A134865 Numbers k meeting the following criterion: if k is a multiple of d, then it is also a multiple of the smallest number with same number of divisors as d.

1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 2520, 5040, 7560, 10080, 15120, 20160, 45360, 50400, 100800, 332640, 352800, 665280, 705600, 4324320, 8648640, 17297280, 21621600, 43243200, 13492656777600

Offset: 1

Author

J. Lowell, Jan 29 2008

Keywords

• more
• nonn

Comments

Note that this is not a subsequence of A002182: 100800 is in this sequence but not in A002182. - J. Lowell, Feb 22 2008

A subset of A005179. - Max Alekseyev, May 19 2008

A number k is in this sequence iff for every divisor d of k, A005179(A000005(d)) (= A140635(d)) is also a divisor of k. So the question of the finiteness of this sequence is closely related to the form of the elements of A005179. - Max Alekseyev, May 19 2008, May 20 2008

Rearrangement of this sequence, forming a subsequence of A005179, is given by A140753. Corresponding indices of elements of A005179 are given by A138394 and A140752. - Max Alekseyev, May 26 2008

A subsequence of A007416 which is a subsequence of A025487, so every term is primally tight and even (after the first term). Thus if d is a divisor of a term, then the least integer with the same prime signature as d (=A046523(d)) is also a divisor. So only the divisors that are in A025487 need be tested. - Ray Chandler

a(32) > 8*10^25 if it exists. - David A. Corneth, Dec 10 2021

Examples

			60 is a multiple of 30 with 8 divisors, but not of 24 (the smallest number with 8 divisors) so 60 is not a term of this sequence.
		

Crossrefs

Cf. A000005, A005179, A007416, A138394, A140752, A140753.

Programs

Mathematica

a = {}; For[n = 1, n < 10000, n++, b = Divisors[n]; c = 1; For[i = 1, i < Length[b] + 1, i++, j = 1; While[Length[Divisors[j]] < Length[Divisors[b[[i]]]], j++ ]; If[ ! Mod[n, j] == 0, c = 0]]; If[c == 1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Feb 05 2008 *)

PARI

isA134865(n)={ n%2 & return(n==1); fordiv(n, d, bigomega(d)>1 || next; nd=numdiv(d); for(k=4, d, numdiv(k)==nd || next; n%k & return; break)); 1 }
for(n=1,10^7,if(isA134865(n),print1(n,", "))); \\ R. J. Mathar, May 17 2008

Formula

a(n) = A005179(A140752(n)). - Max Alekseyev, May 26 2008

Extensions

More terms from Stefan Steinerberger, Feb 05 2008

More terms from J. Lowell, Feb 22 2008

a(22)-a(30) from Don Reble, May 17 2008

a(31)=13492656777600 from Ray Chandler, Jun 30 2008