A326925 -id:A326925 - cp's OEIS frontend

A328610 Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.

-2, 1, 1, 0, 1, 1, 3, 0, 1, 2, 4, 6, 0, 1, 3, 10, 10, 10, 0, 1, 5, 18, 30, 20, 15, 0, 1, 8, 35, 63, 70, 35, 21, 0, 1, 13, 64, 140, 168, 140, 56, 28, 0, 1, 21, 117, 288, 420, 378, 252, 84, 36, 0, 1, 34, 210, 585, 960, 1050, 756, 420, 120, 45, 0, 1, 55, 374

Offset: 1

Clark Kimberling, Oct 24 2019

Let p(n) denote the polynomial (1/n!)*(numerator of n-th derivative of (1-x)/(1-x-x^2)). It is conjectured in A326925 that if n = 2k, then p(n) = f(k)*g(k), where f(k) and g(k) are polynomials of degree k. Row k of the present array shows the coefficients of f(k).

It appears that, after the first term, column 1 consists of the Fibonacci numbers, F(k), for k >= 1; see A000045. It appears that after the first row, the row sums are F(2k+1), and the alternating row sums are (-1)^k F(k).

			First nine rows:
.
  -2,   1;  (coefficients of -2 + x)
   1,   0,   1;  (coefficients of 1 + x^2)
   1,   3,   0,   1;
   2,   4,   6,   0,   1;
   3,  10,  10,  10,   0,   1;
   5,  18,  30,  20,  15,   0,   1;
   8,  35,  63,  70,  35,  21,   0,   1;
  13,  64, 140, 168, 140,  56,  28,   0,   1;
  21, 117, 288, 420, 378, 252,  84,  36,   0,   1;

Clark Kimberling, Table of n, a(n) for n = 1..1325

Cf. A000045, A326925, A328611.

g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
r[n_] := Rest[f[[n]]];
Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]]  (* A328610 *)
Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)

A328611 Irregular triangular array read by rows: row n gives the coefficients of the second of two factors of even-degree polynomials described in Comments.

Original entry on oeis.org

0, -1, 1, 4, -1, 3, 3, 6, -1, 4, 12, 6, 8, -1, 7, 20, 30, 10, 10, -1, 11, 42, 60, 60, 15, 12, -1, 18, 77, 147, 140, 105, 21, 14, -1, 29, 144, 308, 392, 280, 168, 28, 16, -1, 47, 261, 648, 924, 882, 504, 252, 36, 18, -1, 76, 470, 1305, 2160, 2310, 1764, 840

Offset: 1

Clark Kimberling, Oct 24 2019

It appears that, after the first term, column 1 consists of the Lucas numbers, L(k), for k >= 1; see A000032. It appears that after the first row, the row sums are L(2k+1), for k >= 1.

			First nine rows:
.
   0,  -1; (coefficients of -x)
   1,   4,  -1; (coefficients of 1 + 4*x - x^2)
   3,   3,   6,  -1;
   4,  12,   6,   8,  -1;
   7,  20,  30,  10,  10,  -1;
  11,  42,  60,  60,  15,  12,  -1;
  18,  77, 147, 140, 105,  21,  14, -1;
  29, 144, 308, 392, 280, 168,  28, 16, -1;
  47, 261, 648, 924, 882, 504, 252, 36, 18, -1;

Clark Kimberling, Table of n, a(n) for n = 1..1325

Cf. A000032, A326925, A328610.

g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
r[n_] := Rest[f[[n]]];
Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]]  (* A328610 *)
Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)

A328926 Numbers k such that A328925(k) = 1; numbers k such that if we write k = Product_{i=1..t} p_i^e_i, then lcm_{1<=i,j<=t,i!=j} ord(p_i,p_j^e_j) = A002322(k), where ord(a,r) is the multiplicative order of a modulo r, and A002322 is the Carmichael lambda (usually written as psi).

Original entry on oeis.org

1, 2, 6, 10, 12, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 58, 60, 63, 66, 69, 70, 72, 74, 75, 76, 77, 78, 80, 84, 85, 87, 88, 90, 91, 92, 93, 96, 99, 100, 102, 104, 105, 106, 108, 110, 114, 115, 116, 118, 119, 120, 122

Offset: 1

Jianing Song, Oct 31 2019

nonn

Numbers such that A118106(k) = A002322(k).
{a(n)} intersect A000961 = {1, 2}.
{a(n)} union A329062 = N* \ A000961 U {1, 2}.
If k = Product_{i=1..t} p_i^e_i (t > 1), where {p_i} are primes such that p_i is a lambda primitive root modulo p_j^e_j for all i != j (that is to say, ord(p_i,p_j^e_j) = A002322(p_j^e_j), where ord(a,r) is the multiplicative order of a modulo r), then k is here, as A118106(k) = A002322(k). For example, k = 2^5 * 5 * 13.

