A139315 -id:A139315 - cp's OEIS frontend

A357249 a(n) = A139315(n)*n.

2, 6, 24, 60, 360, 840, 10080, 7560, 0, 27720, 332640, 720720, 0, 10810800, 17297280, 36756720, 1102701600, 698377680, 27935107200, 48886437600, 0, 16062686640, 385504479360, 1204701498000, 0, 20238985166400, 4497552259200, 6987268688400, 0, 216605329340400

Offset: 2

J. Lowell, Sep 19 2022

nonn

			a(8) = A139315(8)*8 = 1260*8 = 10080.

Cf. A129902, A139315.

a(n) = A129902(A139315(n)).

More terms from Michel Marcus, Sep 20 2022

A365965 Numbers k such that A139315(k) = 0 but k is not in A138511.

Original entry on oeis.org

30, 50, 68, 76, 90, 92, 98, 116, 124, 132, 148, 150, 154, 160, 164, 165, 172, 174, 182

Offset: 1

J. Lowell, Sep 23 2023

			30 is not in A138511, but A139315(30)=0.

Cf. A139315, A138511, A337709.

A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.

Original entry on oeis.org

2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 6, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4

Offset: 1

Michel Marcus, Sep 15 2020

nonn

The zeros in A139315 are the missing values in this sequence (see A337709).
There are no 1's in this sequence. a(n) = 2 for all odd n and a(n) >= 3 for all even n. - J. Lowell, Sep 15 2020
Empirical observation: A007978(n) - a(n) = 1 for n = 60*A206547(n), = 2 for n = 420*A007310(n), else = 0. - Hugo Pfoertner, Sep 30 2020

			a(1) = 2 because 1 has 1 divisor, 1*2 has 2 divisors, so 2 is the least multiplier to apply to 1 to get twice as many divisors.

Michel Marcus, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Illustration of ratio A007978(n) / a(n), using Plot 2.

Cf. A000005, A129902, A139315, A337709 (missing values).

nn = 105; Do[d[i] = DivisorSigma[0, i], {i, 12 nn}]; Reap[Do[m = 2; While[d[m i] != 2 d[i], m++]; Sow[m ], {i, nn}]][[-1, -1]] (* Michael De Vlieger, Jan 10 2022 *)

a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(n), k++); k;}

a(n) = A129902(n)/n.

A167401 a(n) is the smallest number k such that n*k has twice as many divisors as k.

Original entry on oeis.org

1, 1, 2, 1, 12, 1, 4, 3, 20, 1, 72, 1, 28, 45, 8, 1, 108, 1, 160, 63, 44, 1, 288, 5, 52, 9, 224, 1, 10800, 1, 16, 99, 68, 175, 864, 1, 76, 117, 800, 1, 21168, 1, 352, 675, 92, 1, 1152, 7, 400, 153, 416, 1, 648, 275, 1568, 171, 116, 1, 259200

Offset: 2

J. Lowell, Nov 02 2009

a(n) is 1 for all prime numbers n.
From Robert Israel, Feb 09 2017: (Start)
All prime factors of a(n) divide n.
If n=p^k is a prime power, a(n) = p^(k-1).
If n=p*q with pA006881, a(n) = p^2*q. (End)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A139315.

A167401 := proc(n) if isprime(n) then 1; else for a from 2 do if numtheory[tau](n*a) = 2*numtheory[tau](a) then return a ; end if; end do ; fi; end: seq(A167401(n),n=2..60) ; # R. J. Mathar, Nov 04 2009

tmd[n_]:=Module[{a=1},While[DivisorSigma[0,a*n]!=2DivisorSigma[0,a],a++];a]; Array[tmd,60,2] (* Harvey P. Dale, Apr 20 2013 *)

a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(k), k++); k;} \\ Michel Marcus, Feb 10 2017

Extended by Ray Chandler, Nov 10 2009

Extended beyond a(10) by R. J. Mathar, Nov 04 2009

A337709 Integers m such that A337686(x) = m has no solution.

Original entry on oeis.org

1, 10, 14, 22, 26, 30, 33, 34, 38, 39, 46, 50, 51, 57, 58, 62, 68, 69, 74, 76, 82, 86, 87, 90, 92, 93, 94, 98, 106, 111, 116, 118, 122, 123, 124, 129, 132, 134, 141, 142, 145, 146, 148, 150, 154, 155, 158, 159, 160, 164, 165, 166, 172, 174, 177, 178, 182, 183, 185

Offset: 1

Michel Marcus, Sep 16 2020

nonn

Integers m such that A139315(m) = 0.

Cf. A000005, A139315, A337686.

A138511 is a subsequence.

More terms from Jinyuan Wang, Sep 30 2020

Missing term a(1)=1 inserted by J. Lowell, Jul 14 2022

cp's OEIS Frontend

A357249 a(n) = A139315(n)*n.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A365965 Numbers k such that A139315(k) = 0 but k is not in A138511.

Views

Author

Keywords

Examples

Crossrefs

A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A167401 a(n) is the smallest number k such that n*k has twice as many divisors as k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A337709 Integers m such that A337686(x) = m has no solution.

Views

Author

Keywords

Comments

Crossrefs

Extensions