A002523 -id:A002523 - cp's OEIS frontend

A123665 a(n) = Sum_{k=1..21} n^A001358(k).

22, 471260364628084305, 6457022669043550542502557676, 105149403852520725445003265581519105, 41911381174488637014293971538580334000626

Offset: 1

Jonathan Vos Post, Oct 04 2006

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

Cf. A001358, A002523, A091940, A123177, A123656-A123665.

[1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 +
n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 + n^55 + n^57 + n^58: n in [1..50]]; // G. C. Greubel, Oct 26 2017

Table[1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 +
  n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 +
  n^55 + n^57 + n^58, {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

for(n=1,50, print1(1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 + n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 + n^55 + n^57 + n^58, ", ")) \\ G. C. Greubel, Oct 26 2017

a(n) = 1 +n^4 +n^6 +n^9 +n^10 +n^14 +n^15 +n^21 +n^22 +n^25 +n^26 + n^33 +n^34 +n^35 +n^38 +n^39 +n^46 +n^49 +n^51 +n^55 +n^57 +n^58.

Better name from Joerg Arndt, May 23 2021

A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 3, 2, 6, 2, 4, 3, 3, 2, 2, 2, 4, 3, 3, 2, 4, 6, 3, 2, 2, 4, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 5, 2, 3, 2, 4, 4, 4, 3, 6, 2, 5, 2, 2, 2, 5, 2, 5, 4, 4, 3, 4, 3, 5, 4, 2, 3, 4, 2, 4

Offset: 0

Jonathan Vos Post, Aug 10 2011

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1, and as A060890(n) = n^8+1 is to A002522(n) = n^2 + 1. a(n) = 1 when n^8+1 is prime, iff n is in {1, 2, 4} unless there is a larger Fermat prime than 65537.

			a(10) = 2 because 10^8 + 1 = 100000001 = 17 * 5882353 has 2 prime factors.
a(40) = 6 because 40^8 + 1 = 6553600000001 = 17^2 * 113 * 337 * 641 * 929 has 6 prime factors (with multiplicity) and is the smallest example not squarefree.

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

Cf. A001222, A060890, A002523, A193432, A193562, A193929.

[0] cat [&+[p[2]: p in Factorization(n^8+1)]:n in [1..90]]; // Marius A. Burtea, Feb 09 2020

Join[{0}, Table[Total[Transpose[FactorInteger[n^8 + 1]][[2]]], {n, 50}]]
PrimeOmega[Range[0,90]^8+1] (* Harvey P. Dale, May 27 2018 *)

a(n) = bigomega(n^8+1); \\ Michel Marcus, Feb 09 2020

a(n) = A001222(A060890(n)) = bigomega(n^8+1) or Omega(n^8+1)

A217796 Primes of the form n^4+1 such that (n+2)^4+1 is also prime.

Original entry on oeis.org

17, 257, 4477457, 8503057, 40960001, 59969537, 384160001, 5802782977, 58594980097, 94197431057, 102627966737, 114733948177, 283982410001, 330123790097, 381671897617, 405519334417, 691798081537, 741637881857, 1700843738897, 1749006250001, 2073600000001

Offset: 1

Michel Lagneau, Oct 12 2012

nonn

The corresponding n are in A217795.

			257 is in the sequence because  4^4+1 = 257 and (4+2)^4+1 = 1297 are both prime.

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

Cf. A000068, A002523, A037896, A217795.

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^4+1):else fi:od:

lst={}; Do[p=n^4+1; q=(n+2)^4+1;If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, p]], {n, 0, 3500}];lst
Select[Partition[Table[n^4+1,{n,1300}],3,1],AllTrue[{#[[1]],#[[3]]}, PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 17 2020 *)

A235982 Numbers n of the form p^4 + 1 (for prime p) such that n^4 + 1 is also prime.

Original entry on oeis.org

82, 38950082, 47458322, 131079602, 1982119442, 25856961602, 58120048562, 602425897922, 1053022816562, 1267247769842, 3491998578722, 7181161893362, 7759350084722, 10756569837842, 16948379819282, 28424689653362, 33122338550402, 36562351115762, 50897394646082

Offset: 1

Derek Orr, Jan 17 2014

nonn

All numbers are congruent to 2 mod 20.

