James Burling - cp's OEIS frontend

User: James Burling

James Burling has authored 5 sequences.

A263042 a(n) = Sum_{i >= 1} d_i(n) * prime(i) where d_i(n) is the i-th digit of n in base 10, and prime(i) is the i-th prime.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36

Offset: 0

James Burling, Oct 08 2015

Digits are counted from the right, so d_1(n) is the ones digit, d_2(n) is the tens digit, etc.
d_i(n) can be found using either of the following formulas:
* d_i(n) = floor(n / 10^(i-1)) mod 10;
* d_i(n) = floor(n / 10^(i-1)) - 10 * floor(n / 10^i).
From Derek Orr, Dec 24 2015: (Start)
For n < 1000, this sequence may be written as a series of 10 X 10 subtables:
Subtable 1:
   0,  2,  4,  6,  8, 10, 12, 14, 16, 18
   3,  5,  7,  9, 11, 13, 15, 17, 19, 21
   6,  8, 10, 12, 14, 16, 18, 20, 22, 24
   9, 11, 13, 15, 17, 19, 21, 23, 25, 27
  12, 14, 16, 18, 20, 22, 24, 26, 28, 30
  15, 17, 19, 21, 23, 25, 27, 29, 31, 33
  18, 20, 22, 24, 26, 28, 30, 32, 34, 36
  21, 23, 25, 27, 29, 31, 33, 35, 37, 39
  24, 26, 28, 30, 32, 34, 36, 38, 40, 42
  27, 29, 31, 33, 35, 37, 39, 41, 43, 45
Subtable 2:
   5,  7,  9, 11, 13, 15, 17, 19, 21, 23
   8, 10, 12, 14, 16, 18, 20, 22, 24, 26
  11, 13, 15, 17, 19, 21, 23, 25, 27, 29
  14, 16, 18, 20, 22, 24, 26, 28, 30, 32
  17, 19, 21, 23, 25, 27, 29, 31, 33, 35
  20, 22, 24, 26, 28, 30, 32, 34, 36, 38
  23, 25, 27, 29, 31, 33, 35, 37, 39, 41
  26, 28, 30, 32, 34, 36, 38, 40, 42, 44
  29, 31, 33, 35, 37, 39, 41, 43, 45, 47
  32, 34, 36, 38, 40, 42, 44, 46, 48, 50
Subtable 3:
  10, 12, 14, 16, 18, 20, 22, 24, 26, 28
  13, 15, 17, 19, 21, 23, 25, 27, 29, 31
  16, 18, 20, 22, 24, 26, 28, 30, 32, 34
  19, 21, 23, 25, 27, 29, 31, 33, 35, 37
  22, 24, 26, 28, 30, 32, 34, 36, 38, 40
  25, 27, 29, 31, 33, 35, 37, 39, 41, 43
  28, 30, 32, 34, 36, 38, 40, 42, 44, 46
  31, 33, 35, 37, 39, 41, 43, 45, 47, 49
  34, 36, 38, 40, 42, 44, 46, 48, 50, 52
  37, 39, 41, 43, 45, 47, 49, 51, 53, 55
  ...
Each subtable is 10 X 10. Let T_n(j,k) = the element in the j-th row of the k-th column of subtable n. T_n(1,1) = 5*(n-1). T_n(j,1) = 5*(n-1)+3*(j-1). T_n(1,k) = 5*(n-1)+2*(k-1). Altogether, T_n(j,k) = 5*(n-1)+3*(j-1)+2*(k-1) = 5*n+3*j+2*k-10.
(End)

			For n = 12, the digits are 2 and 1 and the corresponding primes are 2 and 3, so a(12) = (first digit * first prime) + (second digit * second prime) = 2 * 2 + 1 * 3 = 4 + 3 = 7.

James Burling, Table of n, a(n) for n = 0..10000

Similar method, different base for n: A089625 (base 2), A262478 (base 3).

Similar method, uses product instead of sum: A019565 (base 2), A101278 (base 3), A054842 (base 10).

Table[Sum_{m=0}^{infinity} (Floor[n/10^(m)] - 10*Floor[n/10^(m+1)])*Prime(m+1), {n,0,500}] (* G. C. Greubel, Oct 08 2015 *)

a(n) = if (n==0, d = [0], d=Vecrev(digits(n))); sum(i=1,#d, d[i]*prime(i)); \\ Michel Marcus, Oct 10 2015

vector(200,n,n--;sum(i=1,#digits(n),Vecrev(digits(n))[i]*prime(i))) \\ Derek Orr, Dec 24 2015

a(n) = Sum_{i >= 0} prime(i + 1) * (floor(n / 10^i) - 10 * floor(n / 10^(i + 1))).

A262478 a(n) = Sum_{i >= 0} d_i(n) * p_(i + 1) where d_i(n) = i-th digit of n in base 3, and p_i = i-th prime.

Original entry on oeis.org

0, 2, 4, 3, 5, 7, 6, 8, 10, 5, 7, 9, 8, 10, 12, 11, 13, 15, 10, 12, 14, 13, 15, 17, 16, 18, 20, 7, 9, 11, 10, 12, 14, 13, 15, 17, 12, 14, 16, 15, 17, 19, 18, 20, 22, 17, 19, 21, 20, 22, 24, 23, 25, 27, 14, 16, 18, 17, 19, 21, 20, 22, 24, 19, 21, 23, 22, 24, 26, 25, 27, 29, 24, 26, 28, 27

Offset: 0

James Burling, Sep 23 2015

d_i(n) can be found using either of the following formulas:
* d_i(n) = floor(n / 3^i) mod 3;
* d_i(n) = floor(n / 3^i) - 3 * floor(n / 3^(i + 1)).

