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a(n) is the smallest composite k such that d(2)/d(1) + d(3)/d(2) + ... + d(q)/d(q-1) = prime(n), where d(1) < d(2) < ... < d(q) are the q divisors of k, or 0 if no such k exists.
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%I #39 Sep 17 2017 22:29:46

%S 0,0,0,0,50,98,108,338,578,1058,3072,1922,73167,2738,3362,4418,5618,

%T 357735,7442,467985,8978,10658,600285,13778,754635,798735,18818,20402,

%U 21218,22898,1085385,2744445,34322,1217685,1327935,4548765,45602,49298,1526385,55778

%N a(n) is the smallest composite k such that d(2)/d(1) + d(3)/d(2) + ... + d(q)/d(q-1) = prime(n), where d(1) < d(2) < ... < d(q) are the q divisors of k, or 0 if no such k exists.

%C We observe two families of numbers having interesting properties:

%C (i) Integers having 6 divisors. These numbers are of the form m = 2*p^2 == 2 (mod 48) where p is prime and belongs to the set A = {5, 7, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 131, 151, ...}. We find finite multiplicative groups A/qZ. For example, with q = 12 we obtain the group A/12Z = {1, 5, 7, 11}. From the first 5000 values of p, the finite groups are obtained with q = 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 24, 27, 29, 32, 36, 38, 43, 48, 54, 58, 59, 64, ...

%C (ii) Integers having more than 6 divisors. These numbers are 108, 3072, 73167, 357735, 467985, 600285, 1085385, ... with the corresponding numbers of divisors 12, 22, 8, 16, 16, 16, 16, 16, ...

%e a(7) = 108 because the divisors are {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108} => 2/1 + 3/2 + 4/3 + 6/4 + 9/6 + 12/9 + 18/12 + 27/18 + 36/27 + 54/36 + 108/54 = 17 = prime(7).

%p with(numtheory): nn:=36:

%p for n from 5 to nn do:

%p ii:=0:

%p for k from 1 to 4*10^6 while(ii=0) do:

%p d:=divisors(k):n0:=nops(d):

%p s1:=sum(‘d[i+1]/d[i]’, ‘i’=1..n0-1):

%p if type(k,prime)=false and s1=ithprime(n)

%p then

%p ii:=1:printf ( “%d %d \n”,n,k):

%p else

%p fi:

%p od:

%p od:

%o (PARI) a(n) = {m = 2; pn = prime(n); while ((d = divisors(m)) && (sum(k=2, #d, d[k]/d[k-1]) != pn), m++; if (isprime(m), m++)); m;} \\ _Michel Marcus_, Feb 14 2016

%Y Cf. A000040.

%K nonn

%O 1,5

%A _Michel Lagneau_, Feb 13 2016

%E Name edited by _Michel Marcus_, Feb 17 2016