A355593 - OEIS

%I #42 Jan 06 2024 09:21:33

%S 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,

%T 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,

%U 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

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

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

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

%H Robert Israel, <a href="/A355593/b355593.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

%e 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.

%p Alt:= [$1..9, seq(seq(10*i+r - (i mod 2), r=[1,3,5,7,9]),i=1..9)]:

%p V:= Vector(100):

%p for t in Alt do J:= [seq(i,i=t..100,t)]; V[J]:= V[J] +~ 1 od:

%p convert(V,list); # _Robert Israel_, Nov 26 2023

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

%o (Python)

%o from sympy import divisors

%o def p(d): return 0 if d in "02468" else 1

%o def c(n):

%o     if n < 10: return True

%o     s = str(n)

%o     return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))

%o def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))

%o print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Jul 08 2022

%o (PARI) alternate(n,d=digits(n))=for(i=2,#d, if((d[i]-d[i-1])%2==0, return(0))); 1

%o a(n)=sumdiv(n,d,alternate(d)) \\ _Charles R Greathouse IV_, Jul 08 2022

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

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

%K nonn,base

%O 1,2

%A _Bernard Schott_, Jul 08 2022