A141046 -id:A141046 - cp's OEIS frontend

A016744 a(n) = (2*n)^4.

0, 16, 256, 1296, 4096, 10000, 20736, 38416, 65536, 104976, 160000, 234256, 331776, 456976, 614656, 810000, 1048576, 1336336, 1679616, 2085136, 2560000, 3111696, 3748096, 4477456, 5308416, 6250000, 7311616, 8503056, 9834496, 11316496, 12960000, 14776336, 16777216

Offset: 0

N. J. A. Sloane

Suppose the vertices of a triangle are (S(n), S(n+j)), (S(n+2*j), S(n+3*j)) and (S(n+4*j), S(n+5*j)) where S(n) is the n-th square number, A000290(n). Then the area of this triangle will be a(j). A generalization follows: let P(k,n) = the n-th k-gonal number and suppose the vertices of a triangle are (P(k,n), P(k,n+j)), (P(k,n+2*j), P(k,n+3*j)) and (P(k,n+4*j), P(k,n+5*j)). Then the area of this triangle will be (2*k-4)^2*j^4. See also A141046 for k = 3. - Charlie Marion, Mar 26 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016756, A258945.

[(2*n)^4: n in [0..40]]; // Vincenzo Librandi, Sep 05 2011

A016744:=n->(2*n)^4: seq(A016744(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2018

Table[(2*n)^4, {n,0,30}] (* G. C. Greubel, Sep 15 2018 *)

vector(30, n, n--; (2*n)^4) \\ G. C. Greubel, Sep 15 2018

G.f.: 16*x*(x+1)*(x^2+10*x+1)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
E.g.f.: 16*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). - G. C. Greubel, Sep 15 2018
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi^4/1440 (conjecturally A258945).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/11520. (End)

A171607 Expressible as A*B^A in a nontrivial way.

Original entry on oeis.org

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592

Offset: 1

Robert Munafo, Dec 12 2009

nonn

			8=2*2^2. 24=3*2^3. 375=3*5^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. Munafo, Generalized Cullen and Woodall Numbers

Cf. A171606. Union of the "KN^K" sequences A001105, A117642, A141046, ... or of the "NK^N" sequences A036289, A036290, A018215, A036291, ... but omitting the trivial initial terms.

is(n)=if(n<8, return(0)); for(a=2,logint(n\2,2), if(n%a==0 && ispower(n/a,a), return(1))); 0 \\ Charles R Greathouse IV, Feb 19 2017

list(lim)=my(v=List()); if(lim<8,return([])); for(a=2,logint(lim\2,2), for(b=2,sqrtnint(lim\a,a), listput(v,a*b^a))); Set(v) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 2n^2 - O(n^(5/3)). - Charles R Greathouse IV, Feb 19 2017

A211412 a(n) = 4*n^4 + 1.

Original entry on oeis.org

5, 65, 325, 1025, 2501, 5185, 9605, 16385, 26245, 40001, 58565, 82945, 114245, 153665, 202501, 262145, 334085, 419905, 521285, 640001, 777925, 937025, 1119365, 1327105, 1562501, 1827905, 2125765, 2458625, 2829125, 3240001, 3694085, 4194305, 4743685, 5345345, 6002501, 6718465, 7496645

Offset: 1

Alonso del Arte, Feb 10 2013

Except for the first term, all terms are composite. a(n) is divisible by 5 if n is not.
Long before Aurifeuille, Euler discovered that 4n^4 + 1 = (2n^2 + 2n + 1)*(2n^2 - 2n + 1). For example, 325 = 4 * 3^4 + 1 = (2 * 3^2 + 2 * 3 + 1)*(2 * 3^2 - 2 * 3 + 1) = 25 * 13. Euler shared this discovery with Goldbach in a letter dated August 28, 1742. [Euler identity corrected by Graham Holmes, Jun 02 2023]
The terms of the sequence are the arithmetic mean of eight numbers located on concentric circles (see Avilov link). - Nicolay Avilov, Jan 22 2021

Don Knuth, The Art of Computer Programming: Seminumerical Algorithms, 3rd ed., New York: Addison-Wesley Professional (1997), p. 392.
David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 15.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nicolay Avilov, Drawing with circles
P. H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, Saint-Pétersbourg, 1843, p. 145; alternative link. See in particular Lettre XLVI (Euler to Goldbach), Aug 28 1742
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A207262 (subset).
After the first term, subsequence of A121944.
Cf. A053755.

