A255588 -id:A255588 - cp's OEIS frontend

A255689 Convert n to base 4, move the most significant digit to the least significant one and convert back to base 10.

0, 1, 2, 3, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 1, 5, 9, 13, 17, 21, 25

Offset: 0

Paolo P. Lava, Mar 02 2015

a(4*n) = 1.

Fixed points of the transform are listed in A048329.

			11 in base 4 is 23: moving the most significant digit as the least significant one we have 32 that is 14 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..65536 (terms 0..1000 from Paolo P. Lava)

Cf. A006257, A030103, A255588-A255594, A048787, A255690-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 1 to nops(a)-1 do b:=[op(b),a[k]]; od; a:=[a[nops(a)],op(b)];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,4);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Append[Rest@ w, First@ w]]; b = 4; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 70]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateLeft[IntegerDigits[n,4]],4],{n,0,70}] (* Harvey P. Dale, Aug 07 2015 *)

def A255689(n):
    x=A007090(n)
    return int (x[1:]+x[0],4) # Indranil Ghosh, Feb 08 2017

A255693 Convert n to base 8, move the most significant digit to the least significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 1, 9, 17, 25, 33, 41, 49, 57, 2, 10, 18, 26, 34, 42, 50, 58, 3, 11, 19, 27, 35, 43, 51, 59, 4, 12, 20, 28, 36, 44, 52, 60, 5, 13, 21, 29, 37, 45, 53, 61, 6, 14, 22, 30, 38, 46, 54, 62, 7, 15, 23, 31, 39, 47, 55, 63, 1, 9, 17, 25, 33, 41

Offset: 0

Paolo P. Lava, Mar 02 2015

a(8*n) = 1.

Fixed points of the transform are listed in A048333.

			13 in base 8 is 15: moving the most significant digit as the least significant one we have 51 that is 41 in base 10.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A006257, A030107, A255588-A255594, A255689-A255692.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 1 to nops(a)-1 do b:=[op(b),a[k]]; od; a:=[a[nops(a)],op(b)];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,8);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Append[Rest@ w, First@ w]]; b = 8; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 69]) (* Michael De Vlieger, Mar 04 2015 *)

A255594 Convert n to base 9, move least significant digit to most significant digit and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 10, 19, 28, 37, 46, 55, 64, 73, 2, 11, 20, 29, 38, 47, 56, 65, 74, 3, 12, 21, 30, 39, 48, 57, 66, 75, 4, 13, 22, 31, 40, 49, 58, 67, 76, 5, 14, 23, 32, 41, 50, 59, 68, 77, 6, 15, 24, 33, 42, 51, 60, 69, 78, 7, 16, 25, 34, 43, 52, 61

Offset: 0

Paolo P. Lava, Mar 02 2015

Fixed points of the transform are listed in A048334.

For more than 2 base-9 digits, reversal and cyclic shifts of a number start to differ, so this sequence differs from A030108. - Alonso del Arte, Mar 22 2015

			13 in base 9 is 14: moving the least significant digit as the most significant one we have 41 that is 37 in base 10.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A030108, A038572, A048334, A255588-A255593, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,9);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 9; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 69]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateRight[IntegerDigits[n,9]],9],{n,0,80}] (* Harvey P. Dale, Jun 14 2024 *)

a(9*n) = n.

a(9^n) = 9^(n-1).

A048787 Write n in base 3 then rotate left one place.

Original entry on oeis.org

1, 2, 1, 4, 7, 2, 5, 8, 1, 4, 7, 10, 13, 16, 19, 22, 25, 2, 5, 8, 11, 14, 17, 20, 23, 26, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68

Offset: 1

John W. Layman and Anthony C. Hill (hilla(AT)hotmail.com)

A={a(n)} is self-similar in the sense that the subsequence remaining after deleting the first occurrence of each integer is identical to the original sequence A (Kimberling's "upper-trimming" operation).

			a(33)=19 since 33 = 1020(base 3) -> 0201(base 3) = 19.

