A002646 -id:A002646 - cp's OEIS frontend

A000583 Fourth powers: a(n) = n^4.

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921

Offset: 0

N. J. A. Sloane

Figurate numbers based on 4-dimensional regular convex polytope called the 4-measure polytope, 4-hypercube or tesseract with Schlaefli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
Totally multiplicative sequence with a(p) = p^4 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A058649. The inverse binomial transforms yields the (finite) 0, 1, 14, 36, 24, the 4th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Generate Pythagorean triangles with parameters a and b to get sides of lengths x = b^2-a^2, y = 2*a*b, and z = a^2 + b^2. In particular use a=n-1 and b=n for a triangle with sides (x1,y1,z1) and a=n and b=n+1 for another triangle with sides (x2,y2,z2). Then x1*x2 + y1*y2 + z1*z2 = 8*a(n). - J. M. Bergot, Jul 22 2013
For n > 0, a(n) is the largest integer k such that k^4 + n is a multiple of k + n. Also, for n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n^2. - Derek Orr, Sep 04 2014
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
a(n+2)/2 is the area of a trapezoid with vertices at (T(n), T(n+1)), (T(n+1), T(n)), (T(n+1), T(n+2)), and (T(n+2), T(n+1)) with T(n)=A000292(n) for n >= 0. - J. M. Bergot, Feb 16 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
Dov Juzuk, Curiosa 56: An interesting observation, Scripta Mathematica 6 (1939), 218.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 47.

T. D. Noe, Table of n, a(n) for n = 0..1000
Henry Bottomley, Illustration of initial terms
Henry Bottomley, Some Smarandache-type multiplicative sequences
Ralph Greenberg, Math for Poets.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
Eric Weisstein's World of Mathematics, Biquadratic Number.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index entries for sequences related to Benford's law

Cf. A000538, A005917 (first differences), A000332, A014820, A092181, A092182, A092183.
Cf. A004831, A002646.
Cf. A002593, A260810. - Bruno Berselli, Jul 31 2015
Cf. A002415, A000290, A006008, A132366, A039623, A139584, A071270, A047928, A187756.
Cf. A062392, A231303, A267315.

a000583 = (^ 4)
a000583_list = scanl (+) 0 a005917_list
-- Reinhard Zumkeller, Nov 13 2014, Nov 11 2012

[n^4 : n in [0..50]]; // Wesley Ivan Hurt, Sep 05 2014

A000583 := n->n^4: seq(A000583(n), n=0..50);
A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); # Zerinvary Lajos, May 25 2008

Range[0,100]^4 (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)

makelist(n^4,n,0,30); /* Martin Ettl, Nov 12 2012 */

A000583(n) = n^4 \\ Michael B. Porter, Nov 09 2009

def a(n): return n**4
print([a(n) for n in range(34)]) # Michael S. Branicky, Nov 10 2022

a(n) = A123865(n)+1 = A002523(n)-1.
Multiplicative with a(p^e) = p^(4e). - David W. Wilson, Aug 01 2001
G.f.: x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. More generally, g.f. for n^m is Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292).
Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: (x + 7*x^2 + 6*x^3 + x^4)*e^x. More generally, the general form for the e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial of order n. - Franklin T. Adams-Watters, Sep 11 2005
Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - Jaume Oliver Lafont, Sep 20 2009
a(n) = C(n+3,4) + 11*C(n+2,4) + 11*C(n+1,4) + C(n,4). [Worpitzky's identity for powers of 4. See, e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - Bruno Berselli, May 07 2010
a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - Charlie Marion, Jun 13 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - Ant King, Sep 23 2013
From Amiram Eldar, Jan 20 2021: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).
Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)

A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

Original entry on oeis.org

2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177, 4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561, 28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161, 66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, 121937

Offset: 1

N. J. A. Sloane

The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144 + 1 = (145310^65536)^4 + 1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011

Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017

			a(1) =   2 = 1^4 + 1^4.
a(2) =  17 = 1^4 + 2^4.
a(3) =  97 = 2^4 + 3^4.
a(4) = 257 = 1^4 + 4^4.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 22-8 Mu Alpha Theta 1984.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Subsequence of A002313 and of A028916.
Intersection of A004831 and A000040.
Cf. A002646, A000583, A003336, A000290, A256852.