			10756569837842 = 1811^4 + 1 (1811 is prime) and 10756569837842^4 + 1 is prime, so 10756569837842 is a member of this sequence.

Cf. A002523, A000068.

nfp4Q[n_]:=Module[{p=Surd[n-1,4]},AllTrue[{p,n^4+1},PrimeQ]]; Select[ Range[ 2700]^4+ 1,nfp4Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 08 2019 *)

import sympy
from sympy import isprime
{print(n**4+1) for n in range(10000) if isprime(n) if isprime((n**4+1)**4+1)}

A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 1, 3, 3, 1, 0, 0, 2, 4, 4, 4, 1, 0, 1, 2, 5, 9, 5, 5, 1, 0, 0, 3, 6, 10, 16, 6, 6, 1, 0, 1, 1, 7, 12, 17, 25, 7, 7, 1, 0, 0, 2, 8, 13, 20, 26, 36, 8, 8, 1, 0, 1, 2, 9, 27, 21, 30, 37, 49, 9, 9, 1, 0, 0, 3, 10, 28, 64, 31, 42, 50, 64, 10, 10, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

For any n >= 0, the n-th row, k -> T(n, k), corresponds to a polynomial in k with coefficients in {0, 1}.

For any k > 1, the k-th column, n -> T(n, k), corresponds to sums of distinct powers of k.

			Array T(n, k) begins:
  n\k|  0  1   2   3   4    5    6    7    8    9    10    11    12
  ---+-------------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0     0     0     0
    1|  1  1   1   1   1    1    1    1    1    1     1     1     1
    2|  0  1   2   3   4    5    6    7    8    9    10    11    12
    3|  1  2   3   4   5    6    7    8    9   10    11    12    13
    4|  0  1   4   9  16   25   36   49   64   81   100   121   144
    5|  1  2   5  10  17   26   37   50   65   82   101   122   145
    6|  0  2   6  12  20   30   42   56   72   90   110   132   156
    7|  1  3   7  13  21   31   43   57   73   91   111   133   157
    8|  0  1   8  27  64  125  216  343  512  729  1000  1331  1728
    9|  1  2   9  28  65  126  217  344  513  730  1001  1332  1729
   10|  0  2  10  30  68  130  222  350  520  738  1010  1342  1740
   11|  1  3  11  31  69  131  223  351  521  739  1011  1343  1741
   12|  0  2  12  36  80  150  252  392  576  810  1100  1452  1872

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Index entries for sequences related to binary expansion of n

See A342707 for a similar sequence.

Cf. A000035, A000120, A000583, A000695, A001093, A002061, A002378, A002522, A002523, A005836, A007088, A011379, A027444, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A034262, A053698, A071568, A098547, A104258.

T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^e); v }

T(n, n) = A104258(n).
T(n, 0) = A000035(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A005836(n).
T(n, 4) = A000695(n).
T(n, 5) = A033042(n).
T(n, 6) = A033043(n).
T(n, 7) = A033044(n).
T(n, 8) = A033045(n).
T(n, 9) = A033046(n).
T(n, 10) = A007088(n).
T(n, 11) = A033047(n).
T(n, 12) = A033048(n).
T(n, 13) = A033049(n).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^2.
T(5, k) = k^2 + 1 = A002522(k).
T(6, k) = k^2 + k = A002378(k).
T(7, k) = k^2 + k + 1 = A002061(k).
T(8, k) = k^3.
T(9, k) = k^3 + 1 = A001093(k).
T(10, k) = k^3 + k = A034262(k).
T(11, k) = k^3 + k + 1 = A071568(k).
T(12, k) = k^3 + k^2 = A011379(k).
T(13, k) = k^3 + k^2 + 1 = A098547(k).
T(14, k) = k^3 + k^2 + k = A027444(k).
T(15, k) = k^3 + k^2 + k + 1 = A053698(k).
T(16, k) = k^4 = A000583(k).
T(17, k) = k^4 + 1 = A002523(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A349530 Least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2.

