%I #14 May 08 2021 06:21:20
%S 347,1997,2207,2747,2987,2989,3005,3245,3707,3845,4505,4727,4729,5165,
%T 6227,7067,7205,7907,8885,9347,9587,9723,9725,11405,13745,14207,14765,
%U 17147,17987,18125,18587,18827,18843,18845,19547,20147,20477,21485,22187,22983,22985
%N Numbers in A231626 but not in A343302; first of 5 consecutive deficient numbers in arithmetic progression with common difference > 1.
%C Numbers k such that k, k+d, k+2*d, k+3*d and k+4*d are consecutive deficient numbers with some d > 1. Such k with d = 1 are listed in A343302.
%C All known terms have d = 2. If some k is the start of 5 consecutive deficient numbers in arithmetic progression with common difference 3, then k+1, k+4, k+7 and k+10 must be 4 consecutive terms in A096399. This may happen, but each of such k has to be extremely large.
%C If k is an even term here, then none of k, k+d, k+2*d, k+3*d and k+4*d is divisible by 6, so d must be divisible by 3.
%C It seems that most terms are congruent to 5 modulo 6. The smallest term congruent to 1 modulo 6 is a(6) = 2989, and the smallest term congruent to 3 modulo 6 is a(22) = 9723.
%H Jianing Song, <a href="/A343303/b343303.txt">Table of n, a(n) for n = 1..16607</a> (all terms <= 10^7).
%e 347 is here since it is the start of 5 consecutive deficient numbers in arithmetic progression with common difference 2, namely 347, 349, 351, 353 and 355. Indeed, all of 348, 350, 352 and 354 are abundant.
%t DefQ[n_] := DivisorSigma[1, n] < 2 n; m = 2; z1 = 2; cd = 1; a = {}; Do[If[DefQ[n], If[n - z1 == cd, m = m + 1; If[m > 4 && cd != 1, AppendTo[a, n - 4*cd]], m = 2; cd = n - z1]; z1 = n], {n, 3, 50000}]; a (* after the Mathematica program of A231626 *)
%Y Cf. A096399.
%Y Set difference of A231626 by A343302.
%K nonn
%O 1,1
%A _Jianing Song_, Apr 11 2021