))[]}))
        end;
        while nops(w(k)) < n do forget(b);
          l(k):= [l(k)[], (nops(l(k))+1)^k];
          w(k):= sort([select(h-> h

nmax = 12;
pow[n_, k_] := IntegerPartitions[n, {k}, Range[n^(1/k) // Ceiling]^k];
col[k_] := col[k] = Reap[Module[{j = k, n = 1, p}, While[n <= nmax, p = pow[j, k]; If[p =!= {}, Sow[j]; n++]; j++]]][[2, 1]];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 03 2020 *)

cp's OEIS Frontend

A003384 Numbers that are the sum of 6 nonzero 8th powers.

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A003385 Numbers that are the sum of 7 nonzero 8th powers.

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A003387 Numbers that are the sum of 9 nonzero 8th powers.

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A003388 Sum of 10 nonzero 8th powers.

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A003393 Numbers that are the sum of 4 positive 9th powers.

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A004803 Numbers that are the sum of 3 nonzero 10th powers.

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A004804 Numbers that are the sum of 4 nonzero 10th powers.

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A004805 Numbers that are the sum of 5 positive 10th powers.

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