cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Eric Chen, Apr 22 2015

Keywords

Comments

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)).

Examples

			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
...

Links

Crossrefs

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).

Programs

  • Mathematica
    Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]
  • PARI
    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))

Formula

T(m, n) = Phi_m(n)

A243318 Numbers n such that n^10 - n^9 - n^8 - n^7 - n^6 - n^5 - n^4 - n^3 - n^2 - n - 1 is prime.

Original entry on oeis.org

15, 56, 65, 74, 87, 104, 132, 150, 171, 185, 186, 204, 225, 234, 360, 429, 449, 455, 459, 476, 485, 512, 675, 746, 770, 780, 795, 816, 836, 839, 840, 846, 857, 876, 902, 930, 939, 941, 944, 977, 1109, 1152, 1161, 1190, 1262, 1289, 1295, 1316, 1355, 1362, 1374, 1395, 1401, 1425
Offset: 1

Views

Author

Derek Orr, Jun 03 2014

Keywords

Examples

			15^10 - 15^9 - 15^8 - 15^7 - 15^6 - 15^5 - 15^4 - 15^3 - 15^2 - 15 - 1 = 535461077009 is prime. Thus 15 is a member of this sequence.

Links

Crossrefs

Cf. A162862.

Programs

  • Magma
    [n: n in [0..2000] | IsPrime(n^10-n^9-n^8-n^7-n^6- n^5-n^4-n^3-n^2-n-1)]; // Vincenzo Librandi, Dec 06 2016
  • Maple
    A243318:=n->`if`(isprime(n^10-add(n^i, i=0..9)), n, NULL): seq(A243318(n), n=1..2*10^3); # Wesley Ivan Hurt, Dec 05 2016
  • Mathematica
    Select[Range[2000], PrimeQ[#^10 - #^9 - #^8 - #^7 - #^6 - #^5 - #^4 - #^3 - #^2 - # - 1] &] (* Vincenzo Librandi, Dec 06 2016 *)
  • PARI
    for(n=1, 10^4, if(ispseudoprime(n^10-sum(i=0, 9, n^i)), print1(n, ", ")))
  • Python
    import sympy
from sympy import isprime
{print(n, end=', ') for n in range(10**4) if isprime(n**10-n**9-n**8-n**7-n**6-n**5-n**4-n**3-n**2-n-1)}
  • Python
    from sympy import isprime
A243318_list, m = [], [3628800, -16692480, 31651200, -31827600, 18163440, -5826240, 971232, -69720, 1362, -2, -1]
for n in range(1, 10**5+1):
    for i in range(10):
        m[i+1]+= m[i]
    if isprime(m[-1]):
        A243318_list.append(n) # Chai Wah Wu, Nov 06 2014

A261460 Numbers k such that k^11-1 is a semiprime.

Original entry on oeis.org

2, 20, 30, 60, 212, 224, 258, 272, 390, 398, 480, 504, 654, 770, 812, 1040, 1194, 1448, 1698, 1748, 1874, 2000, 2238, 2274, 2294, 2438, 2522, 2664, 2714, 2790, 2802, 3020, 3138, 3168, 3300, 3392, 3434, 3794, 4160, 4232, 4518, 4722, 4968, 5334, 5654, 5658
Offset: 1

Views

Author

Vincenzo Librandi, Aug 21 2015

Keywords

Comments

After 2, numbers k such that k-1 and k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 are both prime.
Intersection of A008864 and A162862. - Michel Marcus, Aug 21 2015

Examples

			20 is in sequence because 20^11-1 = 204799999999999 = 19*10778947368421, where 19 and 10778947368421 are both prime.

Crossrefs

Cf. similar sequences listed in A261435.
Cf. A105122.
Cf. A008864, A162862.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..4000] | IsSemiprime(s) where s is n^11- 1];
  • Mathematica
    Select[Range[6000], PrimeOmega[#^11 - 1] == 2 &]
  • PARI
    isok(n)=bigomega(n^11-1)==2 \\ Anders Hellström, Aug 21 2015

A308238 Nonprimes k such that k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Original entry on oeis.org

1, 20, 21, 30, 60, 86, 172, 195, 212, 224, 258, 268, 272, 319, 339, 355, 365, 366, 390, 398, 414, 480, 504, 534, 539, 543, 567, 592, 626, 654, 735, 756, 766, 770, 778, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1194, 1210, 1239, 1243, 1264, 1309, 1311
Offset: 1

Views

Author

Bernard Schott, May 16 2019

Keywords

Comments

A240693 Union {this sequence} = A162862.
The corresponding prime numbers, (11111111111)_k, are Brazilian primes and belong to A085104 and A285017 (except 11).

Examples

			(11111111111)_20 = (20^11 - 1)/19 = 10778947368421 is prime, thus 20 is a term.

Crossrefs

Cf. A162861, A162862, A240693, A286301.
Intersection of A064108 and A285017.
Similar to A182253 for k^2+k+1, A286094 for k^4+k^3+k^2+k+1, A288939 for k^6+k^5+k^4+k^3+k^2+k+1.

Programs

  • Magma
    [1] cat [n:n in [2..1500]|not IsPrime(n) and IsPrime(Floor((n^11-1)/(n-1)))]; // Marius A. Burtea, May 16 2019
  • Maple
    filter:= n -> not isprime(n) and isprime((n^11-1)/(n-1)) : select(filter, [$2..5000]);
  • Mathematica
    Select[Range@ 1320, And[! PrimeQ@ #, PrimeQ@ Total[#^Range[0, 10]]] &] (* Michael De Vlieger, Jun 09 2019 *)
  • PARI
    isok(n) = !isprime(n) && isprime(polcyclo(11, n)); \\ Michel Marcus, May 19 2019
Previous Showing 31-34 of 34 results.

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