A185208 -id:A185208 - cp's OEIS frontend

A141197 a(n) = the number of divisors of n that are each one less than a power of a prime.

1, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 6, 1, 3, 3, 5, 1, 5, 1, 4, 3, 3, 1, 8, 1, 3, 2, 5, 1, 7, 2, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 6, 1, 4, 3, 3, 1, 10, 2, 3, 2, 5, 1, 5, 1, 6, 2, 3, 1, 10, 1, 3, 4, 5, 1, 6, 1, 3, 2, 5, 1, 11, 1, 2, 3, 3, 2, 6, 1, 8, 2, 3, 1, 9, 1, 2, 2, 6, 1, 8, 2, 4, 3, 2, 1, 11, 1, 3, 2, 5, 1, 5, 1

Offset: 1

Leroy Quet, Jun 12 2008

nonn

A067513(n) <= a(n) <= A000005(n). [From Reinhard Zumkeller, Oct 06 2008]

a(A185208(n)) = 1. - Reinhard Zumkeller, Nov 01 2012

			The divisors of 9 are 1,3,9. 1 is one less than 2, a power of a prime. 3 is one less than 4, a power of a prime. And 9 is one less than 10, not a power of a prime. There are therefore 2 such divisors that are each one less than a power of a prime. So a(9)=2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A141198.

Cf. A049073.

a141197 = sum . map (a010055 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Nov 01 2012

a[n_] := Select[Divisors[n], PrimeNu[# + 1] == 1 &] // Length; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Aug 17 2013 *)
Table[DivisorSum[n, 1 &, PrimePowerQ[# + 1] &], {n, 103}] (* Michael De Vlieger, Aug 29 2017 *)

a(n) = sum (A010055(A027750(n,k)): k=1..A000005(n)). - Reinhard Zumkeller, Nov 01 2012

Added more terms. - Steven Bi (chenhsi(AT)stanford.edu), Dec 22 2008
Added more terms (Terms 27 - 50). Steven Bi (chenhsi(AT)stanford.edu), Jan 09 2009
Corrected and extended by Ray Chandler, Jun 25 2009

A261871 Numbers of the form (2j-1)(2^k-1); j>=1, k>=2.

Original entry on oeis.org

3, 7, 9, 15, 21, 27, 31, 33, 35, 39, 45, 49, 51, 57, 63, 69, 75, 77, 81, 87, 91, 93, 99, 105, 111, 117, 119, 123, 127, 129, 133, 135, 141, 147, 153, 155, 159, 161, 165, 171, 175, 177, 183, 189, 195, 201, 203, 207, 213, 217, 219, 225, 231, 237, 243, 245, 249, 255, 259, 261, 267, 273, 279, 285, 287, 291, 297, 301

Offset: 1

Bob Selcoe, Sep 04 2015

Odd numbers complementary to A185208.

Lim_{n->inf.} a(n)/n > 6/(1 + Sum_{j>=1} (2/(2^(2j+1)-1))) ~ 4.375745.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A185208.

Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

lmt = 310; Take[ Union@ Flatten@ Table[ (2j - 1)(2^k - 1), {j, lmt/4}, {k, 2, 1 + Log2[ lmt/(2j)] }], 68] (* Michael De Vlieger, Sep 04 2015 *) (* and modified by Robert G. Wilson v, Sep 05 2015 *)

list(lim)=my(v=List(),t); for(k=2,logint(lim\1+1,2), t=2^k-1; forstep(j=1,lim\t,2, listput(v,t*j))); Set(v) \\ Charles R Greathouse IV, Sep 05 2015

2n < a(n) < 5n. For n > 51, 4.3n < a(n) < 4.5n. - Charles R Greathouse IV, Sep 05 2015

A049073 LCM of all divisors of d of n such that d+1 is a prime power.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 3, 28, 1, 30, 31, 16, 3, 2, 7, 36, 1, 2, 3, 40, 1, 42, 1, 44, 15, 46, 1, 48, 7, 10, 3, 52, 1, 18, 1, 56, 3, 58, 1, 60, 1, 62, 63, 16, 1, 66, 1, 4, 3, 70, 1, 72, 1, 2, 15, 4, 7, 78, 1, 80, 3, 82, 1, 84, 1

Offset: 1

David E. Daykin

a(A185208(n)) = 1. - Reinhard Zumkeller, Nov 01 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A141197, A027750, A010055.

a049073 = foldl lcm 1 . filter ((== 1) . a010055 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Nov 01 2012

a[n_] := LCM @@ Select[Divisors[n], PrimeNu[# + 1] == 1 &]; Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Aug 17 2013 *)

More terms from Erich Friedman, Jun 03 2001

A219033 Numbers n such that n = x + y, sigma_1(n) = sigma_1(x) + sigma_1(y) and sigma_2(n) = sigma_2(x) + sigma_2(y).

Original entry on oeis.org

434, 2170, 4774, 5642, 7378, 8246, 9982, 10850, 12586, 16058, 17794, 18662, 20398, 23002, 23870, 25606, 26474, 28210, 29078, 30814, 31682, 34286, 36022, 36890, 38626, 41230, 42098, 43834, 44702, 47306, 49042, 49910, 52514, 54250, 55118, 56854, 59458, 60326

Offset: 1

Jon Perry, Nov 10 2012

nonn

Conjecture: This sequence is infinite.
Conjecture: The sequence only consists of even numbers.
Conjecture: The partitions only consist of even numbers.
Conjecture: None satisfy sigma_3(n) = sigma_3(x) + sigma_3(y).
Conjecture: With the lower partition as 6*A185208(n) and the upper partition 214/3 = 71.3333... of this, then the equalities are satisfied.
The first 12 partitions are (428, 6), (2140, 30), (4708, 66), (5564, 78), (7276, 102), (8132, 114), (9844, 138), (10700, 150), (12412, 174), (15836, 222), (17548, 246), (18404, 258).
The first example of this ratio not being used is at a(67) = 103818 where (103554, 264) satisfies the equalities. Here the ratio is 1569/4 = 392.25. - Donovan Johnson, Nov 13 2012

			2140 + 30 = 2170.
sigma_1(2140) + sigma_1(30) = 4536 + 72 = 4608 = sigma_1(2170).
sigma_2(2140) + sigma_2(30) = 6251700 + 1300 = 6253000 = sigma_2(2170).
Hence, 2170 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000203, A001157, A185208.

function divisorSum(n,x) {
c=0;
for (i=1;i<=n;i++) if (n%i==0) c+=Math.pow(i,x);
return c;
}
ds=new Array();
for (j=1;j<40001;j++) {
ds[j]=new Array();
ds[j][0]=divisorSum(j,1);
ds[j][1]=divisorSum(j,2);
}
a=new Array();
ac=0;
for (j=1;j<20000;j++)
for (k=1;k<=j;k++)
if (ds[j][0]+ds[k][0]==ds[j+k][0] && ds[j][1]+ds[k][1]==ds[j+k][1]) a[ac++]=j+", "+k+" ::: ";
a.sort(function(a, b) {return a-b;});
i=0;
while(i++

a(6) corrected and a(13)-a(38) from Donovan Johnson, Nov 10 2012

cp's OEIS Frontend

A141197 a(n) = the number of divisors of n that are each one less than a power of a prime.

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A261871 Numbers of the form (2*j-1)*(2^k-1); j>=1, k>=2.

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A049073 LCM of all divisors of d of n such that d+1 is a prime power.

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A219033 Numbers n such that n = x + y, sigma_1(n) = sigma_1(x) + sigma_1(y) and sigma_2(n) = sigma_2(x) + sigma_2(y).

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A261871 Numbers of the form (2j-1)(2^k-1); j>=1, k>=2.