A030106 -id:A030106 - cp's OEIS frontend

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

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 *)

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 *)

A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

Original entry on oeis.org

1, 9077, 10523, 10838, 30182, 58529, 73273, 77879, 83893, 244022, 303253, 303449, 304853, 329893, 332249, 334001, 334417, 335939, 336083, 346741, 374617, 391187, 504199, 512695, 516982, 595274, 680354, 687142, 758077, 780391, 792214, 854669, 946217, 948539, 995761, 1008487, 1377067, 1389341

Offset: 1

Scott R. Shannon, Jul 05 2021

There are 633 terms below 50 million and 1253 terms below 100 million. All of those have tau(k), the number of divisors of k, equal to 1, 2, 4, 8 or 16. The first term where tau(k) = 2 is n = 93836531, a prime, which is also the first term of A136634. All terms in A136634 will appear in this sequence, as will all terms in A228768(n) for n>=10. The first term with tau(k) = 4 is 9077, the first with tau(k) = 8 is 595274, and the first with tau(k) = 16 is 5170182. It is possible tau(k) must equal 2^i, with i>=0, although this is unknown.

All known terms are squarefree. - Michel Marcus, Jul 07 2021

			9077 is a term as the number of divisors of 9077 = tau(9077) = 4, and this equals the number of divisors of R(9077) when written and then read as a base-j number, with 2 <= j <= 10. See the table below for k = 9077.
.
  base | k_base         | R(k_base)      | R(k_base)_10  | tau(R(k_base)_10)
----------------------------------------------------------------------------------
   2   | 10001101110101 | 10101110110001 | 11185         | 4
   3   | 110110012      | 210011011      | 15421         | 4
   4   | 2031311        | 1131302        | 6002          | 4
   5   | 242302         | 203242         | 6697          | 4
   6   | 110005         | 500011         | 38887         | 4
   7   | 35315          | 51353          | 12533         | 4
   8   | 21565          | 56512          | 23882         | 4
   9   | 13405          | 50431          | 33157         | 4
  10   | 9077           | 7709           | 7709          | 4

Scott R. Shannon, Table of n, a(n) for n = 1..1253

Cf. A000005, A004086.
Cf. A030102, A030103, A030104, A030105, A030106, A030107, A030108.
Cf. A136634 (prime terms), A228768.
Subsequence of A062895.

Select[Range@100000,Length@Union@DivisorSigma[0,Join[{s=#},FromDigits[Reverse@IntegerDigits[s,#],#]&/@Range[2,10]]]==1&] (* Giorgos Kalogeropoulos, Jul 06 2021 *)

isok(k) = {my(t= numdiv(k)); for (b=2, 10, my(d=digits(k, b)); if (numdiv(fromdigits(Vecrev(d), b)) != t, return (0));); return(1);} \\ Michel Marcus, Jul 06 2021

A055954 a(n) = n + reversal of base 7 digits of n (written in base 10).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 8, 16, 24, 32, 40, 48, 56, 16, 24, 32, 40, 48, 56, 64, 24, 32, 40, 48, 56, 64, 72, 32, 40, 48, 56, 64, 72, 80, 40, 48, 56, 64, 72, 80, 88, 48, 56, 64, 72, 80, 88, 96, 50, 100, 150, 200, 250, 300, 350, 64, 114, 164, 214, 264, 314, 364, 78, 128, 178

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base 7 then a(n) is a multiple of 8

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A030106, A056964.

Table[n+FromDigits[Reverse[IntegerDigits[n,7]],7],{n,0,70}] (* Harvey P. Dale, Jan 29 2020 *)

a(n) = n + fromdigits(Vecrev(digits(n, 7)), 7); \\ Michel Marcus, Aug 08 2024

a(n) = n + A030106(n).

A055955 a(n) = n - reversal of base 7 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, -6, -12, -18, -24, -30, 12, 6, 0, -6, -12, -18, -24, 18, 12, 6, 0, -6, -12, -18, 24, 18, 12, 6, 0, -6, -12, 30, 24, 18, 12, 6, 0, -6, 36, 30, 24, 18, 12, 6, 0, 48, 0, -48, -96, -144, -192, -240, 48, 0, -48, -96, -144, -192, -240, 48, 0, -48, -96, -144, -192, -240, 48, 0, -48, -96, -144, -192, -240

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 6.

Cf. A030106, A056965.

Table[n-FromDigits[Reverse[IntegerDigits[n,7]],7],{n,0,80}] (* Harvey P. Dale, Jul 22 2015 *)

a(n) = n - A030106(n).

cp's OEIS Frontend

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

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A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

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PARI

A055954 a(n) = n + reversal of base 7 digits of n (written in base 10).

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A055955 a(n) = n - reversal of base 7 digits of n (written in base 10).

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