A349949 a(n) is the number of divisors of n that are 1 above or 1 below a divisor of either n+1 or n-1.

1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 4, 4, 2, 2, 3, 3, 2, 2, 2, 3, 4, 2, 2, 4, 3, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 5, 4, 3, 3, 2, 3, 3, 2, 2, 2, 2, 2, 4, 3, 3, 3, 2, 5, 4, 2, 2, 4, 3, 2, 3, 3

Offset: 2

Author

Tejo Vrush, Dec 06 2021

Keywords

• nonn

Examples

			a(2) = 1 because 2 and 0 are not divisors of either 1 or 3, but 3 = 2+1 is a divisor of 3.
a(6) = 2 since the divisors of 6 are 1, 2, 3, and 6; those of 5 are 1 and 5; those of 7 are 1 and 7; and, regarding {1, 5, 7}, neither 1-1 = 0 nor 1+1 = 2 are in the set, neither 3-1 = 2 nor 3+1 = 4 is, but 2-1 = 1 is, and 6-1 = 5 is (as is 6+1 = 7).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..20000

Crossrefs

Cf. A000005.

Programs

Mathematica

Table[DivisorSum[n, 1 &, If[# == 1, Or[Mod[n - 1, # + 1] == 0, Mod[n + 1, # + 1] == 0], AnyTrue[# + {-1, 1}, Or[Mod[n - 1, #] == 0, Mod[n + 1, #] == 0] &]] &], {n, 2, 88}] (* Michael De Vlieger, Dec 06 2021 *)

PARI

a(n) = my(sd=setunion(divisors(n-1), divisors(n+1))); sumdiv(n, d, (vecsearch(sd, d-1)>0) || (vecsearch(sd, d+1)>0)); \\ Michel Marcus, Dec 07 2021

Python

from sympy import divisors
def aupton(nn):
    alst, prevdivs, divs, nextdivs = [], set(), {1}, {1, 2}
    for n in range(2, nn+1):
        prevdivs, divs, nextdivs = divs, nextdivs, set(divisors(n+1))
        neighdivs = prevdivs | nextdivs
        an = sum(1 for d in divs if {d-1, d+1} & neighdivs != set())
        alst.append(an)
    return alst
print(aupton(88)) # Michael S. Branicky, Dec 06 2021

Python

def A349949(n): return sum(1 for m in filter(lambda d:not (((n-1) % (d-1) if d > 1 else True) and (n-1) % (d+1) and ((n+1) % (d-1) if d > 1 else True) and (n+1) % (d+1)), divisors(n,generator=True))) # Chai Wah Wu, Dec 30 2021

Formula

a(p) = 2 for odd prime p. - Chai Wah Wu, Dec 30 2021

Extensions

a(6), a(12), a(14), a(18) corrected and a(31) and beyond from Michael S. Branicky, Dec 06 2021