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.

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

1,4

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Example

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

Maple

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

Prog

(PARI) 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

Crossrefs

Cf. A133779, A133829, A134187, A134188.

Sequence in context: A109039 A257656 A109040 * A144907 A185093 A010779

Adjacent sequences: A133825 A133826 A133827 * A133829 A133830 A133831

Keyword

nonn

Author

Leroy Quet, Sep 25 2007

Extensions

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

Status

approved