A055394 -id:A055394 - cp's OEIS frontend

A273498 Numbers that are, at the same time, the sum of: two positive squares, a positive square and a positive cube, and two positive cubes. In other words, intersection of A000404, A003325 and A055394.

2, 65, 72, 128, 468, 730, 793, 1241, 1332, 1458, 2000, 2745, 3528, 4097, 4160, 4608, 4825, 5096, 5840, 5913, 6344, 8125, 8192, 9000, 9325, 9928, 12168, 13357, 13498, 14824, 15626, 15633, 15689, 16354, 17640, 18369, 18737, 19721, 19773, 21953, 22681, 27792, 29449

Offset: 1

Altug Alkan, May 23 2016

Numbers n such that n = x^a + y^b where x,y > 0, is soluble for all 1 < a <= b < 4.

Perfect power terms are 128, 8192, 97344, 140625, 524288, 1500625, ...

			793 is a term because 793 = 3^2 + 28^2 = 8^2 + 9^3 = 4^3 + 9^3.

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

Cf. A000404, A003325, A055394.

isA003325(n)=for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))
isA000404(n) = for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))
isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
lista(nn) = for(n=1, nn, if(isA003325(n) && isA000404(n) && isA055394(n), print1(n, ", ")));

isA000404(n)=my(f=factor(n)); for(i=1, #f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); n>1 && (vecmin(f[,1]%4)==1 || (f[1, 1]==2 && f[1,2]%2))
isA055394(n) = for(k=1, sqrtnint(n-1,3), if(issquare(n-k^3), return(1))); 0
list(lim)=my(v=List(),n3,t); lim\=1; for(n=1,sqrtnint(lim-1,3), n3=n^3; for(m=1,sqrtnint(lim-n3,3), t=n3+m^3; if(isA000404(t) && isA055394(t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, May 31 2016

A111925 Numbers of the form a^2 + b^4, with a,b > 0.

Original entry on oeis.org

2, 5, 10, 17, 20, 25, 26, 32, 37, 41, 50, 52, 65, 80, 82, 85, 90, 97, 101, 106, 116, 117, 122, 130, 137, 145, 160, 162, 170, 181, 185, 197, 202, 212, 225, 226, 241, 250, 257, 260, 265, 272, 277, 281, 290, 292, 305, 306, 320, 325, 337, 340, 356, 362, 370, 377

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

Subsequence of A000404.
Although there are squares, cubes, fifth powers, ... in this sequence, there are no fourth powers. - Altug Alkan, Apr 09 2016
Also, numbers z such that z^5 = x^2 + y^4 for x, y >= 1. - M. F. Hasler, Apr 16 2018
The Friedlander-Iwaniec theorem states that there are infinitely many prime numbers in this sequence. These primes are in A028916. - Bernard Schott, Mar 09 2019

			25 = 3^2 + 2^4, so 25 is an element of the sequence.

R. J. Mathar, Table of n, a(n) for n = 1..1000
J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
Wikipedia, Friedlander-Iwaniec theorem

Cf. A055394, A022549; complement of A111909; subsequence of A000404.

Cf. A028916 (subsequence of primes).

isA111925 := proc(n)
    local a,b ;
    for a from 1 do
        if a^4 >= n then
            return false;
        end if;
        b := n-a^4 ;
        if issqr(b) then
            return true;
        end if;
    end do:
end proc:
A111925 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA111925(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 22 2013

With[{nn=60},Take[Union[First[#]^2+Last[#]^4&/@Tuples[Range[nn],2]],nn]] (* Harvey P. Dale, Jul 09 2014 *)

list(lim)=my(v=List(),t); lim\=1; for(b=1,sqrtnint(lim-1,4), t=b^4; for(a=1,sqrtint(lim-t), listput(v,t+a^2))); Set(v) \\ Charles R Greathouse IV, Jun 07 2016

is(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0 \\ Charles R Greathouse IV, Jun 07 2016

A022549 Sum of a square and a nonnegative cube.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 12, 16, 17, 24, 25, 26, 27, 28, 31, 33, 36, 37, 43, 44, 49, 50, 52, 57, 63, 64, 65, 68, 72, 73, 76, 80, 81, 82, 89, 91, 100, 101, 108, 113, 121, 122, 125, 126, 127, 128, 129, 134, 141, 144, 145

Offset: 1

N. J. A. Sloane

It appears that there are no modular constraints on this sequence; i.e., every residue class of every integer has representatives here. - Franklin T. Adams-Watters, Dec 03 2009

A045634(a(n)) > 0. - Reinhard Zumkeller, Jul 17 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000 - _Reinhard Zumkeller_, Jul 17 2010

Complement of A022550; A002760 and A179509 are subsequences.

