A189974 Numbers m such that d(m-1) = d(m+1) = 4, where d(k) is the number of divisors of k (A000005).

7, 9, 34, 56, 86, 92, 94, 124, 142, 144, 160, 184, 186, 202, 204, 214, 216, 218, 220, 236, 248, 266, 300, 302, 304, 320, 322, 328, 340, 342, 392, 394, 412, 414, 416, 446, 452, 470, 472, 516, 518, 534, 536, 544, 552, 580, 582, 590, 634, 668, 670, 680, 686

Offset: 1

Author

Juri-Stepan Gerasimov, May 03 2011

Keywords

• nonn

Comments

Numbers m such that m-1 and m+1 are both multiplicatively perfect numbers A007422.

Conjecture: all terms but the first two are even numbers. - Harvey P. Dale, Jul 21 2025

Proof of conjecture: if m is odd and > 10 then either m-1 or m+1 is divisible by 4 and > 8 as well. Let t be the number from {m-1, m+1} divisible by 4. Then t is a power of 2 that is > 8 and so has more than two divisors or it has an odd prime divisor such that it has more than 4 divisors. Both exclude the odd m > 8 from the sequence. - David A. Corneth, Aug 05 2025

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A007422.

Programs

Maple

with(numtheory): A189974 := proc(n) option remember: local k: if(n=1)then return 7:else k:=procname(n-1)+1: do if(tau(k-1)=4 and tau(k+1)=4)then return k: fi: k:=k+1: od: fi: end: seq(A189974(n),n=1..60); # Nathaniel Johnston, May 04 2011

Mathematica

Select[Range[2, 754], DivisorSigma[0, # - 1] == DivisorSigma[0, # + 1] == 4 &]
Flatten[Position[Partition[DivisorSigma[0,Range[700]],3,1],?(#[[1]]==#[[3]]==4&),1,Heads->False]]+1 (* _Harvey P. Dale, Jul 21 2025 *)