A355593 a(n) is the number of alternating integers that divide n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 6, 1, 4, 3, 5, 1, 6, 1, 5, 4, 2, 2, 7, 3, 2, 4, 5, 2, 7, 1, 6, 2, 3, 3, 9, 1, 3, 2, 6, 2, 7, 2, 3, 5, 3, 2, 8, 3, 6, 2, 4, 1, 8, 2, 7, 2, 4, 1, 9, 2, 2, 6, 6, 3, 4, 2, 4, 4, 7, 1, 11, 1, 3, 4, 5, 2, 5, 1, 7, 5, 3, 2, 9, 3, 3, 4, 4, 2, 11, 2, 5, 2, 4, 2, 10, 1, 6, 3, 7

Offset

1,2

Comments

This sequence first differs from A355302 at index 13, where a(13) = 1 while A355302(13) = 2.

This sequence first differs from A332268 at index 14, where a(14) = 4 while A332268(14) = 3.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A030141(n) = 5.1... (the sums up to 10^10, 10^11 and 10^12 are 5.1704..., 5.1727... and 5.1738..., respectively). - Amiram Eldar, Jan 06 2024

Example

40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 2 are not alternating integers: {20, 40}, hence a(40) = 8 - 2 = 6.

Maple

Alt:= [$1..9, seq(seq(10*i+r - (i mod 2), r=[1, 3, 5, 7, 9]), i=1..9)]:
V:= Vector(100):
for t in Alt do J:= [seq(i, i=t..100, t)]; V[J]:= V[J] +~ 1 od:
convert(V, list); # Robert Israel, Nov 26 2023

Mathematica

q[n_] := !MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 120] (* Amiram Eldar, Jul 08 2022 *)

Prog

(Python)
from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 08 2022
(PARI) alternate(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1
a(n)=sumdiv(n, d, alternate(d)) \\ Charles R Greathouse IV, Jul 08 2022

Crossrefs

Cf. A030141 (alternating integers), A355594, A355595, A355596.

Similar to A332268 (with Niven numbers) and A355302 (with undulating integers).

Sequence in context: A358099 A095048 A332268 * A355302 A084302 A289872

Adjacent sequences: A355590 A355591 A355592 * A355594 A355595 A355596

Keyword

nonn,base

Author

Bernard Schott, Jul 08 2022

Status

approved