A055394 -id:A055394 - cp's OEIS frontend

A266548 Least prime p such that n + p^3 = x^2 + y^3 for some positive integers x and y, or 0 if no such prime p exists.

71, 2, 2, 5, 2, 3767, 3, 7, 7, 2, 3, 23, 53, 13, 17, 13, 2, 3, 2, 7, 2, 23, 11, 2, 17, 2, 7, 5, 2, 2, 3, 19, 257, 8039, 13, 2, 2, 59, 3, 5, 17, 3, 2, 61, 2, 3, 3, 37, 313, 2, 631, 17, 5, 3, 17, 2, 17, 2, 7, 97, 2, 47, 3, 29, 2, 2, 31, 47, 2, 7, 19

Offset: 0

Zhi-Wei Sun, Dec 31 2015

nonn

Conjecture: (i) Any integer can be written as x^2 + y^3 - p^3, where x and y are positive integers, and p is a prime.
(ii) Each integer can be written as x^2 - y^3 + p^3, where x and y are positive integers, and p is a prime.
See also A266230 and A266277 for related conjectures.
Is every prime in this sequence? - David A. Corneth, Dec 30 2017

			a(0) = 71 since 0 + 71^3 = 588^2 + 23^3 with 71 prime.
a(3) = 5 since 3 + 5^3 = 8^2 + 4^3 with 5 prime.
a(5) = 3767 since 5 + 3767^3 = 214886^2 + 1938^3 with 3767 prime.
a(2966) = 68371 since 2966 + 68371^3 = 17867983^2 + 6992^3 with 68371 prime.
a(7880) = 51137 since 7880 + 51137^3 = 10176509^2 + 31128^3 with 51137 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000040, A000290, A000578, A055394, A266152, A266153, A266230, A266231, A266277, A266314, A266363, A266364, A266528.

p[n_]:=p[n]=Prime[n]
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[x=1;Label[bb];Do[If[SQ[n+p[x]^3-y^3],Print[n," ",p[x]];Goto[aa]],{y,1,(n+p[x]^3-1)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,70}]

isokp(p, n) = {my(s = n+p^3); for (k=1, sqrtnint(s, 3), if ((q=s-k^3) && issquare(q), return (1)););}
a(n) = {p = 2; while(!isokp(p, n), p = nextprime(p+1)); p;} \\ Michel Marcus, Jan 04 2016

a(n, {plim = 2}) = forprime(p = plim, oo, c = n + p^3; for(i = 1, sqrtnint(c, 3), if(issquare(c - i^3) && c - i^3 > 0, return(p))))
first(n, {plim = 100}) = {my(res = vector(n), l = List(), s, i, c);
for(u=1, sqrtint((n+plim^3)\1-1), for(v=1, sqrtnint((n+plim^3)\1-u^2, 3), listput(l, u^2+v^3))); l = Set(l); forprime(p = 2, plim, s = 1; while(l[s] < p^3 + 1, s++); for(i = s, #l, c = l[i] - p^3; if(c <= n, if(res[c] == 0, res[c] = p)
, next(2)))); for(i = 1, n, if(res[i] == 0, res[i] = a(i, plim + 1))); concat([71], res)} \\ David A. Corneth, Dec 30 2017

A066647 Squares of the form a^2 + b^3 with a, b > 0.

Original entry on oeis.org

9, 36, 100, 196, 225, 289, 441, 576, 784, 841, 1225, 1296, 1764, 1849, 2025, 2304, 3025, 3249, 3600, 3844, 3969, 4356, 5776, 6084, 6400, 6561, 8100, 8281, 9801, 11025, 12544, 13924, 14161, 14400, 15129, 16129, 16641, 17424, 18496, 19600, 19881, 20449, 21609, 23409

Offset: 1

Reinhard Zumkeller, Dec 17 2001

nonn

			29^2 = a(9) = 841 = 625 + 216 = 25^2 + 6^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A055394, A066648, A070745.

q[n_] := Length[Reduce[a^2 + b^3 == n && a > 0 && b > 0, {a, b}, Integers]] > 0; Select[Range[140]^2, q] (* Amiram Eldar, Mar 20 2025 *)

a(n) = A070745(n)^2. - Amiram Eldar, Mar 20 2025

A066648 Cubes of the form a^2 + b^3 with a, b > 0.

