A083956 -id:A083956 - cp's OEIS frontend

A094760 Duplicate of A083956.

1, 33, 666, 11110, 166665, 2333331, 31111108, 399999996, 4999999995

Offset: 1

dead

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

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

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

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

A132856 Number of sequences {c(i), i=0..n} that form the initial terms of a self-convolution 6th power of an integer sequence such that 0 < c(n) <= 6*c(n-1) for n>0 with c(0)=1.

Original entry on oeis.org

1, 1, 6, 108, 7614, 2451762, 3773520918, 28927494486144, 1137959521626242430, 234471053096681379609150, 257075108927481255273258364890, 1518584605077301579030226106654776268, 48819910122176867311132781943952677374210562

Offset: 0

Paul D. Hanna, Sep 19 2007, Oct 06 2007

nonn

The minimal path in the 5-convoluted tree is A083956.

Equals the number of nodes at generation n in the 6-convoluted tree, which is defined as follows: tree of all finite sequences {c(k), k=0..n} that form the initial terms of a self-convolution 6th power of some integer sequence such that 0 < c(n) <= 6*c(n-1) for n>0 with a(0)=1.

			a(n) counts the nodes in generation n of the following tree.
Generations 0..3 of the 6-convoluted tree are as follows;
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1];
GEN.1: 1->[6];
GEN.2: 1-6->[3,9,15,21,27,33];
GEN.3:
1-6-3->[2,8,14]
1-6-9->[2,8,14,20,26,32,38,44,50]
1-6-15->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86]
1-6-21->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122]
1-6-27->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122,128,134,140,146,152,158]
1-6-33->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122,128,134,140,146,152,158,164,170,176,182,188,194].
Each path in the tree from the root node forms the initial terms of a self-convolution 6th power of a sequence of integer terms.

Martin Fuller, Computing A132852, A132853, A132854, A132855, A132856

Cf. A132852, A132853, A132854, A132855; A083956.

Extended by Martin Fuller, Sep 24 2007

A124797 Sum of cyclic permutations of 123...n seen as number written in base n+1: ((n+1)^n-1)*(n+1)/2.

Original entry on oeis.org

1, 12, 126, 1560, 23325, 411768, 8388604, 193710240, 4999999995, 142655835300, 4458050224122, 151437553296120, 5556003412779001, 218946945190429680, 9223372036854775800, 413620130943168382080, 19673204037648268787703

Offset: 1

M. F. Hasler, Nov 07 2006

Sequence A083956 becomes "unnatural" for n>9. The present sequence is more universal since 10 is not singled out as a particular value. See A124798 for the sequence of digits of a(n) in base n+1 and for more results.

			a(2) = 12[3] + 21[3] = 110[3] = 12[10] where [b] indicates the basis b in which the number is written;
a(3) = 123[4] + 231[4] + 312[4] = 126[10];
a(4) = 1234[5] + 2341[5] + 3412[5] + 4123[5] = 22220[5] = 1560[10],...

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A083956, A124798.

[((n + 1)^n - 1)*(n + 1) div 2: n in [1..20]]; // Vincenzo Librandi, Jan 09 2013

a:=proc(n) local b,m,i,s; b:=n+1: m:=add(i*b^(n-i),i=1..n): s:=m: for i to n-1 do m:=b^(n-1)*modp(m,b)+iquo(m,b): s:=s+m: od: s end; # or simply # a := n -> (n+1)/2*((n+1)^n-1)

Table[((n+1)^n-1)*(n+1)/2, {n,22}] (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)

a(n) = (n+1)/2*((n+1)^n-1).

A124798 Sequence of digits (least significant digit first) of A124797 (sums of cyclic permutations of 1...n written in base n+1).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 3, 1, 0, 2, 2, 2, 2, 3, 5, 5, 5, 5, 2, 0, 3, 3, 3, 3, 3, 3, 4, 7, 7, 7, 7, 7, 7, 3, 0, 4, 4, 4, 4, 4, 4, 4, 4, 5, 9, 9, 9, 9, 9, 9, 9, 9, 4, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 5, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 13, 13, 13, 13, 13, 13

Offset: 1

M. F. Hasler, Nov 07 2006

Sequence A083956 becomes "unnatural" for n>9. It is easily seen that for n=2k, the sum of permutations A124797(n) is {k:n}0 in base n+1 where {k:n} means n times the digit k; while for n=2k+1 (>1), the sum is k{n:2k}{k+1} (again in base n+1). In particular, this number has n+1 digits (for n>1), such that the digits for A124797(n) start at place n(n+1)/2-1 (for n>1).

			a(1)=1, the sum of cyclic permutations of 1;
a(2..4)=0,1,1 since 12 + 21 = 110 in base 3;
a(5..8)=2,3,3,1 since 123 + 231 + 312 = 1332 in base 4;
a(9..13)=0,2,2,2,2 since 1234 + 2341 + 3412 + 4123 = 22220 in base 5.

Cf. A083956, A124797.

A124797 := n->(n+1)/2*((n+1)^n-1): map(op, [ 'convert(A124797(i),base,i+1)' $ i=1..20 ]);

cp's OEIS Frontend

A094760 Duplicate of A083956.

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

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A132856 Number of sequences {c(i), i=0..n} that form the initial terms of a self-convolution 6th power of an integer sequence such that 0 < c(n) <= 6*c(n-1) for n>0 with c(0)=1.

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A124797 Sum of cyclic permutations of 123...n seen as number written in base n+1: ((n+1)^n-1)*(n+1)/2.

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