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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.

%I A263042 #53 Mar 05 2025 01:12:26
%S A263042 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,
%T A263042 18,20,22,24,9,11,13,15,17,19,21,23,25,27,12,14,16,18,20,22,24,26,28,
%U A263042 30,15,17,19,21,23,25,27,29,31,33,18,20,22,24,26,28,30,32,34,36
%N 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.
%C A263042 Digits are counted from the right, so d_1(n) is the ones digit, d_2(n) is the tens digit, etc.
%C A263042 d_i(n) can be found using either of the following formulas:
%C A263042 * d_i(n) = floor(n / 10^(i-1)) mod 10;
%C A263042 * d_i(n) = floor(n / 10^(i-1)) - 10 * floor(n / 10^i).
%C A263042 From _Derek Orr_, Dec 24 2015: (Start)
%C A263042 For n < 1000, this sequence may be written as a series of 10 X 10 subtables:
%C A263042 Subtable 1:
%C A263042    0,  2,  4,  6,  8, 10, 12, 14, 16, 18
%C A263042    3,  5,  7,  9, 11, 13, 15, 17, 19, 21
%C A263042    6,  8, 10, 12, 14, 16, 18, 20, 22, 24
%C A263042    9, 11, 13, 15, 17, 19, 21, 23, 25, 27
%C A263042   12, 14, 16, 18, 20, 22, 24, 26, 28, 30
%C A263042   15, 17, 19, 21, 23, 25, 27, 29, 31, 33
%C A263042   18, 20, 22, 24, 26, 28, 30, 32, 34, 36
%C A263042   21, 23, 25, 27, 29, 31, 33, 35, 37, 39
%C A263042   24, 26, 28, 30, 32, 34, 36, 38, 40, 42
%C A263042   27, 29, 31, 33, 35, 37, 39, 41, 43, 45
%C A263042 Subtable 2:
%C A263042    5,  7,  9, 11, 13, 15, 17, 19, 21, 23
%C A263042    8, 10, 12, 14, 16, 18, 20, 22, 24, 26
%C A263042   11, 13, 15, 17, 19, 21, 23, 25, 27, 29
%C A263042   14, 16, 18, 20, 22, 24, 26, 28, 30, 32
%C A263042   17, 19, 21, 23, 25, 27, 29, 31, 33, 35
%C A263042   20, 22, 24, 26, 28, 30, 32, 34, 36, 38
%C A263042   23, 25, 27, 29, 31, 33, 35, 37, 39, 41
%C A263042   26, 28, 30, 32, 34, 36, 38, 40, 42, 44
%C A263042   29, 31, 33, 35, 37, 39, 41, 43, 45, 47
%C A263042   32, 34, 36, 38, 40, 42, 44, 46, 48, 50
%C A263042 Subtable 3:
%C A263042   10, 12, 14, 16, 18, 20, 22, 24, 26, 28
%C A263042   13, 15, 17, 19, 21, 23, 25, 27, 29, 31
%C A263042   16, 18, 20, 22, 24, 26, 28, 30, 32, 34
%C A263042   19, 21, 23, 25, 27, 29, 31, 33, 35, 37
%C A263042   22, 24, 26, 28, 30, 32, 34, 36, 38, 40
%C A263042   25, 27, 29, 31, 33, 35, 37, 39, 41, 43
%C A263042   28, 30, 32, 34, 36, 38, 40, 42, 44, 46
%C A263042   31, 33, 35, 37, 39, 41, 43, 45, 47, 49
%C A263042   34, 36, 38, 40, 42, 44, 46, 48, 50, 52
%C A263042   37, 39, 41, 43, 45, 47, 49, 51, 53, 55
%C A263042   ...
%C A263042 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.
%C A263042 (End)
%H A263042 James Burling, <a href="/A263042/b263042.txt">Table of n, a(n) for n = 0..10000</a>
%F A263042 a(n) = Sum_{i >= 0} prime(i + 1) * (floor(n / 10^i) - 10 * floor(n / 10^(i + 1))).
%e A263042 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.
%t A263042 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 *)
%o A263042 (PARI) a(n) = if (n==0, d = [0], d=Vecrev(digits(n))); sum(i=1,#d, d[i]*prime(i)); \\ _Michel Marcus_, Oct 10 2015
%o A263042 (PARI) vector(200,n,n--;sum(i=1,#digits(n),Vecrev(digits(n))[i]*prime(i))) \\ _Derek Orr_, Dec 24 2015
%Y A263042 Similar method, different base for n: A089625 (base 2), A262478 (base 3).
%Y A263042 Similar method, uses product instead of sum: A019565 (base 2), A101278 (base 3), A054842 (base 10).
%K A263042 nonn,base,easy
%O A263042 0,2
%A A263042 _James Burling_, Oct 08 2015