			For k = 115 = 5 * 23, A118106(115) = lcm(ord(23,5),ord(5,23)) = lcm(4,22) = 44 = A002322(115), so 115 is a term.
For k = 973 = 7 * 139, A118106(973) = lcm(ord(139,7),ord(7,139)) = lcm(6,69) = 138 = A002322(973), so 973 is a term.

Cf. A118106, A002322, A326925, A329062, A000961.

isA328926(n) = (A002322(n) == A118106(n)) \\ See A002322 and A118106 for their programs

A329062 Numbers k that are not prime powers (i.e., not in A000961) such that A328925(k) > 1; numbers k such that if we write k = Product_{i=1..t} p_i^e_i , then t > 1, and lcm_{1<=i,j<=t, i!=j} ord(p_i,p_j^e_j) < A002322(k), where ord(a,r) is the multiplicative order of a modulo r, and A002322 is the Carmichael lambda (usually written as psi).

Original entry on oeis.org

14, 34, 39, 46, 55, 62, 65, 68, 82, 86, 94, 95, 98, 111, 112, 117, 123, 124, 133, 136, 142, 145, 146, 153, 155, 158, 164, 172, 175, 178, 183, 194, 201, 203, 205, 206, 209, 218, 219, 221, 224, 226, 248, 253, 254, 259, 272, 274, 275, 287, 291, 292, 295, 299, 301, 302, 305, 309

Offset: 1

Jianing Song, Nov 02 2019

nonn

Write psi = A002322, b = A118106. These are numbers k that are not prime powers such that b(k) < psi(k).
If k = Product_{i=1..t} p_i^e_i (t > 1), where {p_i} are distinct odd primes such that p_i is a quadratic residue modulo p_j for all i != j, then k is here, as b(k) | psi(k)/2.
If k = 2 * Product_{i=1..t} p_i^e_i, where {p_i} are distinct odd primes such that p_i is a quadratic residue modulo p_j for all i != j, and p_i == 1, 7 (mod 8), then k is here. For example, k = 2p, where p == 1, 7 (mod 8).
If k = 2^e * Product_{i=1..t} p_i^e_i (e > 1), where {p_i} are distinct odd primes such that p_i is a quadratic residue modulo p_j for all i != j, and p_i == 1 (mod 8), then k is here. For example, k = 2^e*p, where p == 1 (mod 8).
If k = p_1^e_1 * p_2^e_2, where p_1 is not a primitive root modulo p_2^e_2, p_2 is not a primitive root modulo p_1^e_1, then k is not necessarily here: for k = 3^2 * 67 = 603, 3 is not a primitive root modulo 67, 67 is not a primitive root modulo 3^2, but b(603) = psi(603) = 66. Conversely, if p_1 is a primitive root modulo p_2^e_2, then k can still be here: for k = 3^2 * 17 = 153, 3 is a primitive root modulo 17, but b(153) = 16 while psi(153) = 48.
If k is an odd number, then 4k is here if and only if 8k is also here. Write k = Product_{i=1..t} p_i^e_i, then b(4k) = lcm(lcm_{i=1..t} ord(p_i,4),lcm_{i=1..t} ord(2,p_i^e_i),b(k)), b(8k) = lcm(lcm_{i=1..t} ord(p_i,8),lcm_{i=1..t} ord(2,p_i^e_i),b(k)). It is easy to see that lcm(ord(p_i,4),ord(2,p_i^e_i)) = lcm(ord(p_i,8),ord(2,p_i^e_i)), so b(4k) = b(8k). Note that psi(4k) = psi(8k).
If k is an odd number such that 4k is here, then 16k is also here (but the converse is not true). Write k = Product_{i=1..t} p_i^e_i, N = lcm(lcm_{i=1..t} ord(2,p_i^e_i),b(k)), then b(4k) = lcm(lcm_{i=1..t} ord(p_i,4),N), b(16k) = lcm(lcm_{i=1..t} ord(p_i,16),N) = b(4k) or b(4k)*2 or b(4k)*4. Note that psi(16k) = lcm(psi(4k),4). If we have b(4k) < psi(4k) and b(16k) = psi(16k), let M = psi(k), then:
Case (a): M == 2 (mod 4), then b(16k) = psi(16k) = 2*psi(4k) = 2M, and b(4k) = b(16k)/4 = M/2 which is an odd number, so ord(2,p_i^e_i) is odd for all i, so p_i == 1, 7 (mod 8), which gives ord(p_i,16) <= 2*ord(p_i,4). As a result, b(16k) <= 2*b(4k), a contradiction!
Case (b): M == 0 (mod 4), then b(16k) = psi(16k) = psi(4k) = M. If N is divisible by 4, then b(4k) = b(16k) = N, a contradiction. So 4 does not divide N, we have v(M,2) = v(lcm(lcm_{i=1..t} ord(p_i,16),N),2) <= 2, but 4 | M, so v(M,2) = 2, where v(,2) is the 2-adic valuation. As a result, there must exist p_i congruent to 5 mod 8 to make v(M,2) = 2, then 4 | ord(2,p_i^e_i), so 4 | N, a contradiction!
{a(n)} union A328926 = N* \ A000961 U {1, 2}.

