A248956 -id:A248956 - cp's OEIS frontend

A248348 a(n) = number of polynomials a_kx^k + ... + a_1x + a_0 with k > 0, integer coefficients, only distinct integer roots, and a_0 = p^n (p is a prime).

3, 11, 23, 47, 83, 139, 227, 355, 539, 803, 1175, 1687, 2391, 3343, 4619, 6323, 8571, 11515, 15355, 20323, 26715, 34907, 45339, 58563, 75263, 96255, 122535, 155327, 196087, 246575, 308931, 385691, 479899, 595219, 735979, 907347, 1115483, 1367643, 1672435

Offset: 0

Reiner Moewald, Oct 05 2014

nonn

If D_n = {-p^n, ..., -p^0, p^0, ..., p^n} is the set of all positive and negative divisors of p^n (p is a prime), then a(n) gives the number of all subsets of D_n for which the product of all their elements is a divisor of p^n.

			a(0)= 3: x+1; -x+1; -x^2+1.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..1000

Cf. A248410, A248956.

a(15)-a(38) from Hiroaki Yamanouchi, Nov 21 2014

A248955 Number of polynomials a_kx^k + ... + a_1x + n with k > 0, integer coefficients and distinct positive integer roots and positive integers n.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 9, 5, 9, 3, 17, 3, 9, 9, 13, 3, 17, 3, 17, 9, 9, 3, 31, 5, 9, 9, 17, 3, 29, 3, 19, 9, 9, 9, 35, 3, 9, 9, 31, 3, 29, 3, 17, 17, 9, 3, 49, 5, 17, 9, 17, 3, 31, 9, 31, 9, 9, 3, 61, 3, 9, 17, 27, 9, 29, 3, 17, 9, 29, 3, 67, 3, 9, 17, 17, 9

Offset: 1

Reiner Moewald, Oct 17 2014

nonn

If D_n is the set of all positive divisors of n, then a(n) gives the number of all subsets of D_n for which the product of all their elements is a divisor of n. a(n) depends only on the prime signature of n.

			a(2) = 3: -2x+2; -x+2; x^2 - 3x + 2.

Reiner Moewald, Table of n, a(n) for n = 1..2159

Cf. A248410, A248956.

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {d = divisors(n); nbd = #d; nbts = 2^nbd-1; nbs = 0; for (i=1, nbts, bin = padbin(i, nbd); prd = prod(j=1, nbd,  if (bin[j], d[j], 1)); if (n % prd == 0, nbs++);); nbs;} \\ Michel Marcus, Nov 07 2014

a(p) = 3, for p prime. - Michel Marcus, Nov 07 2014

A381848 Sequence obtained by replacing 3-term subwords of A010060 by 0,1,2,3,4,5 as described in Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 28 2025

nonn

The six 3-term subwords of A010060 are 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0. These are coded as 0,1,2,3,4,5 respectively, and then these numbers replace the corresponding subwords in A010060. The positions of 0,1,2,3,4,5 are given by A248956, A248104, A157971, A157970, A248105, A248057, respectively.

			Starting with A010060 = (0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0,...), the successive 3-term subwords are 0,1,1; 1,1,0; 1,0,1; 0,1,0; 1,0,0 ..., which code as 2,5,4,1,3,... .

Cf. A010060, A383999, A248956, A248104, A157971, A157970, A248105, A248057.

Partition[ThueMorse[Range[0, 200]], 3, 1] /. Thread[{{0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}} -> {0, 1, 2, 3, 4, 5}]  (* Peter J. C. Moses, May 22 2025 *)

cp's OEIS Frontend

A248348 a(n) = number of polynomials a_kx^k + ... + a_1x + a_0 with k > 0, integer coefficients, only distinct integer roots, and a_0 = p^n (p is a prime).

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A248955 Number of polynomials a_kx^k + ... + a_1x + n with k > 0, integer coefficients and distinct positive integer roots and positive integers n.

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A381848 Sequence obtained by replacing 3-term subwords of A010060 by 0,1,2,3,4,5 as described in Comments.

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Mathematica

cp's OEIS Frontend

A248348 a(n) = number of polynomials a_k*x^k + ... + a_1*x + a_0 with k > 0, integer coefficients, only distinct integer roots, and a_0 = p^n (p is a prime).

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A248955 Number of polynomials a_k*x^k + ... + a_1*x + n with k > 0, integer coefficients and distinct positive integer roots and positive integers n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A381848 Sequence obtained by replacing 3-term subwords of A010060 by 0,1,2,3,4,5 as described in Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A248348 a(n) = number of polynomials a_kx^k + ... + a_1x + a_0 with k > 0, integer coefficients, only distinct integer roots, and a_0 = p^n (p is a prime).

A248955 Number of polynomials a_kx^k + ... + a_1x + n with k > 0, integer coefficients and distinct positive integer roots and positive integers n.