A256968 - OEIS

%I #50 Jan 26 2025 20:52:52

%S 0,0,1,2,4,6,9,14,22,35,55,89,142,230,373,609,996,1637,2698,4461,7398,

%T 12301,20503,34253,57348,96198,161659,272124,458789,774616,1309627,

%U 2216968,3757384,6375166,10828012,18409028,31326514,53354259,90945529,155142139

%N Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

%C A001611 is similar but different.

%C Equal to A005579 except for n = 2 and n = 3. The following argument shows that they are equal for n > 3. First note that b(k+1) > b(k). Next, Product_{i=1..k} p_i is 2 times an odd number, i.e., it is not divisible by 4. Similarly since p_i - 1 is even for i > 1, Product_{i=1..k} (p_i - 1) is divisible by 2^(k-1), i.e., it is divisible by 4 for k >= 3. Thus b(k) is not an integer for k >= 3. Since b(3) = 15/4 > 3, this means that a(n) = A005579(n) for n > 3 - _Chai Wah Wu_, Apr 17 2015

%H Popular Computing (Calabasas, CA), <a href="/A256968/a256968.png">Problem 182 (Suggested by Victor Meally)</a>, Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

%e The sequence b(n) for n >= 0 begins 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 2.

%o (Python)

%o from sympy import prime

%o A256968_list, count, bn, bd = [0,0], 2, 1, 1

%o for k in range(1,10**4):

%o     p = prime(k)

%o     bn *= p

%o     bd *= p-1

%o     while bn >= count*bd:

%o         A256968_list.append(k)

%o count += 1 # _Chai Wah Wu_, Apr 17 2015; corrected by _Max Alekseyev_, Jan 26 2025

%Y Cf. A005579, A060753, A038110, A001611.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Apr 17 2015

%E More terms from _Chai Wah Wu_, Apr 17 2015

%E a(32)-a(33) from _Chai Wah Wu_, Apr 19 2015

%E a(0)-a(1) corrected and a(34)-a(39) copied over from A005579 by _Max Alekseyev_, Jan 26 2025