A340346 The largest divisor of n that is a term of A055932 (numbers divisible by all primes smaller than their largest prime factor).

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 16, 1, 2, 1, 12

Offset: 1

Author

Peter Munn, Jan 04 2021

Keywords

• nonn
• easy

Examples

			For n=2: the largest divisor of 2 is 2, and 2 qualifies as divisible by all primes smaller than its largest prime factor, 2 (since there are no smaller primes). So a(2) = 2.
For n=42: of 42's divisors, no multiples of 7 qualify as being divisible by all primes smaller than their largest prime factor (since that factor is 7 and no divisor of 42 is divisible by 5, a smaller prime). The largest of 42's other divisors is 6, which qualifies (since it is divisible by 2, the only prime smaller than 6's largest prime factor, 3). So a(42) = 6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

A003961, A006519, A055932, A064989, A341629 are used in a definition of this sequence.

Sequences with related definitions: A327832, A328479.

Cf. A234959.

Programs

Mathematica

a[?OddQ] = 1; a[n] := Module[{f = FactorInteger[n]}, ind = Position[PrimePi /@ First /@ f - Range @ Length[f], ?(# > 0 &)]; If[ind == {}, n, Times @@ Power @@@ f[[1 ;; ind[[1, 1]] - 1]]]]; Array[a, 100] (* _Amiram Eldar, Jan 14 2021 *)

PARI

is(n) = my(f=factor(n)[, 1]~); f==primes(#f); \\ A055932
a(n) = vecmax(select(is, divisors(n))); \\ Michel Marcus, Jan 19 2021

PARI

A341629(n) = if(1==n,1,my(f=factor(n)[, 1]~); (primepi(f[#f])==#f));
A340346(n) = fordiv(n,d,if(A341629(n/d),return(n/d))); \\ Antti Karttunen, Feb 25 2021

Formula

For n >= 1, a(2n-1) = 1, a(2n) = A006519(2n) * A003961(a(A064989(2n))).

For n >= 1, lcm(A006519(n), A234959(n)) | a(n).