A260901 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.

0, 0, 0, 0, 50, 98, 108, 338, 578, 1058, 3072, 1922, 73167, 2738, 3362, 4418, 5618, 357735, 7442, 467985, 8978, 10658, 600285, 13778, 754635, 798735, 18818, 20402, 21218, 22898, 1085385, 2744445, 34322, 1217685, 1327935, 4548765, 45602, 49298, 1526385, 55778

Offset

1,5

Comments

We observe two families of numbers having interesting properties:

(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, ...

(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, ...

Links

Table of n, a(n) for n=1..40.

Example

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).

Maple

with(numtheory): nn:=36:
for n from 5 to nn do:
ii:=0:
  for k from 1 to 4*10^6 while(ii=0) do:
   d:=divisors(k):n0:=nops(d):
   s1:=sum(‘d[i+1]/d[i]’, ‘i’=1..n0-1):
    if type(k, prime)=false and s1=ithprime(n)
     then
     ii:=1:printf ( “%d %d \n”, n, k):
     else
    fi:
  od:
od:

Prog

(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

Crossrefs

Cf. A000040.

Sequence in context: A228953 A335480 A255585 * A090997 A141757 A248023

Adjacent sequences: A260898 A260899 A260900 * A260902 A260903 A260904

Keyword

nonn

Author

Michel Lagneau, Feb 13 2016

Extensions

Name edited by Michel Marcus, Feb 17 2016

Status

approved