Original entry on oeis.org

1, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 25, 25, 25, 25, 25

Offset: 1

Zhi-Wei Sun, Nov 21 2021

nonn

Conjecture 1: Suppose that 5^a < sqrt(n) <= 5^(a+1). Then a(n) = 3*5^a if sqrt(n) <= sqrt(3)*5^a, and a(n) = 5^(a+1) otherwise.
Conjecture 2: Let f(n) be the least positive integer m such that the n numbers 18k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2. Then f(n) is the least power of 5 not smaller than sqrt(n), except for 25 < n <= 45 (and in this case f(n) = 19).
Conjecture 3: Let n be a positive integer not among 26, 27, 28, 626, 627, 628, 629, 630, and define D(n) as the least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m. If 5^a < n <= 3*5^a, then D(n) = 3*5^a. If 3*5^a < n <= 5^(a+1), then D(n) = 5^(a+1).
We have verified the above conjectures for n up to 10^5.

			a(2) = 3 since the two numbers 1*(1^4+1) = 2 and 2*(2^4+1) = 34 are distinct modulo 3^2, but they are congruent modulo each of 1^2 and 2^2.

Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
Quan-Hui Yang and Lilu Zhao, On a conjecture of Sun involving powers of three, arXiv:2111.02746 [math.NT], 2021.

Cf. A000351, A002523, A208643, A349459, A349537.

f[k_]:=f[k]=k*(k^4+1);
U[m_,n_]:=U[m,n]=Length[Union[Table[Mod[f[k],m^2],{k,1,n}]]]
tab={};s=1;Do[m=s;Label[bb];If[U[m,n]==n,s=m;tab=Append[tab,s];Goto[aa]];m=m+1;Goto[bb];Label[aa],{n,1,80}];Print[tab]

A088666 a(n) = (n^4 + 1) mod 10.

Original entry on oeis.org

1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2, 1, 2, 7, 2, 7, 6, 7, 2, 7, 2

Offset: 0

Cino Hilliard, Nov 23 2003

Periodic, with period 10. - Harvey P. Dale, Jan 31 2015

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A010879 (n mod 10), A002523 (n^4+1).

[(n^4+1) mod 10: n in [0..100]]; // Vincenzo Librandi, Feb 01 2015

PowerMod[Range[0,90],4,10]+1 (* or *) PadRight[{},100,{1,2,7,2,7,6,7,2,7,2}] (* Harvey P. Dale, Jan 31 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 2, 7, 2, 7, 6, 7, 2, 7, 2},90] (* Ray Chandler, Aug 26 2015 *)

a(n) = A010879(A002523(n)). - Michel Marcus, Feb 01 2015

Edited by Don Reble, May 03 2006

Name clarified by Vincenzo Librandi, Feb 01 2015

A100606 a(n) = n^4 + n^3 + n.

Original entry on oeis.org

0, 3, 26, 111, 324, 755, 1518, 2751, 4616, 7299, 11010, 15983, 22476, 30771, 41174, 54015, 69648, 88451, 110826, 137199, 168020, 203763, 244926, 292031, 345624, 406275, 474578, 551151, 636636, 731699, 837030, 953343, 1081376, 1221891, 1375674, 1543535, 1726308

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 30 2004

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, A002523, A027445, A053699, A059826, A071253, A091940, A100019.

[n^4+n^3+n: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

Table[n^4+n^3+n,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,3,26,111,324},40] (* Harvey P. Dale, Apr 25 2015 *)

a(n)=n^4+n^3+n \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=3, a(2)=26, a(3)=111, a(4)=324. - Harvey P. Dale, Apr 25 2015
From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: x*(3 + 11*x + 11*x^2 - x^3)/(1-x)^5.
E.g.f.: x*(3 + 10*x + 7*x^2 + x^3)*exp(x). (End)

A123658 a(n) = 1 + n^4 + n^6 + n^9 + n^10.

