A002523 -id:A002523 - cp's OEIS frontend

A057781 a(n) = n^4+4 = (n^2-2n+2)(n^2+2n+2) = ((n-1)^2+1)((n+1)^2+1).

4, 5, 20, 85, 260, 629, 1300, 2405, 4100, 6565, 10004, 14645, 20740, 28565, 38420, 50629, 65540, 83525, 104980, 130325, 160004, 194485, 234260, 279845, 331780, 390629, 456980, 531445, 614660, 707285, 810004, 923525, 1048580, 1185925

Offset: 0

Henry Bottomley, Nov 04 2000

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, 1997, Vol. 1, exercise 1.2.1, Nr. 11, p. 19. [From Reinhard Zumkeller, Apr 11 2010]

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. A016755, A033999, A053755.

Cf. A000583, A002522, A002523, A272298.

[n^4+4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[n^4+4,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{4,5,20,85,260},40] (* Harvey P. Dale, Aug 20 2020 *)

first(m)=vector(m,i,i--;i^4+4) \\ Anders Hellström, Sep 15 2015

G.f.: -(5*x^4-5*x^3+35*x^2-15*x+4) / (x-1)^5. - Colin Barker, Mar 29 2013
a(n) = A002523(n) + 3.
a(n) = A002522(n-1) * A002522(n+1).
Sum_{k=0..n} A033999(k)*A016755(k)/a(k) = A033999(n)*(n+1)/A053755(n+1), see Knuth reference. - Reinhard Zumkeller, Apr 11 2010
a(n) = (n^2)^2 + 2^2 = (n^2-2)^2 + (2*n)^2. - Thomas Ordowski, Sep 15 2015
a(n) = A272298(3*n)/3^4. - Bruno Berselli, Apr 29 2016
Sum_{n>=0} 1/a(n) = (Pi*coth(Pi) + 1)/8. - Amiram Eldar, Oct 04 2021

A123659 a(n) = 1 + n^4 + n^6 + n^9 + n^10 + n^14.

Original entry on oeis.org

6, 18001, 4862512, 269750529, 6115250626, 78434755921, 678546021756, 4399254736897, 22880667197854, 100011001010001, 379778130741736, 1283985544700161, 3937524853545882, 11112316748827729, 29193541130581876

Offset: 1

Jonathan Vos Post, Oct 05 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001358, A002523, A091940, A123177, A123656-A123659.

[1 + n^4 + n^6 + n^9 + n^10 + n^14: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[1 + n^4 + n^6 + n^9 + n^10 + n^14, {n, 1, 50}] (* G. C. Greubel, Oct 17 2017 *)

for(n=1,25, print1(1 + n^4 + n^6 + n^9 + n^10 + n^14, ", ")) \\ G. C. Greubel, Oct 17 2017

a(n) = 1 + n^4 + n^6 + n^9 + n^10 + n^14.

A113679 Expansion of (1-x-2x^2)/(1-x).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Nov 04 2005

From Gary W. Adamson, Mar 06 2012: (Start)
Signed (1, 0, -2, 2, -2, 2, ...) and convolved with the Toothpick sequence A139250 = A151548: (1, 3, 5, 7, 5, 11, ...).  The inverse of (1, 0, -2, 2, -2, ...) = A151575: (1, 0, 2, -2, 6, -10, 22, ...).
The unsigned sequence convolved with:
(1, 2, 3, ...) = A002523, (n^2 + 1). Convolved with:
(A001045) = .... A097064: (1, 1, 5, 9, 21, 41, ...).
(End)

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A113680.

Cf. A139250, A151548, A151575, A002523. - Gary W. Adamson, Mar 06 2012

CoefficientList[Series[(1-x-2x^2)/(1-x),{x,0,80}],x] (* or *) Join[{1,0}, PadRight[{},80,-2]] (* Harvey P. Dale, Mar 05 2012 *)

a(n) = C(0, n) + 2*C(1, n) - 2.