Cf. A055394, A045634, A055393, A123291, A169618, A123364, A111925.

q=30; imax=q^2; Select[Union[Flatten[Table[x^2+y^3, {y,0,q^(2/3)}, {x,0,q}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

is(n)=for(k=0,sqrtnint(n,3), if(issquare(n-k^3), return(1))); 0 \\ Charles R Greathouse IV, Aug 24 2020

list(lim)=my(v=List(),t); for(k=0,sqrtnint(lim\=1,3), t=k^3; for(n=0,sqrtint(lim-t), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Aug 24 2020

A066649 Primes of the form a^2 + b^3 with a, b > 0.

Original entry on oeis.org

2, 5, 17, 31, 37, 43, 73, 89, 101, 113, 127, 197, 223, 233, 241, 257, 269, 283, 337, 347, 353, 359, 379, 401, 443, 449, 487, 521, 577, 593, 599, 677, 701, 733, 743, 811, 827, 829, 919, 953, 1009, 1019, 1049, 1051, 1097, 1129, 1153, 1213, 1289, 1297, 1361

Offset: 1

Reinhard Zumkeller, Dec 17 2001

nonn

			A000040(26) = 101 = 10^2 + 1^3, therefore 101 is a term.
A000040(51) = a(13) = 233 = 225 + 8 = 15^2 + 2^3.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jori Merikoski, On primes represented by aX^2+bY^3aX^2+bY^3, arXiv preprint (2025). arXiv:2503.05396 [math.NT]

Cf. A055394, A066650.

Cf. A000040, A078393, A078390.

lst={};Do[Do[p=n^2+m^3;If[PrimeQ[p],AppendTo[lst,p]],{n,5!}],{m,5!}];Take[Union[lst],123] (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)

list(lim)=my(v=List()); for(y=1,sqrtnint(lim\=1,3), my(y3=y^3); for(x=1,sqrtint(lim-y3), my(p=y3+x^2); if(isprime(p), listput(v,p)))); Set(v) \\ Charles R Greathouse IV, Mar 11 2025

On Conjecture C_a(1/17), this sequence is infinite and a(n) << n^(6/5) log n, see Merikoski link. - Charles R Greathouse IV, Mar 11 2025

A100291 Numbers of the form a^4 + b^3 with a, b > 0.

Original entry on oeis.org

2, 9, 17, 24, 28, 43, 65, 80, 82, 89, 108, 126, 141, 145, 206, 217, 232, 257, 264, 283, 297, 320, 344, 359, 381, 424, 472, 513, 528, 593, 599, 626, 633, 652, 689, 730, 745, 750, 768, 810, 841, 968, 985, 1001, 1016, 1081, 1137, 1256, 1297, 1304, 1323, 1332

Offset: 1

T. D. Noe, Nov 18 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Gian Cordana Sanjaya and Xiaoheng Wang, On the squarefree values of a^4+b^3, arXiv:2107.10380 [math.NT], 2021.

Cf. A100271 (primes of the form a^4 + b^3).
Cf. A055394 (a^2 + b^3: contains this as subsequence), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4), A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).
Roots of 5th powers are listed in A300565 (z^5 = x^3 + y^4); see also A300564 (z^4 = x^2 + y^3) and A242183, A300566 (z^6 = x^4 + y^5), A300567 (z^7 = x^6 + y^5), A302174.