Indranil Ghosh, Table of n, a(n) for n = 1..59049

Cf. A255588-A255594, A255689-A255693.

Table[ FromDigits[ RotateLeft[ IntegerDigits[n, 3]], 3], {n, 1, 76}] (* Zerinvary Lajos, Mar 25 2007 *)

def A048787(n):
    x=A007089(n)
    return int (x[1:]+x[0],3) # Indranil Ghosh, Feb 08 2017

A255691 Convert n to base 6, move the most significant digit to the least significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 13, 19, 25, 31, 2, 8, 14, 20, 26, 32, 3, 9, 15, 21, 27, 33, 4, 10, 16, 22, 28, 34, 5, 11, 17, 23, 29, 35, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175

Offset: 0

Paolo P. Lava, Mar 02 2015

a(6*n) = 1.

Fixed points of the transform are listed in A048331.

			16 in base 6 is 24: moving the most significant digit as the least significant one we have 42 that is 26 in base 10.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A006257, A030105, A255588-A255594, A255689-A255690, A255692-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 1 to nops(a)-1 do b:=[op(b),a[k]]; od; a:=[a[nops(a)],op(b)];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,6);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Append[Rest@ w, First@ w]]; b = 6; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 65]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateLeft[IntegerDigits[n,6]],6],{n,0,70}] (* Harvey P. Dale, Mar 02 2023 *)

A255692 Convert n to base 7, move the most significant digit to the least significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 15, 22, 29, 36, 43, 2, 9, 16, 23, 30, 37, 44, 3, 10, 17, 24, 31, 38, 45, 4, 11, 18, 25, 32, 39, 46, 5, 12, 19, 26, 33, 40, 47, 6, 13, 20, 27, 34, 41, 48, 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 120, 127, 134

Offset: 0

Paolo P. Lava, Mar 02 2015

a(7*n) = 1.

Fixed points of the transform are listed in A048332.

			11 in base 7 is 14: moving the most significant digit as the least significant one we have 41 that is 29 in base 10.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A006257, A030106, A255588-A255594, A255689-A255691, A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 1 to nops(a)-1 do b:=[op(b),a[k]]; od; a:=[a[nops(a)],op(b)];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,7);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Append[Rest@ w, First@ w]]; b = 7; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 68]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateLeft[IntegerDigits[n,7]],7],{n,0,70}] (* Harvey P. Dale, Oct 20 2020 *)

A255589 Convert n to base 4, move the least significant digit to the most significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 4, 20, 36, 52, 5, 21, 37, 53, 6, 22, 38, 54, 7, 23, 39, 55, 8, 24, 40, 56, 9, 25, 41, 57, 10, 26, 42, 58, 11, 27, 43, 59, 12, 28, 44, 60, 13, 29, 45, 61, 14, 30, 46, 62, 15, 31, 47, 63, 16, 80, 144, 208, 17, 81

Offset: 0

Paolo P. Lava, Mar 02 2015

Fixed points of the transform are listed in A048329.

			11 in base 4 is 23: moving the least significant digit to the most significant one we have 32 that is 14 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..16384 (terms 0..1000 from Paolo P. Lava)

Cf. A030103, A038572, A048329, A255588, A255590-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,4);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 4; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 69]) (* Michael De Vlieger, Mar 04 2015 *)
Array[FromDigits[RotateRight[IntegerDigits[#,4]],4]&,70,0] (* Harvey P. Dale, Mar 01 2016 *)

def A255589(n):
    x=str(A007090(n))
    return int(x[-1]+x[:-1],4) # Indranil Ghosh, Feb 03 2017

a(4*k) = k.

a(4^k) = 4^(k-1).