[4*n^4 + 1: n in [1..50]]; // Vincenzo Librandi, Jun 11 2017

4 Range[44]^4 + 1
Table[4 n^4 + 1, {n, 50}] (* Vincenzo Librandi, Jun 11 2017 *)
LinearRecurrence[{5,-10,10,-5,1},{5,65,325,1025,2501},40] (* Harvey P. Dale, Sep 15 2023 *)

a(n) = 4*n^4+1 \\ Felix Fröhlich, Jun 07 2017

G.f.: -x*(x^4+50*x^2+40*x+5) / (x-1)^5. - Colin Barker, Feb 11 2013
a(n) = A053755(n^2). - Michel Marcus, Sep 18 2015
a(n) = (2*n^2)^2 + 1^2 = (2*n^2-1)^2 + (2*n)^2. - Thomas Ordowski, Sep 18 2015
a(n) = A001844(n) * A001844(n+1) = A141046(n) + 1 = (A000583(n) * 4 ) + 1 = A016742(n) + A173121(n) + 1. - Bruce J. Nicholson, Jun 06 2017
From Amiram Eldar, Jul 26 2022: (Start)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/4 - 1/2.
Sum_{n>=1} (-1)^n/a(n) = 1/2 - sech(Pi/2)*Pi/4. (End)

A211996 Number of ordered pairs (i,j) such that i*j=n and i+j is a square.

Original entry on oeis.org

0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0

Offset: 1

Michel Marcus, Oct 25 2012

nonn

a(n) = 1 for n > 0 in A141046.

a(8820) = 8 and it is the only term in the first 10000 terms that is greater than 6. There are 977 terms in the first 10000 terms that are greater than zero. - Harvey P. Dale, Nov 08 2012

			For n=3, the pairs (a,b) such that a*b=3 are (1,3) and (3,1). Both pairs add up to a square, so a(3) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Clark, An arithmetical function associated with the rank of elliptic curves, Canad. Math. Bull. Vol. 34 (2), (1991), pp. 181-185.
Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, Integers with a sum of co-divisors yielding a square, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; author's copy.

Cf. A010052, A141046, A218381, A218382, A377731.

a211996 n = length [x | x <- [1..n], let (y, m) = divMod n x,
                        m == 0, a010052 (x + y) == 1]
-- Reinhard Zumkeller, Oct 28 2012

nop[n_]:=Module[{divs=Divisors[n]},Count[Thread[{divs,Reverse[divs]}], ?(IntegerQ[Sqrt[Total[#]]]&)]]; Array[nop,90] (* _Harvey P. Dale, Nov 08 2012 *)
a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[# + n/#]] &]; Array[a, 100] (* Amiram Eldar, Nov 05 2024 *)

a(n) = sumdiv(n, d, issquare(d+n/d)); \\ Michel Marcus, Jan 18 2021

Sum_{k=1..n} a(k) = c * n^(3/4) + O(sqrt(n)), where c = A377731 (De Koninck et al., 2024). - Amiram Eldar, Nov 05 2024

A377732 Numbers k such that max{d|k, d <= sqrt(k)} + min{d|k, d >= sqrt(k)} is a square.

Original entry on oeis.org

3, 4, 14, 18, 20, 39, 46, 55, 60, 63, 64, 94, 114, 136, 150, 154, 155, 156, 158, 183, 203, 243, 258, 275, 291, 295, 299, 308, 315, 320, 323, 324, 328, 334, 444, 446, 490, 544, 558, 570, 579, 580, 583, 584, 588, 594, 598, 600, 695, 710, 718, 799, 855, 878, 903, 904, 938, 943, 955, 959, 975, 978, 979, 988, 999

Offset: 1

Amiram Eldar, Nov 05 2024

nonn

Numbers k such that A063655(k) = A033676(k) + A033677(k) is a square.