			The base 3 representation of n = 5 is 12 so a(5) = 2 * 2 + 1 * 3 = 7.
The base 3 representation of n = 12 is 110 so a(12) = 0 * 2 + 1 * 3 + 1 * 5 = 8.

James Burling, Table of n, a(n) for n = 0..10000

Similar method, different base for n: A089625 (base 2).

Similar method, uses product for sum index for multiplication: A019565 (base 2), A101278 (base 3), A054842 (base 10).

Table[Sum[IntegerDigits[n, 3][[-i]] Prime@ i, {i, IntegerLength[n, 3]}], {n, 0, 81}] (* Michael De Vlieger, Sep 24 2015 *)

a(n) = my(d = Vecrev(digits(n, 3))); sum(k=1, #d, d[k]*prime(k)); \\ Michel Marcus, Sep 24 2015

a(n) = Sum_{i >= 0} p_(i + 1) * (floor(n / 3^i) - 3 * floor(n / 3^(i + 1))).

A248591 Numerators of the (simplified) rationals n*2^(n - 1)/(n - 1)! .

Original entry on oeis.org

1, 4, 6, 16, 10, 8, 28, 64, 2, 8, 44, 32, 52, 16, 8, 256, 34, 8, 76, 32, 4, 16, 184, 128, 4, 16, 8, 64, 232, 32, 496, 1024, 2, 8, 4, 32, 148, 16, 8, 128, 164, 16, 344, 64, 8, 32, 752, 512, 4, 16, 8, 64, 424, 32, 16, 256, 8, 32, 944, 128, 976, 64, 32, 4096, 2

Offset: 1

James Burling, Oct 09 2014

James Burling, Another Special Rational Sequence

Cf. A248592 (denominators).

[Numerator(n*2^(n-1)/Factorial(n-1)): n in [1..70]]; // Vincenzo Librandi, Oct 16 2014

Table[Numerator[n 2^(n - 1)/(n - 1)!], {n, 1, 70}] (* Vincenzo Librandi, Oct 16 2014 *)

vector(100, n, numerator(n*2^(n - 1)/(n - 1)!)) \\ Michel Marcus, Oct 09 2014

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

a(n) = numerator(g(1, n)) where g(m, n) = m if m = n; 2*g(m + 1, n)/m otherwise.

More terms from Michel Marcus, Oct 09 2014

A248592 Denominators of the (simplified) rational numbers n*2^(n - 1)/(n - 1)! .

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 45, 315, 35, 567, 14175, 51975, 467775, 868725, 2837835, 638512875, 638512875, 1206079875, 97692469875, 371231385525, 441942125625, 17717861581875, 2143861251406875, 16436269594119375, 5917057053882975, 284473896821296875, 1780595872696265625

Offset: 1

James Burling, Oct 09 2014

James Burling, Another Special Rational Sequence

Cf. A248591 (numerators).

Has same start as A241591 but is a different sequence.

A248592 := proc(n)
    n*2^(n-1)/(n-1)! ;
    denom(%) ;
end proc:
seq(A248592(n),n=1..30) ; # R. J. Mathar, Oct 10 2014

vector(40, n, denominator(n*2^(n - 1)/(n - 1)!)) \\ Michel Marcus, Oct 09 2014

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

a(n) = denom(g(1, n)) where g(m, n) = m if m = n; 2g(m + 1, n)/m otherwise.

More terms from Michel Marcus, Oct 09 2014

A240988 Denominators of the (reduced) rationals (((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), where n is a positive integer.

Original entry on oeis.org

1, 4, 2, 16, 8, 32, 16, 256, 128, 512, 256, 2048, 1024, 4096, 2048, 65536, 32768, 131072, 65536, 524288, 262144, 1048576, 524288, 8388608, 4194304, 16777216, 8388608, 67108864, 33554432, 134217728, 67108864, 4294967296, 2147483648, 8589934592, 4294967296

Offset: 1

James Burling, Aug 06 2014

Numerators for this sequence are the swinging factorial A163590, starting from n = 1.
The terms are all powers of 2 (A000079).
It appears that a(2*n) = 2^A101925(n) and a(2*n+1) = 2^A005187(n). - Robert Israel, Aug 06 2014

			For n = 1, a(1) = 1.
For n = 2, a(2) = 2 * 2 = 4.
For n = 6, a(6) = 2 * 2 * 4 * 2 = 32.

James Burling, The Special Rational Sequence

Cf. A163590 (numerators).

f:= n -> denom(((doublefactorial(n-1)) / (doublefactorial(n)*2^((1+(-1)^n)/2)))^((-1)^n)):
seq(f(n), n=1..100); # Robert Israel, Aug 06 2014

df(n) = prod(i=0, floor((n-1)/2), n-2*i) \\ Double factorial (n!!)
a(n) = denominator(((df(n-1)) / (df(n)*2^((1+(-1)^n)/2)))^((-1)^n))
vector(50, n, a(n)) \\ Colin Barker, Aug 06 2014

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

a(n) = denominator(g(1, n)) where g(m, n) = m if m = n; m/(2 * g(m + 1, n)) otherwise.

More terms from Colin Barker, Aug 06 2014

cp's OEIS Frontend

User: James Burling

A263042 a(n) = Sum_{i >= 1} d_i(n) * prime(i) where d_i(n) is the i-th digit of n in base 10, and prime(i) is the i-th prime.

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A262478 a(n) = Sum_{i >= 0} d_i(n) * p_(i + 1) where d_i(n) = i-th digit of n in base 3, and p_i = i-th prime.

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A248591 Numerators of the (simplified) rationals n*2^(n - 1)/(n - 1)! .

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A248592 Denominators of the (simplified) rational numbers n*2^(n - 1)/(n - 1)! .

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A240988 Denominators of the (reduced) rationals (((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), where n is a positive integer.

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