A022549 -id:A022549 - cp's OEIS frontend

A011379 a(n) = n^2*(n+1).

0, 2, 12, 36, 80, 150, 252, 392, 576, 810, 1100, 1452, 1872, 2366, 2940, 3600, 4352, 5202, 6156, 7220, 8400, 9702, 11132, 12696, 14400, 16250, 18252, 20412, 22736, 25230, 27900, 30752, 33792, 37026, 40460, 44100, 47952, 52022, 56316, 60840

Offset: 0

Glen Burch (gburch(AT)erols.com), Felice Russo

(1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (A000217) - the n-th partial sum of triangular numbers (A000217). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n). - Amarnath Murthy, Nov 05 2002
Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x) = (Product_{i=0..n-1} i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - Benoit Cloitre, Jan 15 2003
Y values of solutions of the equation: (X-Y)^3-X*Y=0. X values are a(n)=n*(n+1)^2 (see A045991) - Mohamed Bouhamida, May 09 2006
a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - Michael Somos, Sep 06 2006
a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. - Mohamed Bouhamida, Sep 05 2009
This sequence is related to A005449 by a(n) = n*A005449(n)-sum(A005449(i), i=0..n-1), and this is the case d=3 in the identity n^2*(d*n+d-2)/2 - Sum_{k=0..n-1} k*(d*k+d-2)/2 = n*(n+d)*(2*d*n+d-3)/6. - Bruno Berselli, Nov 18 2010
Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). - J. M. Bergot, Sep 22 2011
a(n+1) is the sum of n-th row of the triangle in A195437. - Reinhard Zumkeller, Nov 23 2011
Partial sums of A049450. - Omar E. Pol, Jan 12 2013
From Jon Perry, May 11 2013: (Start)
Define a 'stable brick triangle' as:
      -----
      | c |
  ---------
  | a | | b |
  ----------
with a, b, c > 0 and c <= a + b. This can be visualized as two bricks with a third brick on top. The third brick can only be as strong as a+b, otherwise the wall collapses - for example, (1,2,4) is unstable.
a(n) gives the number of stable brick triangles that can be formed if the two supporting bricks are 1 <= a <= n and 1 <= b <= n: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_c 1 = n^3 + n^2 as given in the Adamchuk formula.
So for i=j=n=2 we have 4:
   1    2    3    4
  2 2  2 2  2 2  2 2
For example, n=2 gives 2 from [a=1,b=1], 3 from both [a=1,b=2] and [a=2,b=1] and 4 from [a=2,b=2] so a(2) = 2 + 3 + 3 + 4 = 12. (End)
Define the infinite square array m(n,k) by m(n,k) = (n-k)^2 if n >= k >= 0 and by m(n,k) = (k+n)*(k-n) if 0 <= n <= k. This contains A120070 below the diagonal. Then a(n) = Sum_{k=0..n} m(n,k) + Sum_{r=0..n} m(r,n), the "hook sum" of the terms to the left of m(n,n) and above m(n,n) with irrelevant (vanishing) terms on the diagonal. - J. M. Bergot, Aug 16 2013
a(n) is the sum of all pairs with repetition drawn from the set of odd numbers 2*n-3. This is similar to A027480 but using the odd integers instead. Example using n=3 gives the odd numbers 1,3,5: 1+1, 1+3, 1+5, 3+3, 3+5,5+5 having a total of 36=a(3). - J. M. Bergot, Apr 05 2016
a(n) is the first Zagreb index of the complete graph K[n+1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016
a(n-2) is the maximum sigma irregularity over all trees with n vertices.  The extremal graphs are stars.  (The sigma irregularity of a graph is the sum of squares of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 14 2023

			a(3) = 3^2+3^3 = 36.

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Ivan Gutman and Kinkar Ch. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
J.S. Seneschal, Oblong cuboid illustration
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000290, A000292, A000330, A000578, A002378, A002411, A002412, A002413, A005449, A013661, A022549, A027480, A045991, A049450, A120070, A126890, A195437, A245334.
Cf. A011379, A181617, A270205 (sigma irregularities of maximal k-degenerate graphs).
Cf. A028724, A052289.