A193562 Number of divisors of n^4+1.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 4, 4, 8, 4, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 4, 4, 4, 2, 8, 4, 8, 2, 4, 4, 8, 4, 8, 2, 4, 4, 8, 4, 4, 4, 8, 4, 16, 8, 8, 2, 8, 2, 8, 4, 8, 4, 8, 2, 8, 2, 4, 4, 16, 8, 4, 4, 8, 8, 4, 8, 8, 4, 8, 8, 4, 4, 4, 2, 8, 8, 16, 4, 16, 2, 4, 2, 16, 4

Offset: 0

Jonathan Vos Post, Aug 09 2011

This is to n^4+1 as A193432 is to n^2+1.

a(n) = 2 when n^4+1 is prime, iff n is in A037896.

			a(3) = 4 because 3^4+1 = 82, whose 4 factors are {1, 2, 41, 82}.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A002523, A037896, A193432 (number of divisors of n^2+1).

[NumberOfDivisors(n^4+1):n in [0..90]]; // Marius A. Burtea, Feb 09 2020

DivisorSigma[0,Range[0,90]^4+1] (* Harvey P. Dale, May 05 2013 *)

a(n) = numdiv(n^4+1); \\ Michel Marcus, Feb 09 2020

a(n) = A000005(A002523(n)) = d(n^4+1) (also called tau(n^4+1) or sigma_0(n^4+1)), the number of divisors of n^4+1.

A193929 Number of prime factors of n^4 + 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 2, 3, 2, 4, 3, 3, 1, 3, 1, 3, 2, 3, 2, 3, 1, 3, 1, 2, 2, 4, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 1, 3, 3, 4, 2, 4, 1, 2, 1, 4, 2, 4, 2, 3, 1, 3, 1, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 1, 3, 2, 3, 3, 4, 2, 2, 2, 2, 2, 3, 1

Offset: 0

Jonathan Vos Post, Aug 09 2011

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1. a(n) = 2 when n^4+1 is prime, iff n is in A037896.

			a(9) = 3 because 9^4+1 = 6562 = 2 * 17 * 193, which has 3 prime factors, counted with multiplicity

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A002523, A193432, A193562.

[0] cat [&+[p[2]: p in Factorization(n^4+1)]:n in [1..120]]; // Marius A. Burtea, Feb 09 2020

Join[{0}, Table[Total[Transpose[FactorInteger[n^4 + 1]][[2]]], {n, 100}]] (* T. D. Noe, Aug 10 2011 *)
Join[{0},Table[PrimeOmega[n^4+1],{n,120}]] (* Harvey P. Dale, Sep 25 2012 *)

a(n) = bigomega(n^4+1); \\ Michel Marcus, Feb 09 2020

a(n) = A001222(A002523(n)) = bigomega(n^4+1) or Omega(n^4+1).

A217795 Numbers n such that n^4+1 and (n+2)^4+1 are both prime.

Original entry on oeis.org

2, 4, 46, 54, 80, 88, 140, 276, 492, 554, 566, 582, 730, 758, 786, 798, 912, 928, 1142, 1150, 1200, 1236, 1404, 1540, 1552, 1610, 1644, 1650, 1932, 1942, 2044, 2102, 2204, 2222, 2224, 2238, 2254, 2374, 2436, 2486, 2510, 2640, 2674, 2698, 2732, 2734, 3244, 3286

Offset: 1

Michel Lagneau, Oct 12 2012

			4 is in the sequence because 4^4+1 = 257 and 6^4+1 = 1297 are both prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000068, A002523, A037896.

[n: n in [0..3300] | IsPrime(n^4 + 1) and IsPrime((n + 2)^4 + 1)]; // Vincenzo Librandi, Oct 13 2012

for n from 0 by 2 to 3500 do: if type(n^4+1,prime)=true and type((n+2)^4+1,prime)=true then printf(`%d, `,n):else fi:od:

lst={}; Do[p=n^4+1; q=(n+2)^4+1;If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, n]], {n, 0, 3000}];lst
Select[Range[3500],AllTrue[{#^4+1,(#+2)^4+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 29 2015 *)

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A258806 a(n) = n^7 + 1.

Original entry on oeis.org

1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 10000001, 19487172, 35831809, 62748518, 105413505, 170859376, 268435457, 410338674, 612220033, 893871740, 1280000001, 1801088542, 2494357889, 3404825448, 4586471425, 6103515626, 8031810177

Offset: 0

Vincenzo Librandi, Jun 11 2015

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Subsequence of A004864.
Sequences of the type n^k+1: A002522 (k=2), A001093 (k=3), A002523 (k=4), A002561 (k=5), A002604 (k=6), this sequence (k=7), A060890 (k=8).
Cf. A300785.