The square terms of this sequence are the positive numbers of the form A141046(m) = 4*m^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, Integers with a sum of co-divisors yielding a square, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; author's copy.

Cf. A033676, A033677, A063655, A377731.

Subsequences: A141046 \ {0}, A377733, A377736.

q[k_] := If[IntegerQ[Sqrt[k]], IntegerQ[Sqrt[2*Sqrt[k]]], Module[{d = Divisors[k], nh}, nh = Length[d]/2; IntegerQ[Sqrt[d[[nh]] + d[[nh + 1]]]]]]; Select[Range[1000], q]

is(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A377732_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        d = (a:=divisors(k))[len(a)-1>>1]
        if is_square(d+k//d):
            yield k
A377732_list = list(islice(A377732_gen(),30)) # Chai Wah Wu, Nov 06 2024

c * x^(3/4) / log(x) < R(x) < 2 * c * x^(3/4) / log(x) for sufficiently large x, where R(x) is the number of terms that do not exceed x, and c = A377731 (De Koninck et al., 2024).

A227855 Numbers of the form x^4 + 4*y^4.

Original entry on oeis.org

0, 1, 4, 5, 16, 20, 64, 65, 80, 81, 85, 145, 256, 260, 320, 324, 325, 340, 405, 580, 625, 629, 689, 949, 1024, 1025, 1040, 1105, 1280, 1296, 1300, 1360, 1620, 1649, 2320, 2401, 2405, 2465, 2500, 2501, 2516, 2581, 2725, 2756, 3125, 3425, 3796, 4096, 4100, 4160

Offset: 1

Alonso del Arte, Oct 31 2013

Since 4 is even, either x or y or both may be negative integers, because their fourth powers will then be positive.

The only prime term in this sequence is 5; this can be proved using Sophie Germain's identity.

			80 = 2^4 + 4 * 2^4.
81 = 3^4 + 4 * 0^4.
85 = 3^4 + 4 * 1^4.

Titu Andreescu and Rǎzvan Gelca, Mathematical Olympiad Challenges, New York, Birkhäuser (2009), p. 48.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Graeme Taylor, Identity of Sophie Germain, April 1, 2006.

Subsequences include A000583, A141046 and A001589.

nn = 10; Union[Select[Flatten[Table[x^4 + 4*y^4, {x, 0, nn}, {y, 0, nn}]], # <= nn^4 &]] (* T. D. Noe, Nov 08 2013 *)

list(lim)=my(v=List(),t); for(y=0,sqrtnint(lim\4,4), for(x=0, sqrtnint(lim\1-(t=4*y^4),4), listput(v,t+x^4))); Set(v) \\ Charles R Greathouse IV, Nov 07 2013

x^4 + 4y^4 = (x^2 - 2xy + 2y^2)(x^2 + 2xy + 2y^2). This is Sophie Germain's identity.

A244730 a(n) = 2*n^4.

Original entry on oeis.org

0, 2, 32, 162, 512, 1250, 2592, 4802, 8192, 13122, 20000, 29282, 41472, 57122, 76832, 101250, 131072, 167042, 209952, 260642, 320000, 388962, 468512, 559682, 663552, 781250, 913952, 1062882, 1229312, 1414562, 1620000, 1847042, 2097152, 2371842, 2672672

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A141046, A219056.

Magma
```
[2*n^4: n in [0..40]];
```

I:=[0,2,32,162, 512]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Table[2 n^4, {n, 0, 40}] (* or *) CoefficientList[Series[2(x + 11 x^2 + 11 x^3 + x^4)/(1 - x)^5, {x, 0, 40}], x]
LinearRecurrence[{5,-10,10,-5,1},{0,2,32,162,512},40] (* Harvey P. Dale, Jun 17 2022 *)

G.f.: 2*(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4.
a(n) = (A082044(n) + A099761(n+1)-2)/2. - Bruce J. Nicholson, Jun 12 2017

A377733 Numbers k such that k and k+1 are both terms in A377732.