Original entry on oeis.org

512, 1000, 2744, 21952, 32768, 35937, 64000, 175616, 185193, 274625, 357911, 373248, 405224, 474552, 729000, 1157625, 1404928, 1481544, 2000376, 2097152, 2197000, 2299968, 2744000, 3241792, 3652264, 3723875, 4096000, 5451776, 7189057, 8000000, 10218313, 10360232

Offset: 1

Reinhard Zumkeller, Dec 17 2001

nonn

			8^3 = a(0) = 512 = 169 + 343 = 13^2 + 7^3;
10^3 = a(1) = 1000 = 784 + 216 = 28^2 + 6^3.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A055394, A066647, A228946.

q[n_] := Length[Reduce[a^2 + b^3 == n && a > 0 && b > 0, {a, b}, Integers]] > 0; Select[Range[220]^3, q] (* Amiram Eldar, Mar 20 2025 *)

a(n) = A228946(n)^3. - R. J. Mathar, Dec 03 2015

A078383 Sum of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 3, 7, 5, 17, 5, 15, 9, 31, 14, 5, 37, 43, 13, 7, 15, 22, 10, 18, 19, 5, 73, 21, 7, 43, 89, 20, 7, 101, 5, 113, 63, 12, 127, 2, 46, 69, 50, 34, 39, 10, 21, 30, 43, 24, 22, 34, 62, 42, 10, 9, 197, 22, 105, 15, 38, 18, 223, 8, 115, 31, 233, 241, 46, 12, 257, 20, 16, 58, 269

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008472, A055394, A078377, A078382, A078384.

Total[FactorInteger[#][[;; , 1]]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A008472(A055394(n)).

A078384 Sum of all prime factors of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 6, 7, 7, 17, 9, 15, 11, 31, 14, 10, 37, 43, 15, 12, 17, 22, 13, 18, 21, 12, 73, 23, 13, 43, 89, 20, 14, 101, 13, 113, 63, 15, 127, 14, 46, 69, 50, 34, 41, 15, 25, 30, 45, 24, 25, 34, 62, 42, 16, 18, 197, 24, 105, 21, 38, 20, 223, 16, 115, 35, 233, 241, 46, 17, 257, 22

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001414, A055394, A078378, A078382, A078383.

Total[Times @@@ FactorInteger[#]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A001414(A055394(n)).

A078386 Squarefree kernels of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 3, 10, 6, 17, 6, 26, 14, 31, 33, 6, 37, 43, 22, 10, 26, 57, 21, 65, 34, 6, 73, 38, 10, 82, 89, 91, 10, 101, 6, 113, 122, 42, 127, 2, 129, 134, 141, 145, 74, 30, 38, 161, 82, 170, 57, 174, 177, 185, 21, 14, 197, 102, 206, 26, 217, 110, 223, 15, 226, 58, 233, 241

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

			A055394(12) = 3^2+3^3 = 36, a(12) = A007947(A055394(12)) = A007947(36) = A007947(2^2*3^2) = 2*3 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A055394, A078387.

Times @@ FactorInteger[#][[;; , 1]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A007947(A055394(n)).

A111909 Numbers that cannot be represented as a^4 + b^2 with a, b > 0.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 84, 86, 87, 88

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

			3 cannot be represented as a^4 + b^2 and thus is in this sequence while 10 = 1^4 + 3^2 is not.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A111925 (complement), A055394, A022549. A022544 is a subsequence.

list(lim)=lim\=1; my(v=List(),u=vectorsmall(lim),m2); for(m=1,sqrtint(lim-1), m2=m^2; for(n=1,sqrtnint(lim-m2,4), u[m2+n^4]=1)); for(i=1,#u, if(!u[i], listput(v,i))); Set(v) \\ Charles R Greathouse IV, Sep 01 2015

a(n) = n + O(n^(3/4)). - Charles R Greathouse IV, Sep 01 2015

A123053 Sum of a positive square, a positive cube and a positive fourth power.

Original entry on oeis.org

3, 6, 10, 11, 13, 18, 21, 25, 26, 27, 28, 29, 32, 33, 34, 37, 38, 40, 42, 44, 45, 47, 49, 51, 52, 53, 58, 59, 60, 64, 66, 68, 69, 73, 74, 77, 79, 81, 83, 84, 86, 88, 89, 90, 91, 92, 93, 96, 98, 101, 102, 105, 107, 109, 112, 114, 116, 117, 118, 123, 124, 125

Offset: 1

Jonathan Vos Post, Sep 25 2006

			a(1) = 3 = 1^4 + 1^3 + 1^2.
a(2) = 6 = 1^4 + 1^3 + 2^2.
a(3) = 10 = 1^4 + 2^3 + 1^2.
a(4) = 11 = 1^4 + 1^3 + 3^2.
a(5) = 13 = 1^4 + 2^3 + 2^2.
a(6) = 18 = 1^4 + 1^3 + 4^2 = 1^4 + 2^3 + 3^2 = 2^4 + 1^3 + 1^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A055394 (numbers that are the sum of a positive square and a positive cube).

isA123053 := proc(n)
    local x,y,z ;
    for x from 1 do
        if x^2 > n then
            return false;
        end if;
        for y from 1 do
            if x^2+y^3> n then
                break;
            end if;
            for z from 1 do
                if x^2+y^3+z^4 > n then
                    break;
                elif x^2+y^3+z^4 = n then
                    return true;
                end if;
            end do:
        end do:
    end do:
end proc:
n := 1 ;
for c from 0 to 10000 do
    if isA123053(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Sep 07 2020

Select[ Union[ Total /@ Tuples[{Range[64]^2, Range[8]^4, Range[16]^3}]], # < 200 &] (* Giovanni Resta, Jun 12 2016 *)

{A000290 \ 0} + {A000578 \ 0} + {A000583}. {a^4 + b^3 + c^2 for a,b,c>0}.