a002645 n = a002645_list !! (n-1)
a002645_list = 2 : (map a000040 $ filter ((> 1) . a256852) [1..])
-- Reinhard Zumkeller, Apr 11 2015

nn = 100000; Sort[Reap[Do[n = a^4 + b^4; If[n <= nn && PrimeQ[n], Sow[n]], {a, nn^(1/4)}, {b, a}]][[2, 1]]]
With[{nn=20},Select[Union[Flatten[Table[x^4+y^4,{x,nn},{y,nn}]]],PrimeQ[ #] && #<=nn^4+1&]] (* Harvey P. Dale, Aug 10 2021 *)

upto(lim)=my(v=List(2),t);forstep(x=1,lim^.25,2,forstep(y=2,(lim-x^4)^.25,2,if(isprime(t=x^4+y^4),listput(v,t))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 05 2011

list(lim)=my(v=List([2]),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; forstep(y=1+x%2,min(sqrtnint(lim-x4,4), x-1),2, if(isprime(t=x4+y^4), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017

A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006

A256852(A049084(a(n))) > 1 for n > 1. - Reinhard Zumkeller, Apr 11 2015

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002

A282867 Primes of the form x^2 + y^2 with x > y such that x^2 - y^2 is a square and x^4 + y^4 is a prime.

Original entry on oeis.org

41, 313, 3593, 4481, 32633, 42961, 66361, 67073, 165233, 198593, 237161, 266921, 378953, 462073, 465041, 487073, 559001, 594161, 750353, 757633, 815401, 1157033, 1414081, 1416161, 1687393, 2439881, 2793481, 2866121, 2947561, 3344161, 3577913, 3759713, 4295281, 4617073, 4795481, 5654641

Offset: 1

Thomas Ordowski and Altug Alkan, Feb 23 2017

nonn

Primes of the form (u^4 + v^4)/2 with u and v odd and (u^8 + 6*u^4*v^4 + v^8)/8 prime. - Robert Israel, Feb 24 2017

			For prime 41 = 5^2 + 4^2 is 5^2 - 4^2 = 3^2 and 5^4 + 4^4 = 881 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A002646.

Cf. A002144, A002645.

N:= 10^7: # to get all terms <= N Res:= {}:
for w from 1 to floor((2*N)^(1/4)) by 2 do
  for u from 1 to min(w-1, floor((2*N-w^4)^(1/4))) by 2 do
    p:= (u^4 + w^4)/2;
    if isprime(p) and isprime((u^8 + 6*u^4*w^4 + w^8)/8) then
      Res:= Res union {p}
    fi;
od od:
sort(convert(Res,list)); # Robert Israel, Feb 24 2017

Select[Total[#^2]&/@Select[Subsets[Range[3000],{2}],IntegerQ[Sqrt[#[[2]]^2-#[[1]]^2]] && PrimeQ[ Total[#^4]]&],PrimeQ]//Union (* Harvey P. Dale, Jul 23 2024 *)

a(n) == 1 (mod 8).

a(n) == 1 or 33 (mod 40).

A290780 Half-octavan primes: primes of the form (x^8 + y^8)/2.

Original entry on oeis.org

198593, 21523361, 107182721, 407865361, 429388721, 3487882001, 11979660241, 39155495921, 84785726833, 141217650641, 141321947681, 250123401793, 253611085201, 289278699121, 391337974721, 426445714033, 426448401121

Offset: 1

Charles R Greathouse IV, Aug 20 2017

nonn

			a(1) = (5^8 + 3^8)/2 = 198593.
a(2) = (9^8 + 1^8)/2 = 21523361.
a(3) = (11^8 + 3^8)/2 = 107182721.
a(4) = (13^8 + 1^8)/2 = 407865361.
a(5) = (13^8 + 9^8)/2 = 429388721.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.

Cf. A006686, A002646.

N:= 10^12: # to get all terms <= N
sort(convert(select(isprime, {seq(seq((x^8+y^8)/2, y= (x mod 2)..min(x,floor((2*N-x^8)^(1/8))),2),x=1..floor((2*N)^(1/8)))}),list)); # Robert Israel, Aug 21 2017

Sort[Select[Total/@(Union[Sort/@Tuples[Range[0, 50], 2]]^8)/2, PrimeQ]] (* or *) lst={};Do[If[PrimeQ[(a^8 + b^8) / 2], AppendTo[lst, (a^8 + b^8) / 2]], {a, 100}, {b, a, 100}]; Sort[lst] (* Vincenzo Librandi, Aug 21 2017 *)

list(lim)=my(v=List(),x8,t); forstep(x=1,sqrtnint(lim\=1,8),2, x8=x^8; forstep(y=1,min(sqrtnint(lim-x8,8), x-1),2, if(isprime(t=(x8+y^8)/2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017

cp's OEIS Frontend

A000583 Fourth powers: a(n) = n^4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Python

Formula

A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A282867 Primes of the form x^2 + y^2 with x > y such that x^2 - y^2 is a square and x^4 + y^4 is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A290780 Half-octavan primes: primes of the form (x^8 + y^8)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI