A108118 -id:A108118 - cp's OEIS frontend

A099477 Numbers having no divisors d such that also d+2 is a divisor.

1, 2, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 110, 113, 115, 118, 119, 121, 122, 125, 127, 130, 131, 133

Offset: 1

Reinhard Zumkeller, Oct 18 2004

nonn

Except for 3, all primes are in this sequence. - Alonso del Arte, Jun 13 2014

			10 is in the sequence because its divisors are 1, 2, 5, 10, none of which is 2 less than another.
11 is in the sequence as are all primes other than 3.
12 is not in the sequence because its divisors are 1, 2, 3, 4, 6, 12, of which 2 and 4 are 2 less than another divisor.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A059267.

Cf. A099475, A108118.

twinDivsQ[n_] := Union[ IntegerQ[ # ] & /@ (n/(Divisors[n] + 2))][[ -1]] == True; Select[ Range[133], !twinDivsQ[ # ] &] (* Robert G. Wilson v, Jun 09 2005 *)
d2noQ[n_]:=Module[{d=Divisors[n]},Intersection[d,d+2]=={}]; Select[ Range[ 150],d2noQ] (* Harvey P. Dale, Feb 15 2019 *)

A099475(a(n)) = 0.

A285440 Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.

Original entry on oeis.org

3, 4, 8, 9, 15, 16, 20, 21, 27, 28, 32, 33, 39, 40, 44, 45, 51, 52, 56, 57, 63, 64, 68, 69, 75, 76, 80, 81, 87, 88, 92, 93, 99, 100, 104, 105, 111, 112, 116, 117, 123, 124, 128, 129, 135, 136, 140, 141, 147, 148, 152, 153, 159, 160, 164, 165, 171, 172, 176, 177

Offset: 1

Paolo P. Lava, Apr 19 2017

Numbers with no sum equal to n are listed in A108118, with two sums equal to n are listed in A017593 and with three sums equal to n in A008594.
First difference has period 4: {1,4,1,6}.
Numbers that are congruent to {3, 4, 8, 9} mod 12. - Amiram Eldar, Dec 31 2021

			20 is in the sequence because:
gcd(k,20) = 1 for k = 1, 3, 7, 9, 11, 13, 17, 19: sum is 80.
gcd(k,20) = 2 for k = 2, 6, 14, 18: sum is 40.
gcd(k,20) = 4 for k = 4, 8, 12, 16: sum is 40.
gcd(k,20) = 5 for k = 5, 15: sum is 20.
gcd(k,20) = 10 for k = 10: sum is 10.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008594, A017593, A108118.

P:=proc(q) local a,k,n,t;
for n from 1 to q do a:=array(1..n-1); for k from 1 to n-1 do a[k]:=0; od;
for k from 1 to n-1 do a[gcd(n,k)]:=a[gcd(n,k)]+k; od; t:=0;
for k from 1 to n-1 do if a[k]=n then t:=t+1; fi; od; if t=1 then print(n); fi;
od; end: P(10^6);

Flatten@ Position[#, k_ /; Length@ k == 1] &@ Table[Select[Transpose@ {Values@ #, Keys@ #} &@ Map[Total, PositionIndex@ Map[GCD @@ {n, #} &, Range[n - 1]]], First@ # == n &][[All, -1]], {n, 180}] (* Michael De Vlieger, Apr 28 2017, Version 10 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 4, 8, 9, 15}, 60] (* Amiram Eldar, Dec 31 2021 *)

a(n) = n--; [3, 4, 8, 9][n%4+1] + 12*(n\4) \\ David A. Corneth, Apr 28 2017

is(n) = {my(d=divisors(n), map=vector(d[#d-1]), v=vector(#d-1)); for(i=1,#d-1, map[d[i]]=i); for(i=1,n-1,v[map[gcd(i, n)]]+=i); sum(i=1,#v,v[i]==n)==1} \\ David A. Corneth, Apr 28 2017

is(n) = vecsort(concat([3, 4, 8, 9], [n%12]), ,8)==[3, 4, 8, 9] \\ David A. Corneth, Apr 28 2017

From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x*(3*x^4 + x^3 + 4*x^2 + x + 3)/(x^5 - x^4 - x + 1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3-sqrt(3))*Pi/36. - Amiram Eldar, Dec 31 2021

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A099477 Numbers having no divisors d such that also d+2 is a divisor.

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A285440 Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.

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