Original entry on oeis.org

3, 63, 154, 155, 323, 579, 583, 903, 978, 1023, 2019, 2499, 3503, 5174, 5183, 5379, 8234, 9603, 11534, 12415, 14718, 16383, 20454, 20538, 26243, 31930, 39999, 46814, 58563, 69719, 82943, 90218, 93995, 96663, 102943, 114243, 117998, 118979, 124118, 135814, 138490, 149879

Offset: 1

Amiram Eldar, Nov 05 2024

nonn

This sequence is infinite. All the positive numbers of the form A141046(m) - 1 = 4*m^4 -1 are terms (De Koninck et al., 2024, section 6).

Amiram Eldar, Table of n, a(n) for n = 1..5242 (terms below 10^10)
Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, Integers with a sum of co-divisors yielding a square, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; author's copy.

Subsequence of A377732.
A377736 is a subsequence.
Cf. A141046.

q[k_] := q[k] = If[IntegerQ[Sqrt[k]], IntegerQ[Sqrt[2*Sqrt[k]]], Module[{d = Divisors[k], nh}, nh = Length[d]/2; IntegerQ[Sqrt[d[[nh]] + d[[nh + 1]]]]]]; Select[Range[150000], q[#] && q[#+1] &]

is1(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

A305290 Numbers k such that 4*k + 1 is a perfect cube, sorted by absolute values.

Original entry on oeis.org

0, -7, 31, -86, 182, -333, 549, -844, 1228, -1715, 2315, -3042, 3906, -4921, 6097, -7448, 8984, -10719, 12663, -14830, 17230, -19877, 22781, -25956, 29412, -33163, 37219, -41594, 46298, -51345, 56745, -62512, 68656, -75191, 82127, -89478, 97254, -105469, 114133, -123260, 132860

Offset: 1

Bruno Berselli, May 29 2018

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A016755.
Cf. A000290: k such that 4*k is a square.
Cf. A002378: k such that 4*k+1 is a square.
Cf. A033431: k such that 4*k is a nonnegative cube.
Cf. A305291: k such that 4*k+3 is a cube.
Cf. A141046: k such that 4*k is a fourth power.
Cf. 4*A219086: k such that 4*k+1 is a fourth power.

seq(coeff(series(x^2*(-7+10*x-7*x^2)/((1-x)*(1+x)^4), x,50),x,n),n=1..45); # Muniru A Asiru, May 31 2018

LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -7, 31, -86, 182}, 45] (* Jean-François Alcover, Jun 04 2018 *)

concat(0, Vec(-x^2*(7 - 10*x + 7*x^2) / ((1 - x)*(1 + x)^4) + O(x^40))) \\ Colin Barker, Jun 04 2018

G.f.: x^2*(-7 + 10*x - 7*x^2)/((1 - x)*(1 + x)^4).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = (-1 - A016755(n-1)*(-1)^n)/4.
a(n) + a(-n) = (-1)^n*2^((1-(-1)^n)/2).
(n - 2)*(4*n^2 - 16*n + 19)*a(n) + (12*n^2 - 36*n + 31)*a(n-1) - (n - 1)*(4*n^2 - 8*n + 7)*a(n-2) = 0.
From Colin Barker, May 30 2018: (Start)
a(n) = n*(4*n^2 + 6*n + 3)/2 for n even.
a(n) = -(n + 1)*(4*n^2 + 2*n + 1)/2 for n odd.
(End)

cp's OEIS Frontend

A016744 a(n) = (2*n)^4.

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A171607 Expressible as A*B^A in a nontrivial way.

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A211412 a(n) = 4*n^4 + 1.

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A211996 Number of ordered pairs (i,j) such that i*j=n and i+j is a square.

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A377732 Numbers k such that max{d|k, d <= sqrt(k)} + min{d|k, d >= sqrt(k)} is a square.

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A227855 Numbers of the form x^4 + 4*y^4.

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A244730 a(n) = 2*n^4.

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