A048329 -id:A048329 - 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

A059711 Smallest base in which n is a repdigit.

Original entry on oeis.org

2, 2, 3, 2, 3, 4, 5, 2, 3, 8, 4, 10, 5, 3, 6, 2, 7, 16, 5, 18, 9, 4, 10, 22, 5, 24, 3, 8, 6, 28, 9, 2, 7, 10, 16, 6, 8, 36, 18, 12, 3, 40, 4, 6, 10, 8, 22, 46, 7, 48, 9, 16, 12, 52, 8, 10, 13, 7, 28, 58, 9, 60, 5, 2, 15, 12, 10, 66, 16, 22, 9, 70, 11, 8, 36, 14, 18, 10, 12, 78, 3, 26, 40, 82, 11, 4

Offset: 0

Erich Friedman, Feb 19 2001

Numbers n such that a(n) < n - 1 correspond to Brazilian numbers (A125134); conversely, positive numbers n such that a(n) >= n - 1 correspond to A220570. - Rémy Sigrist, Apr 04 2018

			a(13) = 3 since 13 in base 3 is 111.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A010785, A125134, A048328, A048329, A048330, A048331, A048332, A048333, A048334, A048335, A048336, A048337, A048338, A048339, A048340, A220570.

a(n) = for (b=2, oo, if (#Set(digits(n, b))<=1, return (b))) \\ Rémy Sigrist, Apr 04 2018

From Rémy Sigrist, Apr 04 2018: (Start)
a(n) <= n - 1 for any n >= 3.
a(2^n-1) = 2 for any n >= 0.
a(A048328(n)) <= 3 for any n >= 0.
a(A048329(n)) <= 4 for any n >= 0.
a(A048330(n)) <= 5 for any n >= 0.
a(A048331(n)) <= 6 for any n >= 0.
a(A048332(n)) <= 7 for any n >= 0.
a(A048333(n)) <= 8 for any n >= 0.
a(A048334(n)) <= 9 for any n >= 0.
a(A010785(n)) <= 10 for any n >= 0.
a(A048335(n)) <= 11 for any n >= 0.
a(A048336(n)) <= 12 for any n >= 0.
a(A048337(n)) <= 13 for any n >= 0.
a(A048338(n)) <= 14 for any n >= 0.
a(A048339(n)) <= 15 for any n >= 0.
a(A048340(n)) <= 16 for any n >= 0.
(End)

Example clarified by Harvey P. Dale, Oct 11 2015

Terms a(0) = 2, a(1) = 2 and a(2) = 3 prepended by Rémy Sigrist, Apr 04 2018

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).

A226542 Primes p such that p - 1 can be represented as a repdigit number in some base < p which is a power of two.

Original entry on oeis.org

11, 19, 37, 43, 67, 103, 131, 137, 199, 239, 293, 331, 397, 439, 463, 521, 547, 661, 683, 727, 859, 911, 991, 1033, 1093, 1171, 1291, 1301, 1543, 1549, 1951, 2053, 2081, 2341, 2731, 2861, 3079, 3121, 3251, 3511, 3613, 3823, 4099, 4129, 4229, 4903, 5419, 6151

Offset: 1

Arkadiusz Wesolowski, Jun 10 2013

It is believed that this is a supersequence of A001220 (Wieferich primes).

			103 is in the sequence because it is prime and 102 = 66 (base 16).
463 is in the sequence because it is prime and 462 = ee (base 32).
7 is not in the sequence since 6 = 6 (base 8) and 8 > 7.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
John Blythe Dobson, A note on the two known Wieferich Primes
Wells Johnson, On the nonvanishing of Fermat quotients (mod p), Journal f. die reine und angewandte Mathematik 292, (1977): 196-200.
Wikipedia, Repdigit

Cf. A001220, A010785, A048329, A048333, A048340.

lst = {}; r = 13; Do[If[PrimeQ[p] && Length@Union@IntegerDigits[p - 1, 2^b] == 1, AppendTo[lst, p]], {b, 2, r - 1}, {p, 2^b + 1, 2^r - 1, 2}]; Union[lst]

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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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A059711 Smallest base in which n is a repdigit.

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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.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

Python

Formula

A226542 Primes p such that p - 1 can be represented as a repdigit number in some base < p which is a power of two.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica