A138940 -id:A138940 - cp's OEIS frontend

A138937 Indices k such that A019325(k)=Phi[k](7) is prime, where Phi is a cyclotomic polynomial.

5, 6, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531, 581, 650, 654, 730, 740, 759, 1026, 1047, 1065, 1460, 1660, 1699, 1959, 2067, 2260, 2380, 2665, 2890, 3238, 4020

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..95
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019325, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 7]] &]

for( i=1,999, ispseudoprime( polcyclo(i,7)) && print1( i",")) /* use ...subst(polcyclo(i),x,7)... in PARI < 2.4.2 */

a(40)-a(53) from Robert Price, Apr 18 2012

A138939 Indices k such that A019327(k)=Phi[k](9) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

6, 12, 18, 20, 30, 36, 54, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, 624, 720, 762, 1066, 1094, 1098, 1170, 1230, 1254, 1320, 1428, 1546, 2018, 2574, 2724, 2804, 2920, 3074, 3316, 3646, 4124, 4132, 4186, 4620, 4802

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..64
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019327, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 9]] &]

for( i=1,999, ispseudoprime( polcyclo(i,9)) && print1( i",")) /* use ...subst( polcyclo(i),x,9)... in PARI < 2.4.2 */

a(25)-a(46) from Robert Price, Apr 28 2012

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)

A138921 Indices k such that A020508(k)=Phi[k](-9) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 9, 12, 15, 20, 27, 33, 36, 59, 69, 91, 152, 207, 223, 232, 264, 336, 340, 380, 381, 492, 533, 540, 547, 549, 585, 615, 624, 627, 720, 773, 1009, 1287, 1320, 1428, 1537, 1823, 2093, 2401, 2724, 2733, 2804, 2920, 3316, 3803, 4124, 4132, 4620, 7143, 7520, 7708

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Most terms of this sequence are multiples of 3, exceptions are 20, 59, 91, 152, 223, 232, 340, 380, 533, 547, 773... corresponding to a(n) with n=5, 9, 11, 12, 14, 15, 18, 19, 22, 24, 31...

Robert Price, Table of n, a(n) for n = 1..68
Yves Gallot, Cyclotomic polynomials and prime numbers

Cf. A020508, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, -9]] &]

for( i=1,999, ispseudoprime( polcyclo(i,-9)) && print1( i",")) /* use ...subst(polcyclo(i),x,-9)... in PARI < 2.4.2 */

a(32)-a(51) by Robert Price, Mar 22 2012

A138923 Indices k such that A020506(k)=Phi[k](-7) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 9, 10, 15, 17, 23, 25, 26, 27, 28, 29, 36, 42, 47, 48, 61, 76, 84, 110, 126, 148, 210, 224, 262, 280, 288, 296, 298, 325, 327, 332, 352, 365, 456, 513, 528, 740, 1062, 1162, 1445, 1460, 1518, 1619, 1660, 2094, 2130, 2260, 2380, 3398, 3447, 3918

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

There are only 6 terms between 365 and 1445 (exclusive).

Robert Price, Table of n, a(n) for n = 1..97
Yves Gallot, Cyclotomic polynomials and prime numbers

Cf. A020506, A138920-A138940.

Select[ Range[ 2400], PrimeQ[ Cyclotomic[#, -7]] &] (* Robert G. Wilson v, Mar 25 2012 *)

for( i=1,999, ispseudoprime( polcyclo(i,-7)) && print1( i",")) /* use ...subst(polcyclo(i),x,-7)... in PARI < 2.4.2 */

A138924 Indices k such that A020505(k)=Phi[k](-6) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 4, 6, 8, 9, 11, 14, 15, 21, 24, 25, 31, 42, 43, 45, 47, 58, 59, 77, 107, 124, 142, 144, 177, 192, 254, 279, 360, 407, 437, 480, 525, 542, 551, 579, 764, 811, 822, 891, 917, 1018, 1028, 1150, 1326, 1376, 1464, 1468, 1650, 1719, 1924, 2096, 2098, 2176, 2226

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..104
Yves Gallot, Cyclotomic polynomials and prime numbers

Cf. A020505, A138920-A138940.

Select[ Range[ 2000], PrimeQ[ Cyclotomic[#, -6]] &] (* Robert G. Wilson v, Mar 25 2012 *)

for( i=1,999, ispseudoprime( polcyclo(i,-6)) && print1( i",")) /* use ...subst(polcyclo(i),x,-6)... in PARI < 2.4.2 */

a(51)-a(54) from Robert Price, Apr 02 2012

A138925 Indices k such that A020504(k)=Phi[k](-5) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

5, 6, 12, 14, 22, 24, 26, 28, 45, 48, 55, 56, 67, 88, 92, 94, 98, 99, 101, 103, 108, 114, 116, 120, 229, 236, 248, 254, 265, 282, 288, 298, 322, 347, 362, 384, 399, 420, 500, 536, 567, 615, 620, 714, 835, 992, 1047, 1064, 1238, 1794, 1858, 1962, 2313, 2397

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..114
Yves Gallot, Cyclotomic polynomials and prime numbers

Cf. A020504, A138920-A138940.

