A083952 -id:A083952 - cp's OEIS frontend

A083953 Least integer coefficients of A(x), where 1<=a(n)<=3, such that A(x)^(1/3) consists entirely of integer coefficients.

1, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 2, 3, 3, 1, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 1, 3, 3, 2, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, "least integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. Is this sequence periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..5000.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A083954, A083945, A083946.

a[0]=1; a[n_] :=a[n] = Block[{k=1, s=Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/3), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v, Jul 25 2005 *)

a(k)=1 at k=0, 3, 12, 21, 51, 57, 60, 63, 66, ...; a(k)=2 at k=15, 18, 24, 30, 39, 42, 48, 54, ...

More terms from Robert G. Wilson v, Jul 25 2005

A083954 Least integer coefficients of A(x), where 1<=a(n)<=4, such that A(x)^(1/4) consists entirely of integer coefficients.

Original entry on oeis.org

1, 4, 2, 4, 3, 4, 4, 4, 1, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 2, 4, 2, 4, 4, 4, 4, 4, 3, 4, 2, 4, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 2, 4, 2, 4, 1, 4, 4, 4, 1, 4, 2, 4, 4, 4, 4, 4, 1, 4, 2, 4, 3, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 4, 1, 4, 4, 4, 1, 4, 2, 4, 3

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, "least integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m>0. Is this sequence periodic?
From M. F. Hasler, Jan 27 2025: (Start)
The sequence does not seem to become periodic.
Positions of '1's are: (0, 8, 60, 64, 72, 96, 100, 112, 116, 148, 160, 176, 184, 200, 240, 248, 268, 288, 304, 328, 336, 360, 376, 380, 384, 400, 408, 420, 424, 448, 460, 472, ...). All seem to be multiples of 4, mostly multiples of 8.
Positions of '3's are: (4, 12, 16, 28, 36, 40, 76, 84, 104, 124, 136, 172, 192, 196, 208, 212, 220, 232, 252, 260, 284, 296, 312, 364, 368, 392, 404, 428, 432, 436, 452, 456, 468, 488, 492, ...). All seem to be (mostly odd) multiples of 4.
The proportions of '1's, '2's, '3's and '4's among the terms are approximately: 6.5%, 18%, 6.5%, 69%. (Roughly the same values for the first 500 or 5000 terms.) (End)

Robert G. Wilson v, Table of n, a(n) for n = 0..5000.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A083953, A083945, A083946.

a[0] = 1; a[n_] :=a[n] = Block[{k=1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/4), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v, Jul 26 2005 *)

A083954_upto(N=99)=vector(N+1, n, if(n>1, for(k=1,4, denominator(polcoeff(sqrtn(O(x^n)+N+=x^(n-1), 4), n-1))>1|| [n=k, break]); n, N=1)) \\ _M. F. Hasler, Jan 27 2025

More terms from Robert G. Wilson v, Jul 26 2005

A083948 Integer coefficients of A(x), where 1<=a(n)<=8, such that A(x)^(1/8) consists entirely of integer coefficients.

Original entry on oeis.org

1, 8, 4, 8, 2, 8, 4, 8, 7, 8, 8, 8, 4, 8, 8, 8, 3, 8, 8, 8, 2, 8, 8, 8, 1, 8, 8, 8, 8, 8, 8, 8, 6, 8, 4, 8, 6, 8, 4, 8, 6, 8, 8, 8, 4, 8, 8, 8, 4, 8, 8, 8, 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 6, 8, 8, 8, 8, 8, 4, 8, 6, 8, 4, 8, 8, 8, 8, 8, 6, 8, 8, 8, 7, 8, 4, 8, 8, 8, 4, 8, 3, 8, 4, 8, 4, 8, 4, 8, 3

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally the sequence, "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083947, A083949, A083950.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/8), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083949 Integer coefficients of A(x), where 1<=a(n)<=9, such that A(x)^(1/9) consists entirely of integer coefficients.

Original entry on oeis.org

1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 9, 9, 6, 9, 9, 6, 9, 9, 9, 9, 9, 6, 9, 9, 6, 9, 9, 9, 9, 9, 3, 9, 9, 3, 9, 9, 2, 9, 9, 6, 9, 9, 6, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 5, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 6, 9, 9, 6, 9, 9, 5, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 1, 9, 9, 6, 9, 9, 6, 9, 9, 7, 9, 9, 6, 9, 9

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, the sequence, "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083947, A083948, A083950.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/9), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083950 Integer coefficients of A(x), where 1<=a(n)<=10, such that A(x)^(1/10) consists entirely of integer coefficients.