			Let ord(a,r) be the multiplicative order of a modulo r.
For k = 175 = 5^2 * 7, b(175) = lcm(ord(7,5^2),ord(5,7)) = lcm(4,6) = 12, while psi(175) = lcm(20,6) = 60, so 175 is a term.
For k = 410 = 2 * 5 * 41, b(410) = lcm(ord(5,2),ord(41,2),ord(2,5),ord(41,5),ord(2,41),ord(5,41)) = 20, while psi(410) = lcm(1,4,40) = 40, so 410 is a term.

Cf. A118106, A002322, A326925, A326926, A000961.

isA329062(n) = !isprimepower(n) && (A002322(n) > A118106(n)) \\ See A002322 and A118106 for their programs

A327314 Rectangular array read by descending antidiagonals: the distinct rows of the quotient array, A307693, of A003188.

Original entry on oeis.org

1, 3, 1, 2, 2, 1, 6, 4, 3, 1, 7, 5, 2, 2, 1, 5, 3, 5, 4, 2, 1, 4, 8, 6, 3, 5, 2, 1, 12, 9, 4, 7, 4, 4, 3, 1, 13, 10, 10, 9, 3, 3, 2, 3, 1, 15, 7, 11, 8, 9, 8, 6, 2, 2, 1, 14, 6, 12, 6, 10, 9, 7, 6, 5, 2, 1, 10, 16, 8, 5, 11, 7, 5, 5, 4, 4, 2, 1, 11, 17, 9

Offset: 1

Clark Kimberling, Nov 01 2019

Each row of the quotient array, A307693, occurs infinitely many times. Specifically, if p is a prime (A000040), then for every multiple m*p of p, the rows numbered m*p are identical. In the present array only the first occurrence of each row of A307693 is retained; these are the prime-numbered rows of A307693. Every row is a permutation of the positive integers, so that every positive integer occurs infinitely many times.

			Northwest corner:
  1   3   2   6   7   5   4   12  13  15
  1   2   4   5   3   8   9   10   7   6
  1   3   2   5   6   4  10   11  12   8
  1   2   4   3   7   9   8    6   5  14
  1   2   5   4   3   9  10   11   8   7
  1   2   4   3   8   9   7    6   5  15

Cf. A000040, A003188, A307693.

s = Table[BitXor[n, Floor[n/2]], {n, 2000}]; (* A003188 *)
g[n_] := Flatten[Position[Mod[s, n], 0]];
u[n_] := s[[g[Prime[n]]]]/Prime[n];
Column[Table[Take[u[n], 20], {n, 1, 20}]]  (* A326925 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten  (* A326925 sequence *)

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A328610 Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.

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A328611 Irregular triangular array read by rows: row n gives the coefficients of the second of two factors of even-degree polynomials described in Comments.

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A328926 Numbers k such that A328925(k) = 1; numbers k such that if we write k = Product_{i=1..t} p_i^e_i, then lcm_{1<=i,j<=t,i!=j} ord(p_i,p_j^e_j) = A002322(k), where ord(a,r) is the multiplicative order of a modulo r, and A002322 is the Carmichael lambda (usually written as psi).

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A327314 Rectangular array read by descending antidiagonals: the distinct rows of the quotient array, A307693, of A003188.

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Mathematica