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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.
4
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
1,2
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
FORMULA
a(n) = A005179(A140752(n)). - Max Alekseyev, May 26 2008
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
more,nonn
AUTHOR
J. Lowell, Jan 29 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
STATUS
approved