List([0..40], n-> n^2*(n+1) ); # G. C. Greubel, Aug 10 2019

a011379 n = a000290 n + a000578 n  -- Reinhard Zumkeller, Apr 28 2013

[n^2+n^3: n in [0..40]]; // Vincenzo Librandi, May 02 2011

A011379:=n->n^2*(n+1); seq(A011379(n), n=0..40); # Wesley Ivan Hurt, Feb 25 2014

Table[n^3+n^2, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009, modified by G. C. Greubel, Aug 10 2019 *)
LinearRecurrence[{4,-6,4,-1},{0,2,12,36},40] (* Harvey P. Dale, Sep 13 2018 *)

a(n)=n^3+n^2 \\ Charles R Greathouse IV, Apr 06 2016

[n^2*(n+1) for n in (0..40)] # G. C. Greubel, Aug 10 2019

a(n) = 2*A002411(n).
a(n) = Sum_{j=1..n} (Sum_{i=1..n} (i+j)), row sums of A126890 skipping numbers in the first column. - Alexander Adamchuk, Oct 12 2004
Sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340... [Jolley eq 272] - Gary W. Adamson, Dec 22 2006
a(n) = 2*n*binomial(n+1,2) = 2*n*A000217(n). - Arkadiusz Wesolowski, Feb 10 2012
G.f.: 2*x*(1 + 2*x)/(1 - x)^4. - Arkadiusz Wesolowski, Feb 11 2012
a(n) = A000330(n) + A002412(n) = A000292(n) + A002413(n). - Omar E. Pol, Jan 11 2013
a(n) = A245334(n+1,2), n > 0. - Reinhard Zumkeller, Aug 31 2014
Sum_{n>=1} 1/a(n) = A013661-1. - R. J. Mathar, Oct 18 2019 [corrected by Jason Yuen, Aug 04 2024]
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 - 2*log(2). - Amiram Eldar, Jul 04 2020
E.g.f.: exp(x)*x*(2 + 4*x + x^2). - Stefano Spezia, May 20 2021
a(n) = n*A002378(n) = A000578(n) + A000290(n). - J.S. Seneschal, Jun 18 2024

A055394 Numbers that are the sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, 68, 72, 73, 76, 80, 82, 89, 91, 100, 101, 108, 113, 122, 126, 127, 128, 129, 134, 141, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204, 206, 208, 217, 220, 223

Offset: 1

Henry Bottomley, May 12 2000

This sequence was the subject of a question in the German mathematics competition Bundeswettbewerb Mathematik 2017 (see links). The second round contained a question A4 which asks readers to "Show that there are an infinite number of a such that a-1, a, and a+1 are members of A055394". - N. J. A. Sloane, Jul 04 2017 and Oct 14 2017

This sequence was also the subject of a question in the 22nd All-Russian Mathematical Olympiad 1996 (see link). The 1st question of the final round for Grade 9 asked competitors "What numbers are more frequent among the integers from 1 to 1000000: those that can be written as a sum of a square and a positive cube, or those that cannot be?" Answer is that there are more numbers not of this form. - Bernard Schott, Feb 18 2022

			a(5)=17 since 17=3^2+2^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Bundeswettbewerb Mathematik 2017, Der Wettbewerb in der 47 Runde
Bundeswettbewerb Mathematik 2017, Aufgaben und Lösungen
The IMO Compendium, Problem 1, 22nd All-Russian Mathematical Olympiad 1996.
Index to sequences related to Olympiads and other Mathematical competitions.

Cf. A022549, A055393, A078360. Complement of A066650.

isA055394 := proc(n)
    local a,b;
    for b from 1 do
        if b^3 >= n then
            return false;
        end if;
        asqr := n-b^3 ;
        if asqr >= 0 and issqr(asqr) then
            return true;
        end if;
    end do:
    return;
end proc:
for n from 1 to 1000 do
    if isA055394(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 03 2015

r[n_, y_] := Reduce[x > 0 && n == x^2 + y^3, x, Integers]; ok[n_] := Catch[Do[If[r[n, y] =!= False, Throw[True]], {y, 1, Ceiling[n^(1/3)]}]] == True; Select[Range[300], ok] (* Jean-François Alcover, Jul 16 2012 *)
solQ[n_] := Length[Reduce[p^2 + q^3 == n && p > 0 && q > 0, {p, q}, Integers]] > 0; Select[Range[224], solQ] (* Jayanta Basu, Jul 11 2013 *)
isQ[n_] := For[k = 1, k <= (n-1)^(1/3), k++, If[IntegerQ[Sqrt[n-k^3]], Return[True]]; False];
Select[Range[1000], isQ] (* Jean-François Alcover, Apr 06 2021, after Charles R Greathouse IV *)

list(lim)=my(v=List()); for(n=1,sqrtint(lim\1-1), for(m=1,sqrtnint(lim\1-n^2,3), listput(v,n^2+m^3))); Set(v) \\ Charles R Greathouse IV, May 15 2015

is(n)=for(k=1,sqrtnint(n-1,3), if(issquare(n-k^3), return(1))); 0 \\ Charles R Greathouse IV, May 15 2015

a(n) >> n^(6/5). - Charles R Greathouse IV, May 15 2015

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

A002760 Squares and cubes.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849

