A322036 a(n) = A322035(n) - A322034(n).

1, 1, 2, 1, 4, 1, 6, 1, 5, 2, 10, 1, 12, 3, 3, 1, 16, 5, 18, 1, 13, 5, 22, 1, 19, 6, 14, 3, 28, 3, 30, 1, 7, 8, 27, 5, 36, 9, 25, 1, 40, 13, 42, 5, 8, 11, 46, 1, 41, 19, 11, 3, 52, 7, 43, 3, 37, 14, 58, 3, 60, 15, 34, 1, 51, 7, 66, 4, 15, 27, 70

Offset: 1

Author

N. J. A. Sloane and David James Sycamore, Nov 28 2018

Keywords

• nonn
• frac

Comments

Let s be the fraction defined in A322034 and A322035. Then for n >= 2, 1-s is a(n)/A322035(n).

Note that a(n) >= 1, see A322034.

Examples

			Let s be the fraction defined in A322034 and A322035. The fractions 1-s for n >= 2 are 1/2, 2/3, 1/4, 4/5, 1/3, 6/7, 1/8, 5/9, 2/5, 10/11, 1/6, 12/13, 3/7, 3/5, 1/16, 16/17, 5/18, 18/19, 1/5, 13/21, 5/11, 22/23, 1/12, 19/25, 6/13, 14/27, ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Crossrefs

Cf. A006022, A027746, A322034, A322035.

Programs

Maple

# This generates the terms starting at n=2:
P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
f0:=[]; f1:=[]; f2:=[];
for n from 2 to 120 do
a:=0; b:=1; t1:=[P(n)];
for i from 1 to nops(t1) do b:=b/t1[i]; a:=a+b; od;
f0:=[op(f0),a]; f1:=[op(f1), numer(a)]; f2:=[op(f2),denom(a)]; od:
f0;    # s
f1;    # A322034
f2;    # A322035
f2-f1; # A322036

Mathematica

f[x_] := Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]]; {1}~Join~Table[Denominator[#] - Numerator[#] &@ Total@ Table[1/Times @@ #[[;; i]], {i, Length[#]}] &@ f[n], {n, 2, 120}] (* Michael De Vlieger, Jun 20 2025 *)