A061077 - OEIS

%I #50 Mar 21 2022 11:48:27

%S 1,4,9,16,25,26,29,34,41,50,52,58,68,82,100,103,112,127,148,175,179,

%T 191,211,239,275,280,295,320,355,400,406,424,454,496,550,557,578,613,

%U 662,725,733,757,797,853,925,934,961,1006,1069,1150,1150,1150,1150

%N a(n) is the sum of the products of the digits of the first n odd numbers.

%D Amarnath Murthy, Smarandache friendly numbers and a few more sequences, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

%H Luca Onnis, <a href="https://arxiv.org/abs/2203.07227">On the general Smarandache's sigma product of digits</a>, arXiv:2203.07227 [math.GM], 2022.

%F a(n) = Sum_{k = 1..n} (product of the digits of 2k-1).

%F From _Luca Onnis_, Mar 20 2022: (Start)

%F a(5*10^n) = (25/44)*(45^(n+1)-1).

%F a(n) <= (25/44)*(45^(log(n/5)+1)-1) for all n.

%F a(n) ~ (5/4)*A061078(n) as n -> infinity. (End)

%e a(7) = 1 + 3 + 5 + 7 + 9 + 1*1 + 1*3 = 29.

%t Accumulate[Times @@@ IntegerDigits[Range[1, 99, 2]]] (* _Luca Onnis_, Mar 20 2022 *)

%o (PARI) pd(n) = my(d = digits(n)); prod(i=1, #d, d[i]);

%o a(n) = sum(k=1, n, pd(2*k-1)); \\ _Michel Marcus_, Feb 01 2015

%o (PARI) a(n) = {m=digits(2*n - 1); p=1; d=#m; for(i=1, #m, if(m[i]==0, d=i-1; break));

%o (25/44) * (45^(#m-1)-1) + sum(i=1, #m-1, (prod(j=1,#m-i-1,m[j])) * (m[#m-i]) * (m[#m-i]-1) * (5^(i + 1) * 9^(i-1)) / 2)+prod(k=1,#m-1,m[k])*(((m[#m]+1)^2)/4)} \\ _Luca Onnis_, Mar 20 2022

%o (Python)

%o from math import prod

%o def A061077(n): return sum(prod(int(d) for d in str(2*i+1)) for i in range(n)) # _Chai Wah Wu_, Mar 21 2022

%Y Cf. A061076, A061078.

%K nonn,base,easy

%O 1,2

%A _Amarnath Murthy_, Apr 14 2001

%E More terms from _Matthew Conroy_, Apr 16 2001

%E Offset corrected by _Charles R Greathouse IV_, Feb 01 2015

%E Incorrect formula removed by _Luca Onnis_, Mar 20 2022