A255590 Convert n to base 5, move the least significant digit to the most significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 11, 16, 21, 2, 7, 12, 17, 22, 3, 8, 13, 18, 23, 4, 9, 14, 19, 24, 5, 30, 55, 80, 105, 6, 31, 56, 81, 106, 7, 32, 57, 82, 107, 8, 33, 58, 83, 108, 9, 34, 59, 84, 109, 10, 35, 60, 85, 110, 11, 36, 61, 86, 111, 12, 37, 62, 87, 112, 13, 38, 63, 88

Offset: 0

Paolo P. Lava, Mar 02 2015

a(5*n) = n.
a(5^n) = 5^(n-1).
Fixed points of the transform are listed in A048330.

			14 in base 5 is 24: moving the least significant digit to the most significant one we have 42 that is 22 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..15625 (terms 0..1000 from Paolo P. Lava)

Cf. A030104, A038572, A048330, A255588, A255589, A255591-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,5);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 5; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 68]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateRight[IntegerDigits[n,5]],5],{n,0,100}] (* Harvey P. Dale, Jun 11 2025 *)

def A255590(n):
    x=str(A007091(n))
    return int(x[-1]+x[:-1], 5) # Indranil Ghosh, Feb 03 2017

A255591 Convert n to base 6, move least significant digit to most significant digit and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 13, 19, 25, 31, 2, 8, 14, 20, 26, 32, 3, 9, 15, 21, 27, 33, 4, 10, 16, 22, 28, 34, 5, 11, 17, 23, 29, 35, 6, 42, 78, 114, 150, 186, 7, 43, 79, 115, 151, 187, 8, 44, 80, 116, 152, 188, 9, 45, 81, 117, 153, 189, 10, 46, 82, 118, 154, 190, 11

Offset: 0

Paolo P. Lava, Mar 02 2015

a(6*n) = n.

Fixed points of the transform are listed in A048331.

			16 in base 6 is 24: moving the least significant digit to the most significant one we have 42 that is 26 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..46656 (terms 0..1000 from Paolo P. Lava)

Cf. A030105, A038572, A048331, A255588-A255590, A255592-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,6);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 6; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 66]) (* Michael De Vlieger, Mar 04 2015 *)

def A255591(n):
    x=A007092(n)
    return int(x[-1]+x[:-1],6) # Indranil Ghosh, Feb 03 2017

A255592 Convert n to base 7, move least significant digit to most significant digit and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 15, 22, 29, 36, 43, 2, 9, 16, 23, 30, 37, 44, 3, 10, 17, 24, 31, 38, 45, 4, 11, 18, 25, 32, 39, 46, 5, 12, 19, 26, 33, 40, 47, 6, 13, 20, 27, 34, 41, 48, 7, 56, 105, 154, 203, 252, 301, 8, 57, 106, 155, 204, 253, 302, 9, 58, 107, 156

Offset: 0

Paolo P. Lava, Mar 02 2015

a(7*n) = n.
a(7^n) = 7^(n-1).
Fixed points of the transform are listed in A048332.

			11 in base 7 is 14: moving the least significant digit to the most significant one we have 41 that is 29 in base 10.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A030106, A038572, A048332, A255588-A255591, A255593, A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,7);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 7; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 66]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateRight[IntegerDigits[n,7]],7],{n,0,100}] (* Harvey P. Dale, Jan 27 2023 *)

cp's OEIS Frontend

A255689 Convert n to base 4, move the most significant digit to the least significant one and convert back to base 10.

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A255693 Convert n to base 8, move the most significant digit to the least significant one and convert back to base 10.

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A255594 Convert n to base 9, move least significant digit to most significant digit and convert back to base 10.

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A048787 Write n in base 3 then rotate left one place.

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A255691 Convert n to base 6, move the most significant digit to the least significant one and convert back to base 10.

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A255692 Convert n to base 7, move the most significant digit to the least significant one and convert back to base 10.

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A255589 Convert n to base 4, move the least significant digit to the most significant one and convert back to base 10.

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