Original entry on oeis.org

5, 1617, 79543, 1315073, 11735001, 70591825, 322948907, 1208225793, 3874742893, 11001010001, 28297158495, 67080151297, 148467846593, 309923269713, 615105191251, 1168247947265, 2134605998037, 3768860634193, 6453801131783, 10752064160001, 17474246985385

Offset: 1

Jonathan Vos Post, Oct 04 2006

			a(40) = 1+40^(A001358(1))+40^(A001358(2))+40^(A001358(3))+40^(A001358(4)) = 1+40^4+40^6+40^9+40^10 = 10747908098560001.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

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

[1+n^4+n^6+n^9+n^10: n in [0..50]]; // G. C. Greubel, Oct 17 2017

Table[1+n^4+n^6+n^9+n^10, {n,1,50}] (* G. C. Greubel, Oct 17 2017 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{5,1617,79543,1315073,11735001,70591825,322948907,1208225793,3874742893,11001010001,28297158495},30] (* Harvey P. Dale, Jul 11 2025 *)

a(n)=1+n^4+n^6+n^9+n^10 \\ Charles R Greathouse IV, Oct 07 2015

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

G.f.: x*(x^10 -8*x^9 +615*x^8 +33654*x^7 +381288*x^6 +1242534*x^5 +1378908*x^4 +528210*x^3 +62031*x^2 +1562*x +5)/(1-x)^11. - Colin Barker, May 27 2012

A260534 Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 7, 2, 1, 1, 1, 6, 1, 13, 5, 3, 1, 1, 1, 7, 1, 21, 10, 11, 1, 1, 1, 1, 8, 1, 31, 17, 31, 1, 4, 1, 1, 1, 9, 1, 43, 26, 69, 1, 23, 3, 1, 1, 1, 10, 1, 57, 37, 131, 1, 94, 21, 5, 1, 1, 1, 11

Offset: 0

Peter Luschny, Sep 20 2015

A parametrization of Stern's diatomic series (which is here T(1,k)). (For other generalizations of Dijkstra's fusc function see the Luschny link.)

			Array starts:
n\k[0, 1,  2, 3,  4,  5,   6, 7,    8,    9,    10]
[0] 1, 1,  1, 1,  1,  1,   1, 1,    1,    1,     1, ...
[1] 1, 1,  2, 1,  3,  2,   3, 1,    4,    3,     5, ... [A002487]
[2] 1, 1,  3, 1,  7,  5,  11, 1,   23,   21,    59, ... [A101624]
[3] 1, 1,  4, 1, 13, 10,  31, 1,   94,   91,   355, ...
[4] 1, 1,  5, 1, 21, 17,  69, 1,  277,  273,  1349, ... [A101625]
[5] 1, 1,  6, 1, 31, 26, 131, 1,  656,  651,  3881, ...
[6] 1, 1,  7, 1, 43, 37, 223, 1, 1339, 1333,  9295, ...
[7] 1, 1,  8, 1, 57, 50, 351, 1, 2458, 2451, 19559, ...
[8] 1, 1,  9, 1, 73, 65, 521, 1, 4169, 4161, 37385, ...
-,-,-,-,A002061,A002522,A071568,-,-,A059826,-,A002523,

Chai Wah Wu, Table of n, a(n) for n = 0..10010
Peter Luschny, Rational Trees and Binary Partitions.

Cf. A002061, A002487, A002522, A002523, A059826, A071568, A101624, A101625.

T := (n,k) -> add(modp(binomial(k-j,j),2)*n^j, j=0..k):
seq(lprint(seq(T(n,k),k=0..10)),n=0..5);

Table[If[(n - k) == 0, 1, Sum[(n - k)^j Mod[Binomial[k - j, j], 2], {j, 0, k}]], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, Sep 21 2015 *)

def A260534_T(n,k):
    return sum(0 if ~(k-j) & j else n**j for j in range(k+1)) # Chai Wah Wu, Feb 08 2016

cp's OEIS Frontend

A123665 a(n) = Sum_{k=1..21} n^A001358(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A217796 Primes of the form n^4+1 such that (n+2)^4+1 is also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A235982 Numbers n of the form p^4 + 1 (for prime p) such that n^4 + 1 is also prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A349530 Least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A088666 a(n) = (n^4 + 1) mod 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions