A343303 Numbers in A231626 but not in A343302; first of 5 consecutive deficient numbers in arithmetic progression with common difference > 1.

347, 1997, 2207, 2747, 2987, 2989, 3005, 3245, 3707, 3845, 4505, 4727, 4729, 5165, 6227, 7067, 7205, 7907, 8885, 9347, 9587, 9723, 9725, 11405, 13745, 14207, 14765, 17147, 17987, 18125, 18587, 18827, 18843, 18845, 19547, 20147, 20477, 21485, 22187, 22983, 22985

Offset: 1

Author

Jianing Song, Apr 11 2021

Keywords

• nonn

Comments

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.

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.

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.

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.

Examples

			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.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..16607 (all terms <= 10^7).

Crossrefs

Cf. A096399.

Set difference of A231626 by A343302.

Programs

Mathematica

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 *)