A140752 -id:A140752 - cp's OEIS frontend

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

J. Lowell, Jan 29 2008

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

			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.

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

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 *)

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

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

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

A138394 Smallest number with a(n) divisors is in A134865.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 16, 20, 24, 30, 48, 60, 64, 72, 80, 84, 100, 108, 126, 162, 189, 192, 224, 384, 448, 512, 576, 672, 11520

Offset: 1

J. Lowell, May 08 2008

The indices of elements of A005179 that belong to A134865. Sorted version of the sequence A140752. - Max Alekseyev, May 26 2008

			8 is in this sequence because the smallest number with 8 divisors (24) is a member of A134865.

Cf. A000005, A005179, A134865, A140752, A140753.

a(n) = A000005(A140753(n)) - Max Alekseyev, May 26 2008

a(31)=11520 from Ray Chandler, Jun 30 2008

A140753 Subsequence of elements of A005179 that appear in A134865.

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, May 26 2008

Rearrangement of elements of A134865 in order of their appearance in A005179. Hence A134865 represents a sorted variant of this sequence.

Cf. A000005, A005179, A134865, A138394, A140752.

a(n) = A005179(A138394(n))

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

cp's OEIS Frontend

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.

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A138394 Smallest number with a(n) divisors is in A134865.

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A140753 Subsequence of elements of A005179 that appear in A134865.

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