A135060 -id:A135060 - cp's OEIS frontend

A136096 Ratios of consecutive terms of A135060, or 0 if quotient is not an integer.

2, 3, 2, 5, 2, 7, 1, 3, 1, 11, 2, 13, 1, 0, 2, 17, 1, 19, 1, 1, 1, 23, 3, 0, 1, 1, 0, 29, 1, 31, 0, 1, 1, 0, 0, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 0, 1, 1, 1, 53, 1, 0, 1, 1, 1, 59, 1, 61, 1, 0, 1, 1, 1, 67, 1, 1, 1, 71, 2, 73, 1, 1, 1, 0, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 97

Offset: 2

J. Lowell, May 10 2008

nonn

			a(8) = 1 because both A135060(7) and A135060(8) are 840

Ray Chandler, Table of n, a(n) for n=2..130

Cf. A135060.

Changed definition. - J. Lowell, Oct 30 2009

Extended by Ray Chandler, Nov 10 2009

A139315 a(n) is the smallest integer k such that n*k is the smallest multiple of k with twice as many divisors as k, or 0 if no such number is possible.

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 1260, 840, 0, 2520, 27720, 55440, 0, 720720, 1081080, 2162160, 61261200, 36756720, 1396755360, 2327925600, 0, 698377680, 16062686640, 48188059920, 0, 749592043200, 160626866400, 240940299600, 0, 6987268688400

Offset: 2

J. Lowell, Jun 07 2008

nonn

Proof that a(10)=0: In order for 10*k to have twice as many divisors as k, it must be either a multiple of 20 but not of 40 or 100 (in which case 8*k has twice as many divisors) or a multiple of 50 but not of 100 or 250 (in which case 4*k has twice as many divisors). In both cases, 10*k is not the smallest number with twice as many divisors as k and so a(10)=0.
Generalizing above result, a(pq)=0 for distinct primes p,q with p < q if p^2 < q. - Ray Chandler, Dec 03 2009
That is, a(m)=0 for m in A138511, but there are also other zeros, such as those at n = 30, 50, 68, 76, 90, 92, 98, ... - Michel Marcus, Sep 14 2020
a(n) is the least k such that A337686(k) = n, or 0 if there is no such k. - Michel Marcus, Sep 16 2020

			a(8) = 1260 because it must be a multiple of 4 but not of 8. It cannot be 4 because 4*3=12 has twice as many divisors as 4. It cannot be 12 because 12*5=60 has twice as many divisors as 12. It cannot be 60 because 60*6=360 has twice as many divisors as 60. It cannot be 180 because 180*7=1260 has twice as many divisors as 180. It must be 1260.

Ray Chandler, Table of n, a(n) for n = 2..100.

Cf. A129902, A135060, A138511, A337686.

a(14)-a(31) from Ray Chandler, Dec 03 2009
Name corrected by J. Lowell, Sep 14 2020
Name edited by Michel Marcus, Sep 15 2020

A138570 Numbers that set records for the number by which they must be multiplied to double the number of divisors.

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 840, 2520, 27720, 55440, 720720, 1081080, 2162160, 36756720, 698377680, 16062686640, 48188059920, 160626866400, 240940299600, 6987268688400, 216605329340400, 288807105787200, 1010824870255200

Offset: 1

J. Lowell, May 12 2008

nonn

A number is in this sequence exactly once if and only if it is in A135060 at least once.

			840 is a term of this sequence because 840 sets a new record (9) for the number by which it must be multiplied to double its number of divisors (840 has 32 divisors; the smallest number with 64 divisors is 840 * 9 = 7560).
120's record of 7 is not exceeded by any number from 121 to 839.

Ray Chandler, Table of n, a(n) for n = 1..52

Cf. A135060.

f[ n_ ] := f[ n ] = Module[ {i = 2}, While[ 2 * Length[ Divisors[ n ] ] != Length[ Divisors[ i * n ] ], i++ ]; Return[ i ] ]; A138570 = {}; searchMax = 100000; currHigh = 0; i = 1; While[ i < searchMax, n = f[ i ]; If[ n > currHigh, A138570 = {A138570, i}; currHigh = n ]; i++ ]; A138570 = Flatten[ A138570 ]

a(1) to a(8) verified and a(9) to a(12) added by Alonso del Arte, Oct 25 2009

Extended by Ray Chandler, Nov 10 2009

A140297 Values of records set by A138570.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16, 17, 19, 23, 24, 25, 28, 29, 31, 32, 35, 36, 37, 41, 43, 47, 49, 53, 55, 59, 61, 63, 67, 71, 72, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 113, 121, 127, 131

Offset: 1

J. Lowell, May 25 2008

nonn

			The value of 9 in this sequence corresponds to the record of 9 set by 840 in A138570

Cf. A135060, A138570.

a(12)=16 and a(15)-a(52) from Ray Chandler, Nov 10 2009

cp's OEIS Frontend

A136096 Ratios of consecutive terms of A135060, or 0 if quotient is not an integer.

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A139315 a(n) is the smallest integer k such that n*k is the smallest multiple of k with twice as many divisors as k, or 0 if no such number is possible.

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A138570 Numbers that set records for the number by which they must be multiplied to double the number of divisors.

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Mathematica

Extensions

A140297 Values of records set by A138570.

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