List([0..30],n->n^7+1); # Muniru A Asiru, Oct 24 2018

Magma
```
[n^7+1: n in [0..40]];
```

I:=[1,2,129,2188, 16385,78126,279937,823544]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6) + 8*Self(n-7)-Self(n-8): n in [1..40]];

seq(n^7+1,n=0..30); # Muniru A Asiru, Oct 24 2018

Table[n^7 + 1, {n, 0, 40}] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 2, 129, 2188, 16385, 78126, 279937, 823544}, 40]

a(n)=n^7+1 \\ Charles R Greathouse IV, Jun 11 2015

[n^7+1 for n in (1..40)] # Bruno Berselli, Jun 11 2015

G.f.: (1 - 6*x + 141*x^2 + 1156*x^3 + 2451*x^4 + 1170*x^5 + 127*x^6)/(1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) = (n + 1)*(n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
a(n) = Sum_{k=0..n} A300785(n,k). - Kolosov Petro, Oct 23 2018
E.g.f.: (1 +x +63*x^2 +301*x^3 +350*x^4 +140*x^5 +*21*x^6 +x^7)*exp(x). - G. C. Greubel, Oct 24 2018

A354051 Decimal expansion of Sum_{k>=0} 1 / (k^4 + 1).

Original entry on oeis.org

1, 5, 7, 8, 4, 7, 7, 5, 7, 9, 6, 6, 7, 1, 3, 6, 8, 3, 8, 3, 1, 8, 0, 2, 2, 1, 9, 3, 2, 4, 5, 7, 1, 9, 2, 3, 5, 0, 4, 6, 6, 7, 2, 2, 1, 7, 3, 2, 7, 2, 9, 1, 3, 2, 7, 5, 8, 7, 4, 8, 6, 6, 4, 5, 7, 9, 3, 8, 0, 8, 4, 4, 8, 0, 6, 1, 6, 8, 1, 1, 1, 7, 4, 5, 7, 3, 1, 9, 4, 3, 5, 4, 1, 6, 6, 6, 2, 8, 6, 3, 8, 3, 1, 6, 6

Offset: 1

Vaclav Kotesovec, May 16 2022, following a suggestion from Bernard Schott

Apart from leading digits the same as A256920. - R. J. Mathar, May 20 2022

			1.578477579667136838318022193245719235046672217327291327587486645793808...

Cf. A256920, A113319, A354052, A354053.

Cf. A002523, A255434, A354004.

evalf(1/2 + (Pi*(sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi))) / (2*sqrt(2)*(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))), 105);

RealDigits[Chop[N[Sum[1/(k^4 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^4 + 1))
```

Equals 1/2 + (Pi*(sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi))) / (2*sqrt(2)*(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))).

A123657 a(n) = 1 + n^4 + n^6 + n^9.

Original entry on oeis.org

4, 593, 20494, 266497, 1969376, 10125649, 40473658, 134483969, 387958492, 1001010001, 2359733894, 5162787073, 10609354744, 20668614737, 38454800626, 68736319489, 118612097588, 198393407569, 322734873982, 512064160001

Offset: 1

Jonathan Vos Post, Oct 04 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001358, A002523, A091940, A123177, A123656-A123659.

[1 + n^4 + n^6 + n^9: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[1 + n^4 + n^6 + n^9, {n, 1, 50}] (* G. C. Greubel, Oct 17 2017 *)

a(n)=n^9+n^6+n^4+1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 1 + n^4 + n^6 + n^9 = 1001010001 (base n).

G.f.: -x*(x^9 -8*x^8 -406*x^7 -14592*x^6 -88496*x^5 -156316*x^4 -87762*x^3 -14744*x^2 -553*x -4)/(x-1)^10. - Colin Barker, May 27 2012

cp's OEIS Frontend

A057781 a(n) = n^4+4 = (n^2-2*n+2)*(n^2+2*n+2) = ((n-1)^2+1)*((n+1)^2+1).

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A123659 a(n) = 1 + n^4 + n^6 + n^9 + n^10 + n^14.

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A113679 Expansion of (1-x-2x^2)/(1-x).

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A193562 Number of divisors of n^4+1.

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A193929 Number of prime factors of n^4 + 1, counted with multiplicity.

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A217795 Numbers n such that n^4+1 and (n+2)^4+1 are both prime.

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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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A057781 a(n) = n^4+4 = (n^2-2n+2)(n^2+2n+2) = ((n-1)^2+1)((n+1)^2+1).