Offset: 1

N. J. A. Sloane

nonn

Catalan's Conjecture states that 8 and 9 are the only pair of consecutive numbers in this sequence. The conjecture was established in 2003 by Mihilescu.

Subsequence of A022549. - Reinhard Zumkeller, Jul 17 2010

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 236.

Michael De Vlieger, Table of n, a(n) for n = 1..10443 (first 1000 terms from Zak Seidov)
Yuri F. Bilu, Catalan's Conjecture (After Mihilescu), Astérisque, No. 294, 1-26, 2004.
Yuri F. Bilu, Catalan Without Logarithmic Forms (after Bugeaud, Hanrot and Mihailescu), J. Théor. Nombres Bordeaux 17, 69-85, 2005.
David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 4.
Tauno Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.
Preda Mihǎilescu, A Class Number Free Criterion for Catalan's Conjecture, J. Number Th. 99 225-231, 2003.
Preda Mihǎilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. MR 2076124.
Paulo Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), p. 1-11.
Paulo Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.

Cf. A131799; union of A000290 and A000578.
First differences in A075052. [From Zak Seidov, May 10 2010]
Cf. A002117, A013661, A013664.

[n: n in [0..1600] | IsIntegral(n^(1/3)) or IsIntegral(n^(1/2))]; // Bruno Berselli, Feb 09 2016

nMax=2000;Union[Range[0,nMax^(1/2)]^2,Range[0,nMax^(1/3)]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
nxt[n_] := Min[ Floor[1 + Sqrt[n]]^2, Floor[1 + n^(1/3)]^3]; NestList[ nxt, 0, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isok(n) = issquare(n) || ispower(n, 3); \\ Michel Marcus, Mar 29 2016

from math import isqrt
from sympy import integer_nthroot
def A002760(n):
    def f(x): return n-1+x+integer_nthroot(x,6)[0]-integer_nthroot(x,3)[0]-isqrt(x)
    m, k = n-1, f(n-1)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 09 2024

Sum_{n>=2} 1/a(n) = zeta(2) + zeta(3) - zeta(6). - Amiram Eldar, Dec 19 2020

A045634 Number of ways in which n can be partitioned as a sum of a square and cube.

Original entry on oeis.org

1, 2, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Felice Russo

nonn

a(A022550(n))=0; a(A179509(n))=1; a(A022549(n))>0; a(A060861(n))=n. [From Reinhard Zumkeller, Jul 17 2010]

			a(9)=2 because 9=2^3+1^2 and 9=3^2+0^3.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A022549, A060861, A135910, A135911, A135912.

M:=100; M2:=M^2; t0:=array(0..M2);
for i from 0 to M2 do t0[i]:=0; od:
for a from 0 to M do for b from 0 to M do
i:=a^2+b^3; if i <= M2 then t0[i]:=t0[i]+1; fi; od: od:
[seq(t0[i],i=0..M2)];

max = 100; Clear[a]; a[_] = 0;
Do[n = i^2 + j^3; a[n] += 1, {i, 0, Sqrt[max]}, {j, 0, max^(1/3)}];
Table[a[n], {n, 0, max}] (* Jean-François Alcover, Aug 02 2018 *)

More terms from Erich Friedman

A022550 Numbers that are not the sum of a square and a nonnegative cube.

Original entry on oeis.org

3, 6, 7, 11, 13, 14, 15, 18, 19, 20, 21, 22, 23, 29, 30, 32, 34, 35, 38, 39, 40, 41, 42, 45, 46, 47, 48, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 66, 67, 69, 70, 71, 74, 75, 77, 78, 79

Offset: 1

N. J. A. Sloane

nonn

Complement of A022549; A045634(a(n))=0. - Reinhard Zumkeller, Jul 17 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000

A123364 Primes of the form a^2 + b^3 (with repetition).

