A340346 - OEIS

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

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OFFSET

1,2

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

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).

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

Sequences with related definitions: A327832, A328479.

Cf. A234959.

Sequence in context: A161510 A329379 A328479 * A193267 A327832 A083258

Adjacent sequences: A340343 A340344 A340345 * A340347 A340348 A340349

KEYWORD

nonn,easy

AUTHOR

Peter Munn, Jan 04 2021

STATUS

approved