Select[ Range[ 2000], PrimeQ[ Cyclotomic[#, -5]] &] (* Robert G. Wilson v, Mar 25 2012 *)

for( i=1,999, ispseudoprime( polcyclo(i,-5)) && print1( i",")) /* use ...subst(polcyclo(i),x,-5)... in PARI < 2.4.2 */

a(53)-a(54) from Robert Price, Apr 05 2012

A138926 Indices k such that A020503(k)=Phi[k](-4) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 4, 8, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, 1004, 1228, 1268, 2240, 2532, 3060, 3796, 3824, 3944, 5096, 5540, 7476, 7700, 8544, 9800, 14628, 15828, 16908, 18480, 20260, 21924, 24656, 38456

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

It appears that for all k>1, a(k) is a multiple of 4.

It also appears that all Cyclotomic Polynomials, Phi[k](x), where k is a multiple of 4 have no odd powers of x. For example, Phi[20](x)=x^8-x^6+x^4-x^2+1. This implies that Phi[k](x)=Phi[k](-x), where k is a multiple of 4. - Robert Price, Apr 14 2012

Robert Price, Table of n, a(n) for n = 1..49
Yves Gallot, Cyclotomic polynomials and prime numbers

Cf. A020503, A138920-A138940.

Select[ Range[3, 5000], PrimeQ[ Cyclotomic[#, -4]] &] (* Robert G. Wilson v, Mar 25 2012 *)

for( i=1,999, ispseudoprime( polcyclo(i,-4)) && print1( i",")) /* use ...subst(polcyclo(i),x,-4)... in PARI < 2.4.2 */

a(36)-a(49) from Robert Price, Apr 07 2012

A138927 Indices k such that A020502(k)=Phi[k](-3) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 5, 6, 7, 12, 13, 14, 18, 23, 24, 26, 30, 35, 36, 40, 42, 43, 60, 65, 66, 72, 77, 108, 126, 132, 142, 206, 215, 236, 276, 281, 286, 290, 304, 322, 359, 364, 391, 464, 487, 510, 522, 528, 529, 535, 558, 574, 577, 672, 680, 713, 731, 760, 799, 828, 892, 984

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..157
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A020502, A138920-A138940.

Select[ Range[3, 1000], PrimeQ[ Cyclotomic[#, -3]] &] (* Robert G. Wilson v, Mar 25 2012 *)

for( i=1,999, ispseudoprime( polcyclo(i,-3)) && print1( i","))

A138928 Indices n such that A020501(n) = phi(n)(-2) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 26, 30, 31, 32, 34, 38, 39, 40, 43, 45, 49, 54, 56, 61, 62, 63, 66, 75, 79, 80, 85, 87, 98, 101, 117, 120, 122, 127, 130, 138, 154, 161, 167, 170, 178, 183, 184, 186, 187, 191, 192, 199, 205, 207, 208

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

While the sequence is not very interesting up to a(n)<300, there are only 4 values in the interval [400,599].

Robert Price, Table of n, a(n) for n = 1..227
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A020501, A138920-A138940.

Select[ Range[3, 1000], PrimeQ[ Cyclotomic[#, -2]] &] (* Robert G. Wilson v, Mar 25 2012 *)

for( i=1,999, isprime( polcyclo(i,-2)) && print1( i",")) /* for PARI < 2.4.2 use ...subst(polcyclo(i),x,-2)...*/

cp's OEIS Frontend

A138937 Indices k such that A019325(k)=Phi[k](7) is prime, where Phi is a cyclotomic polynomial.

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A138939 Indices k such that A019327(k)=Phi[k](9) is prime, where Phi is a cyclotomic polynomial.

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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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A138921 Indices k such that A020508(k)=Phi[k](-9) is prime, where Phi is a cyclotomic polynomial.

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A138923 Indices k such that A020506(k)=Phi[k](-7) is prime, where Phi is a cyclotomic polynomial.

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A138924 Indices k such that A020505(k)=Phi[k](-6) is prime, where Phi is a cyclotomic polynomial.

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A138925 Indices k such that A020504(k)=Phi[k](-5) is prime, where Phi is a cyclotomic polynomial.

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A138926 Indices k such that A020503(k)=Phi[k](-4) is prime, where Phi is a cyclotomic polynomial.

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