38, 86, and 93 added and 108 deleted by Giovanni Resta, Jun 12 2016

A274035 Numbers k such that k^7 = a^2 + b^3 for positive integers a and b.

Original entry on oeis.org

2, 5, 8, 9, 10, 12, 15, 17, 24, 26, 28, 31, 33, 36, 37, 40, 43, 44, 46, 50, 52, 54, 56, 57, 63, 65, 68, 69, 72, 73, 76, 80, 82, 89, 91, 98, 100, 101, 108, 113, 122, 126, 127, 128, 129, 134, 136, 141, 145, 148, 150, 152, 161, 164, 168, 170, 171, 174, 177, 183, 185, 189, 192, 196, 197

Offset: 1

Charles R Greathouse IV, Jun 06 2016

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..3001 (all terms from Charles R Greathouse IV except for a(58)=174)
Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, Twists of X(7) and primitive solutions to x^2+y^3=z^7, arXiv:math/0508174 [math.NT], 2005; Duke Math. J. 137:1 (2007), pp. 103-158.

Cf. A055394, A174115.

okQ[n_] := Module[{a, b}, For[b = 1, b < n^(7/3), b++, If[IntegerQ[a = Sqrt[n^7 - b^3]] && a > 0, Print["n = ", n, ", a = ", a, ", b = ", b]; Return[True]]]; False];
Reap[For[n = 1, n < 200, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)

isA055394(n)=for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0
is(n)=isA055394(n^7)

# Sage cannot handle n = 123, 174, ... without the fallback, even with descent_second_limit = 1000.
def fallback(n):
    return gp("my(n=" + str(n) + ");for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0")
def isA055394(z):
    z7 = z^7
    E = EllipticCurve([0,z7], descent_second_limit = 1000)
    try:
        for c in E.integral_points():
            if c[0] < 0 and c[1] != 0:
                return True
        return False
    except RuntimeError:
        return fallback(z7)
[x for x in range(1, 201) if isA055394(x)]

Missing term 174 inserted by Jean-François Alcover, Jan 30 2019

A295787 Positive integers m such that m, m + 1 and m + 2 are a sum of a positive square and a positive cube.

Original entry on oeis.org

126, 127, 350, 351, 441, 485, 511, 848, 1431, 1568, 2024, 2752, 2843, 3024, 3844, 4697, 5489, 7120, 7343, 7399, 8125, 8126, 8623, 9430, 9800, 10703, 10842, 11474, 12176, 12335, 12742, 12743, 13748, 14191, 14911, 15254, 16128, 16640, 16857, 17067, 17207, 18095, 18567

Offset: 1

David A. Corneth, Dec 30 2017

nonn

Is a(n) >= c*n^e for some constants c and e? For terms in the b-file, we'd have e > 2.1598. - David A. Corneth, Mar 15 2019

			126 and 127 are terms because: 126 = 1^2 + 5^3, 127 = 10^2 + 3^3, 128 = 8^2 + 4^3, 129 = 11^2 + 2^3. - _Bernard Schott_, Mar 17 2019

David A. Corneth, Table of n, a(n) for n = 1..14685 (Terms <= 10^9)
Bundeswettbewerb Mathematik 2017, Die Aufgaben der 2. Runde 2017
Bundeswettbewerb Mathematik 2017, Aufgaben und Lösungen 2. Runde 2017
David A. Corneth, PARI program

Cf. A055394.

s = Union@ Flatten@ Table[s^2 + c^3, {s, 141}, {c, 27}]; First@# & /@ Select[Partition[s, 3, 1], #[[1]] + 2 == #[[3]] &] (* Robert G. Wilson v, Jan 07 2018 *)
With[{mx=19000},Select[Partition[Union[Flatten[Table[a^2+b^3,{a,Ceiling[ Sqrt[mx]]},{b,Ceiling[Surd[mx,3]]}]]],3,1],Differences[#]=={1,1}&]][[All,1]] (* Harvey P. Dale, Sep 07 2020 *)

is_a055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0 \\ after Charles R Greathouse IV
is(n) = is_a055394(n) && is_a055394(n+1) && is_a055394(n+2) \\ Felix Fröhlich, Jan 08 2018

See Corneth Link \\ David A. Corneth, Mar 15 2019

cp's OEIS Frontend

A266548 Least prime p such that n + p^3 = x^2 + y^3 for some positive integers x and y, or 0 if no such prime p exists.

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A066647 Squares of the form a^2 + b^3 with a, b > 0.

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A066648 Cubes of the form a^2 + b^3 with a, b > 0.

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A078383 Sum of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.

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A078384 Sum of all prime factors of numbers which can be written as sum of a positive square and a positive cube.

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A078386 Squarefree kernels of numbers which can be written as sum of a positive square and a positive cube.

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A111909 Numbers that cannot be represented as a^4 + b^2 with a, b > 0.

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PARI

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A123053 Sum of a positive square, a positive cube and a positive fourth power.

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Maple