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Oct 12 2006

nonn

Primes in A022549, A123291. Cf. A066649 Primes of the form a^2 + b^3 (without repetition), with a, b > 0.

			Each of 17, 89, 233 appears two times because 17=3^2+2^3=4^2+1^3, 89=5^2+4^3=9^2+2^3, 233=13^2+4^3=15^2+2^3;
2089 appears three times because 2089=19^2+12^3=33^2+10^3=45^2+4^3;
65537 appears four times because 65537=122^2+37^3=219^2+26^3=255^2+8^3=256^2+1^3.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A022549, A066649, A123291.

t={}; Do[AppendTo[t, n^2+m^3], {n, 100},{m, 100}]; Take[Select[Sort[t], PrimeQ], 60] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

A055393 Sum of a square and a nonnegative cube in more than one way.

Original entry on oeis.org

1, 9, 17, 36, 64, 65, 89, 100, 108, 129, 145, 196, 225, 233, 252, 289, 297, 316, 388, 441, 449, 464, 505, 512, 537, 548, 576, 577, 593, 633, 729, 730, 737, 745, 784, 792, 793, 801, 841, 873, 1000, 1025, 1088, 1090, 1116, 1225, 1289, 1296, 1304, 1305, 1367

Offset: 1

Henry Bottomley, May 12 2000

			a(12) = 225 since 225 = 6^3 + 3^2 = 5^3 + 10^2 = 0^3 + 15^2.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A022549, A055394.

nn = 2000; t = Table[0, {nn}]; Do[n = i^2 + j^3; If[0 < n <= nn, t[[n]]++], {i, 0, Sqrt[nn]}, {j, 0, nn^(1/3)}]; Flatten[Position[t, ?(# > 1 &)]] (* _T. D. Noe, Dec 08 2012 *)

Extended by T. D. Noe, Dec 08 2012

A214891 Numbers that are not the sum of two squares and two fourth powers.

Original entry on oeis.org

23, 44, 71, 79, 184, 368, 519, 599, 704, 1136, 1264, 2944, 4024, 5888, 8304, 9584, 11264, 18176, 20224, 47104, 64384, 94208, 132864, 153344, 180224, 290816, 323584, 753664, 1030144, 1507328, 2125824, 2453504, 2883584, 4653056, 5177344, 12058624, 16482304

Offset: 1

Joerg Arndt, Jul 29 2012

nonn

From XU Pingya, Feb 07 2018: (Start)
When n is a term, 16n is also. This can be proved as follows:
(1) If w is odd, then 16n - w^4 == 7 (mod 8), and it follows from Legendre's three-square theorem that the equation x^2 + y^2 + z^4 + w^4 = 16n has no solution (it is the same when x, y or z are odd numbers).
(2) If x, y, z and w are even numbers (x = 2a, y = 2b, z = 2c, w = 2d) such that x^2 + y^2 + z^4 + w^4 = 16n, then a^2 + b^2 = 4(n - c^4 - d^4). So there are integers u and v satisfying u^2 + v^2 = n - c^4 - d^4. i.e. u^2 + v^2 + c^4 + d^4 = n, which is a contradiction.
(End)
Conjecture: The set {a(n): n > 0} coincides with {16^k*m: k = 0, 1, 2, ... and m = 23, 44, 71, 79, 184, 519, 599, 4024}. - Zhi-Wei Sun, Jan 27 2022

Donovan Johnson, Table of n, a(n) for n = 1..52 (terms <= 4*10^9)
Zhi-Wei Sun, On w^4+x^4+y^2+z^2 over a number field, Question 414791 at MathOverflow, Jan. 27, 2022.

Cf. A001481, A004999, A022549, A346643, A347865, A350857, A350860.

N=10^6;  x='x+O('x^N);
S(e)=sum(j=0, ceil(N^(1/e)), x^(j^e));
v=Vec( S(4)^2 * S(2)^2 );
for(n=1,#v,if(!v[n],print1(n-1,", ")));

a(29)-a(37) from Donovan Johnson, Jul 29 2012

cp's OEIS Frontend

A011379 a(n) = n^2*(n+1).

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A055394 Numbers that are the sum of a positive square and a positive cube.

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

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A002760 Squares and cubes.

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A045634 Number of ways in which n can be partitioned as a sum of a square and cube.

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A022550 Numbers that are not the sum of a square and a nonnegative cube.

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A123364 Primes of the form a^2 + b^3 (with repetition).

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