lst={}; Do[p=a^4+b^3; If[p<2000, AppendTo[lst, p]], {a, 64}, {b, 256}]; Union[lst]
With[{nn=20},Select[Union[#[[1]]^4+#[[2]]^3&/@Tuples[Range[20],2]],#<= nn^3+1&]] (* Harvey P. Dale, May 27 2020 *)

is(n)=for(a=1, sqrtnint(n-1, 4), ispower(n-a^4, 3) && return(a)) \\ Returns a > 0 if n is in the sequence, or 0 otherwise. - M. F. Hasler, Apr 25 2018

list(lim)=my(v=List());for(b=1,sqrtnint(lim\=1,3), my(b3=b^3); for(a=1,sqrtnint(lim-b3,4), listput(v,a^4+b3))); Set(v) \\ Charles R Greathouse IV, Jul 26 2021

Edited by M. F. Hasler, Apr 25 2018

A100292 Numbers of the form a^5 + b^2 with a, b > 0.

Original entry on oeis.org

2, 5, 10, 17, 26, 33, 36, 37, 41, 48, 50, 57, 65, 68, 81, 82, 96, 101, 113, 122, 132, 145, 153, 170, 176, 197, 201, 226, 228, 244, 247, 252, 257, 259, 268, 279, 288, 290, 292, 307, 321, 324, 325, 343, 356, 362, 364, 387, 393, 401, 412, 432, 439, 442, 468, 473

Offset: 1

T. D. Noe, Nov 18 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A100272 (primes of the form a^5 + b^2).

Cf. A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4), A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).

lst={}; Do[p=a^5+b^2; If[p<1000, AppendTo[lst, p]], {a, 16}, {b, 1024}]; Union[lst]

is(n, m=5)=for(a=1, sqrtnint(n-1, m), issquare(n-a^m) && return(a)) \\ M. F. Hasler, Apr 25 2018

A100294 Numbers of the form a^5 + b^4 with a, b > 0.

Original entry on oeis.org

2, 17, 33, 48, 82, 113, 244, 257, 259, 288, 324, 499, 626, 657, 868, 1025, 1040, 1105, 1280, 1297, 1328, 1539, 1649, 2320, 2402, 2433, 2644, 3126, 3141, 3206, 3381, 3425, 3750, 4097, 4128, 4339, 4421, 5120, 5526, 6562, 6593, 6804, 7221, 7585, 7777, 7792

Offset: 1

T. D. Noe, Nov 18 2004

nonn

In view of computing A300566, it would be interesting to have an efficient way to check whether a given (large) n is in this sequence. - M. F. Hasler, Apr 25 2018

Cf. A100274 (primes of the form a^5 + b^4).
Subsequence of A100292 (a^5 + b^2); see also A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100293 (a^5 + b^3), A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).
Roots of 6th powers are listed in A300566 (z such that z^6 = x^5 + y^4 for some x, y >= 1); see also A300564 (z^4 = x^3 + y^2) and A242183, A300565 (z^5 = x^4 + y^3), A300567 (z^7 = x^6 + y^5), A302174.

lst={}; Do[p=a^5+b^4; If[p<15000, AppendTo[lst, p]], {a, 16}, {b, 32}]; Union[lst]

A100294_vec(L=10^6, k=4, m=5, S=List())={for(a=1, sqrtnint(L-1, m), for(b=1, sqrtnint(L-a^m, k), listput(S, a^m+b^k))); Set(S)} \\ all terms up to limit L. - M. F. Hasler, Apr 25 2018

is(n, k=4, m=5)=for(a=1, sqrtnint(n-1, m), ispower(n-a^m,k) && return(a)) \\ Returns a > 0 if n is in the sequence, or 0 otherwise. - M. F. Hasler, Apr 25 2018

A303375 Numbers of the form a^5 + b^6, with integers a, b > 0.

Original entry on oeis.org

2, 33, 65, 96, 244, 307, 730, 761, 972, 1025, 1088, 1753, 3126, 3189, 3854, 4097, 4128, 4339, 5120, 7221, 7777, 7840, 8505, 11872, 15626, 15657, 15868, 16649, 16808, 16871, 17536, 18750, 20903, 23401, 32432, 32769, 32832, 33497, 36864, 46657, 46688, 46899, 47680, 48393

Offset: 1

M. F. Hasler, Apr 22 2018

Although it is easy to produce many terms of this sequence, it is nontrivial to check whether a very large number is of this form.

This sequence is among others motivated by the hard-to-compute sequence A300567 = numbers z such that z^7 = x^5 + y^6 for some x, y >= 1.

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

Cf. A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).
Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6).
See also A300567: numbers z such that z^7 = x^5 + y^6 for some x, y >= 1.

is(n,k=5,m=6)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,n)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
A303375_vec(L=10^5,k=5,m=6,S=List())={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k), listput(S,a^m+b^k)));Set(S)} \\ all terms up to limit L

a(n) >> n^(30/11). Probably this is the correct asymptotic order. - Charles R Greathouse IV, Jan 23 2025

A303374 Numbers of the form a^4 + b^6, with integers a, b > 0.

Original entry on oeis.org

2, 17, 65, 80, 82, 145, 257, 320, 626, 689, 730, 745, 810, 985, 1297, 1354, 1360, 2025, 2402, 2465, 3130, 4097, 4112, 4160, 4177, 4352, 4721, 4825, 5392, 6497, 6562, 6625, 7290, 8192, 10001, 10064, 10657, 10729, 14096, 14642, 14705, 15370, 15626, 15641, 15706, 15881

Offset: 1

M. F. Hasler, Apr 22 2018

A subsequence of A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^4 + b^2), A100291 (a^4 + b^3), A303372 (a^2 + b^6).

Although it is easy to produce many terms of this sequence, it is nontrivial to check whether a very large number is of this form. Maybe the most efficient way is to consider decompositions of n into sums of two positive squares (see sum2sqr in A133388), and check if one of the terms is a third power and the other a fourth power.

Cf. A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).

Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303375 (a^5 + b^6).

Take[Flatten[Table[a^4+b^6,{a,20},{b,20}]]//Union,50] (* Harvey P. Dale, Jul 17 2025 *)

is(n,k=4,m=6)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,k)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
is(n,L=sum2sqr(n))={for(i=1,#L,L[i][1]&&for(j=1,2,ispower(L[i][j],3)&&issquare(L[i][3-j])&&return(L[i][j])))} \\ See A133388 for sum2sqr(). Much faster than the above for n >> 10^30.
A303374(L=10^5,k=4,m=6,S=[])={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k),S=setunion(S,[a^m+b^k])));S}

A303372 Numbers of the form a^2 + b^6, with integers a, b > 0.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 50, 65, 68, 73, 80, 82, 89, 100, 101, 113, 122, 128, 145, 164, 170, 185, 197, 208, 226, 233, 257, 260, 289, 290, 320, 325, 353, 362, 388, 401, 425, 442, 464, 485, 505, 530, 548, 577, 593, 626, 640, 677, 689, 730, 733, 738, 740, 745, 754, 765, 778

Offset: 1

M. F. Hasler, Apr 22 2018

A subsequence of A055394, the numbers of the form a^2 + b^3.

Although it is easy to produce many terms of this sequence, it is nontrivial to check whether a very large number is of this form.

			The first terms are 1^2 + 1^6 = 2, 2^2 + 1^6 = 5, 3^2 + 1^6 = 10, 4^2 + 1^6 = 17, 5^2 + 1^6 = 26, ..., 8^2 + 1^6 = 1^2 + 2^6 = 65, 2^2 + 2^6 = 68, 3^2 + 2^6 = 73, ...

Cf. A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).

Cf. A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).

is(n,k=2,m=6)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,k)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
A303372_vec(L=10^5,k=2,m=6,S=List())={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k),listput(S,a^m+b^k)));Set(S)} \\ List of all terms up to limit L

cp's OEIS Frontend

A273498 Numbers that are, at the same time, the sum of: two positive squares, a positive square and a positive cube, and two positive cubes. In other words, intersection of A000404, A003325 and A055394.

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A111925 Numbers of the form a^2 + b^4, with a,b > 0.

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A022549 Sum of a square and a nonnegative cube.

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A066649 Primes of the form a^2 + b^3 with a, b > 0.

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A100291 Numbers of the form a^4 + b^3 with a, b > 0.

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Extensions

A100292 Numbers of the form a^5 + b^2 with a, b > 0.

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A100294 Numbers of the form a^5 + b^4 with a, b > 0.

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A303375 Numbers of the form a^5 + b^6, with integers a, b > 0.

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