A206768 a(n) = smallest number k such that sigma(k-n) = sigma(k) - n, with k > n+1.

3, 5, 5, 7, 7, 11, 81, 11, 11, 13, 13, 17, 4431, 17, 17, 19, 19, 23, 25, 23, 23, 29

Offset

1,1

Comments

This sequence begins

3, 5, 5, 7, 7, 11, 81, 11, 11, 13, 13, 17, 4431, 17, 17, 19, 19, 23, 25, 23, 23, 29, ?, 29, ?, 29, 29, 31, 31, 37, ?, 37, 51, 37, 37, 41, 81, 41, 41, 43, 43, 47, ?, 47, 47, 53, ?, 53, 3364, 53, 53, 59, ?, 59, ?, 59, 59, 61, 61, 67, ?, 67, ?, 67, 67, 71, ?, 71, 71, 73, 73, 79, 91, 79, ?, 79, 79, 83, ?, 83, 83, 89, ?, 89, ?, 89, 89, 101, ?, 97, ?, 97, 125, 97, 97, 101, ?, 101, 101, 103, 103, 107... where the other missing terms (designated by "?") are > 10^6, if they exist.

For a given n, n being even, among the integers k satisfying the property sigma(k-n) = sigma(k)-n, we will find prime numbers p, such that p and p-n are primes. This is because in that case sigma(p-n) = (p-n)+1 = (p+1)-n = sigma(p)-n. For instance, when n is even, for n=2 to 14, a(n) is the first term of A006512, A046132, A046117, A092402, A092146, A092216, A098933. If we restrict to composite numbers, then see A084293. - Michel Marcus, Feb 16 2013

For the missing terms mentioned in first comment, a(n) is > 10^7. - Michel Marcus, Sep 21 2013

Links

Table of n, a(n) for n=1..22.

Example

a(13) = 4431 because 4431 is the minimum number for which sigma(4431-13) = sigma(4418)= 6771 and sigma(4431) - 13 = 6784 -13 = 6771.
a(19) = 25 because 25 is the minimum number for which sigma(25-19) = sigma(6) = 12 and sigma(25) - 19 = 31 -19 = 12.

Maple

A206768:=proc(q)
local k, n;
for n from 1 to q do
  for k from n+1 to q do
  if sigma(-n+k)=sigma(k)-n then print(k); break; fi;
od; od; end:
A206768(1000000000);

Crossrefs

Cf. A015886.

Sequence in context: A204894 A109258 A088081 * A168322 A138475 A195990

Adjacent sequences: A206765 A206766 A206767 * A206769 A206770 A206771

Keyword

nonn,more,hard

Author

Paolo P. Lava, Jan 10 2013

Status

approved