A134188 -id:A134188 - cp's OEIS frontend

A134187 a(0)=1. a(n) = the number of terms of the sequence (from among terms a(0) through a(n-1)) which equal any "non-isolated divisors" of (2n). A divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

1, 1, 2, 3, 3, 3, 6, 3, 3, 8, 3, 3, 10, 3, 3, 13, 3, 3, 14, 3, 3, 17, 3, 3, 18, 3, 3, 20, 4, 3, 23, 3, 3, 23, 3, 3, 27, 3, 3, 27, 4, 3, 31, 3, 3, 32, 3, 3, 34, 3, 5, 33, 3, 3, 37, 4, 4, 35, 3, 3, 43, 3, 3, 40, 3, 3, 45, 3, 3, 43, 8, 3, 50, 3, 3, 48, 3, 3, 53, 3, 8, 49, 3, 3, 59, 3, 3, 53, 3, 3, 62, 5

Offset: 0

Leroy Quet, Oct 12 2007

nonn

			The positive divisors of 2*12=24 are 1,2,3,4,6,8,12,24. Of these, 1,2,3,4 are the non-isolated divisors of 24. There are 2 terms among the earlier terms of the sequence that equal 1, 1 term that equals 2, 7 terms which equal 3 and 0 terms which equal 4. So a(12) = 2+1+7+0 = 10.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A132747, A132881, A134188.

up_to = 91;
A134187list(up_to) = { my(v=vector(1+up_to)); v[1] = 1; for(n=1,up_to,v[1+n] = sum(k=0,n-1,my(u=v[1+k]); !((2*n)%u) && ((!((2*n)%(1+u))) || ((u>1)&&(!((2*n)%(u-1))))))); (v); };
v134187 = A134187list(up_to);
A134187(n) = v134187[1+n]; \\ Antti Karttunen, Apr 06 2021

Extended by Ray Chandler, Jun 25 2008

A133828 a(n) = the smallest "isolated divisor" of n, or 0 if no such divisor exists. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

1, 0, 1, 4, 1, 6, 1, 4, 1, 5, 1, 6, 1, 7, 1, 4, 1, 6, 1, 10, 1, 11, 1, 6, 1, 13, 1, 4, 1, 10, 1, 4, 1, 17, 1, 6, 1, 19, 1, 8, 1, 14, 1, 4, 1, 23, 1, 6, 1, 5, 1, 4, 1, 6, 1, 4, 1, 29, 1, 10, 1, 31, 1, 4, 1, 6, 1, 4, 1, 5, 1, 6, 1, 37, 1, 4, 1, 6, 1, 8, 1, 41, 1, 12, 1, 43, 1, 4, 1, 15, 1, 4, 1, 47, 1, 6, 1

Offset: 1

Leroy Quet, Sep 25 2007

nonn

a(2n-1) = 1 for all positive integers n. 2 has no isolated divisors. a(2) is 0 only as a placeholder.

			a(18)=6 because the isolated divisors of 18 are 6,9 and 18.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A133779, A133829, A134187, A134188.

with(numtheory): a:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1, div)=false and member(div[i]+1, div) = false then ISO := `union`(ISO, {div[i]}) end if end do end proc: 1, 0, seq(a(j)[1],j=3..80); # Emeric Deutsch, Oct 16 2007
A133828 := proc(n) local divs,k,i ; divs := sort(convert(numtheory[divisors](n),list)) ; for i from 1 to nops(divs) do k := op(i,divs) ; if not k-1 in divs and not k+1 in divs then RETURN(k) ; fi ; od: RETURN(0) ; end: seq(A133828(n),n=1..100) ; # R. J. Mathar, Oct 19 2007

a[n_] := If[OddQ[n], 1, For[d = 2, d <= n, d++, If[Divisible[n, d] && !Divisible[n, d-1] && !Divisible[n, d+1], Return[d]]]] /. Null -> 0;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2024 *)

A133828(n) = if(n%2,1,fordiv(n,d,if((d>1)&&(n%(d-1))&&(n%(d+1)), return(d))); (0)); \\ Antti Karttunen, Apr 01 2021

More terms from Emeric Deutsch and R. J. Mathar, Oct 16 2007

cp's OEIS Frontend

A134187 a(0)=1. a(n) = the number of terms of the sequence (from among terms a(0) through a(n-1)) which equal any "non-isolated divisors" of (2n). A divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

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