A084293 -id:A084293 - cp's OEIS frontend

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

Paolo P. Lava, Jan 10 2013

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

			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.

Cf. A015886.

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);

A084292 a(n) = 6n + A054904(n).

Original entry on oeis.org

110, 77, 38, 104, 74, 161, 87, 111, 94, 159, 122, 142, 374, 209, 178, 206, 206, 253, 326, 302, 206, 302, 471, 249, 519, 341, 346, 303, 354, 481, 542, 377, 2057, 533, 386, 411, 5138, 662, 846, 527, 386, 437, 1034, 519, 794, 689, 626, 493, 566, 629, 873, 527, 638

Offset: 1

Labos Elemer, May 26 2003

nonn

Composite solutions y to sigma(y-6n) = sigma(y) - 6n. For terms x of A054904, where sigma(x+6n) = sigma(x) + 6n, replacing x+6n = y, x = y-6n, we get sigma(y) - 6n = sigma(y-6n).

Amiram Eldar, Table of n, a(n) for n = 1..724

Cf. A000203 (sigma), A054904, A084293.

For several analogous sequences such corresponding "mirror-solutions" can be easily constructed. See, e.g., A015913-A015918, A050507, A054799, A054903-A054906, A054982-A054987, A059118, A055009, A055458, A063500, etc.

cp's OEIS Frontend

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

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