Original entry on oeis.org

1, 10, 5, 10, 10, 2, 5, 10, 10, 10, 3, 10, 5, 10, 10, 2, 10, 10, 10, 10, 5, 10, 5, 10, 5, 8, 5, 10, 5, 10, 8, 10, 10, 10, 10, 4, 5, 10, 10, 10, 7, 10, 10, 10, 5, 2, 10, 10, 5, 10, 7, 10, 5, 10, 5, 4, 10, 10, 10, 10, 7, 10, 10, 10, 10, 2, 5, 10, 5, 10, 9, 10, 5, 10, 5, 6, 5, 10, 10, 10, 8

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, the sequence, "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083952, A083956, A083947, A083948, A083949.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/10), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 80}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083946 Least integer coefficients of A(x), where 1<=a(n)<=6, such that A(x)^(1/6) consists entirely of integer coefficients.

Original entry on oeis.org

1, 6, 3, 2, 3, 6, 6, 6, 3, 4, 6, 6, 6, 6, 3, 4, 6, 6, 3, 6, 6, 2, 3, 6, 6, 6, 3, 4, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 3, 4, 6, 6, 4, 6, 6, 2, 6, 6, 4, 6, 3, 2, 3, 6, 6, 6, 3, 4, 3, 6, 3, 6, 3, 4, 6, 6, 2, 6, 3, 6, 3, 6, 1, 6, 6, 4, 6, 6, 2, 6, 6, 2, 6, 6, 3, 6, 3, 4, 6, 6, 1, 6, 6, 6, 6, 6, 6, 6, 3, 2, 6, 6, 6, 6, 3

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, "least integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m>0. Is this sequence periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A083953, A083954, A083945.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/6), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083947 Integer coefficients of A(x), where 1<=a(n)<=7, such that A(x)^(1/7) consists entirely of integer coefficients.

Original entry on oeis.org

1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 2, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 3, 7, 7, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 2, 7, 7, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally the sequence, "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083948, A083949, A083950.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/7), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A084067 Integer coefficients of A(x), where 1<=a(n)<=12, such that A(x)^(1/12) consists entirely of integer coefficients.

Original entry on oeis.org

1, 12, 6, 4, 9, 12, 4, 12, 12, 8, 6, 12, 6, 12, 12, 12, 12, 12, 8, 12, 9, 12, 12, 12, 12, 12, 6, 12, 6, 12, 10, 12, 6, 12, 12, 12, 2, 12, 6, 8, 6, 12, 12, 12, 12, 4, 12, 12, 8, 12, 12, 8, 3, 12, 4, 12, 12, 4, 12, 12, 9, 12, 6, 4, 6, 12, 4, 12, 12, 12, 12, 12, 2, 12, 6, 12, 3, 12, 6, 12, 3, 8

Offset: 0

Paul D. Hanna, May 10 2003

nonn

More generally, the sequence: "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m>0. Are these sequences ever periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083947 A083948, A083949, A083950, A084066.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/12), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 81}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083945 Least integer coefficients of A(x), where 1<=a(n)<=5, such that A(x)^(1/5) consists entirely of integer coefficients.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, "least integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m>0. Is this sequence periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A083953, A083954, A083946.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/5), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A084066 Least integer coefficients of A(x), where 1<=a(n)<=11, such that A(x)^(1/11) consists entirely of integer coefficients.

Original entry on oeis.org

1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 4, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 5, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 5, 11, 11, 11, 11, 11

Offset: 0

Paul D. Hanna, May 10 2003

nonn

More generally, the sequence: "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m>0. Are these sequences ever periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083947, A083948, A083949, A083950, A084067.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/11), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 71}] (* Robert G. Wilson v *)

a(k)=0 (mod 11) when k not= 0 (mod 11); a(0)=1, a(11)=1, a(22)=7, a(33)=4, a(44)=9, a(55)=5, a(66)=5, ...

More terms from Robert G. Wilson v, Jul 26 2005

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A083953 Least integer coefficients of A(x), where 1<=a(n)<=3, such that A(x)^(1/3) consists entirely of integer coefficients.

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A083954 Least integer coefficients of A(x), where 1<=a(n)<=4, such that A(x)^(1/4) consists entirely of integer coefficients.

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A083948 Integer coefficients of A(x), where 1<=a(n)<=8, such that A(x)^(1/8) consists entirely of integer coefficients.

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A083949 Integer coefficients of A(x), where 1<=a(n)<=9, such that A(x)^(1/9) consists entirely of integer coefficients.

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A083950 Integer coefficients of A(x), where 1<=a(n)<=10, such that A(x)^(1/10) consists entirely of integer coefficients.

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A083946 Least integer coefficients of A(x), where 1<=a(n)<=6, such that A(x)^(1/6) consists entirely of integer coefficients.

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A083947 Integer coefficients of A(x), where 1<=a(n)<=7, such that A(x)^(1/7) consists entirely of integer coefficients.

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A084067 Integer coefficients of A(x), where 1<=a(n)<=12, such that A(x)^(1/12) consists entirely of integer coefficients.

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