A097218 -id:A097218 - cp's OEIS frontend

A076078 a(n) is the number of nonempty sets of distinct positive integers that have a least common multiple of n.

1, 2, 2, 4, 2, 10, 2, 8, 4, 10, 2, 44, 2, 10, 10, 16, 2, 44, 2, 44, 10, 10, 2, 184, 4, 10, 8, 44, 2, 218, 2, 32, 10, 10, 10, 400, 2, 10, 10, 184, 2, 218, 2, 44, 44, 10, 2, 752, 4, 44, 10, 44, 2, 184, 10, 184, 10, 10, 2, 3748, 2, 10, 44, 64, 10, 218, 2, 44, 10, 218, 2, 3392, 2, 10

Offset: 1

Amarnath Murthy, Oct 05 2002

a(n)=1 iff n=1, a(p^k)=2^k, a(p*q)=10; where p & q are unique primes. a(n) cannot equal an odd number >1. - Robert G. Wilson v
If m has more divisors than n, then a(m) > a(n). - Matthew Vandermast, Aug 22 2004
If n is of the form p^r*q^s where p & q are distinct primes and r & s are nonnegative integers then a(n)=2^(rs)*(2^(r+s+1) -2^r-2^s+1); for example f(1400846643)=f(3^5*7^8)=2^(5*8)*(2^ (5+8+1)-2^5-2^8+1)=17698838672310272. Also if n=p_1^r_1*p_2^r_2*...*p_k^r_k where p_1,p_2,...,p_k are distinct primes and r_1,r_2,...,r_k are natural numbers then 2^(r_1*r_2*...*r_k)||a(n). - Farideh Firoozbakht, Aug 06 2005
None of terms is divisible by Mersenne numbers 3 or 7. For any n, a(n) is congruent to A008836(n) mod 3. Since A008836(n) is always 1 or -1, this implies that A000225(2)=3 never divides a(n). - Matthew Vandermast, Oct 12 2010
There are terms divisible by larger Mersenne numbers. For example, a(2*3*5*7*11*13*19*23^3) is divisible by 31. - Max Alekseyev, Nov 18 2010

			a(6) = 10. The sets with LCM 6 are {6}, {1,6}, {2,3}, {2,6}, {3,6}, {1,2,3}, {1,2,6}, {1,3,6}, {2,3,6}, {1,2,3,6}.

David Wasserman, Table of n, a(n) for n = 1..1000
Index entries for sequences related to prime signature

Cf. A076413, A097210-A097218, A097416, A002235.

Cf. A069626.

with(numtheory): seq(add(mobius(n/d)*(2^tau(d)-1), d in divisors(n)), n=1..80); # Ridouane Oudra, Mar 12 2024

f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; Table[ f[n], {n, 75}] (* Robert G. Wilson v *)

a(n) = local(f, l, s, t, q); f = factor(n); l = matsize(f)[1]; s = 0; forvec(v = vector(l, i, [0, 1]), q = sum(i = 1, l, v[i]); t = (-1)^(l - q)*2^prod(i = 1, l, f[i, 2] + v[i]); s += t); s; \\ Definition corrected by David Wasserman, Dec 26 2007

2^d(n) - 1 = Sum_{m|n} a(m), where d(n) = A000005(n) is the number of divisors of n, so a(n) = Sum_{m|n} mu(n/m)*(2^d(m) - 1).

a(n) = 2*A069626(n), for n > 1. - Ridouane Oudra, Mar 12 2024

Edited by Dean Hickerson, Oct 08 2002
Definition corrected by David Wasserman, Dec 26 2007
Edited by Charles R Greathouse IV, Aug 02 2010
Edited by Max Alekseyev, Nov 18 2010

A097217 Odd numbers n such that A076078(n) > n, where A076078(n) equals the number of sets of distinct positive integers with a least common multiple of n.

Original entry on oeis.org

105, 135, 165, 195, 225, 315, 405, 495, 525, 567, 585, 675, 693, 765, 819, 825, 855, 945, 975, 1035, 1071, 1125, 1155, 1197, 1215, 1275, 1287, 1305, 1323, 1365, 1395, 1425, 1449, 1485, 1575, 1665, 1683, 1701, 1725, 1755, 1785, 1827, 1845, 1881, 1925

Offset: 1

Matthew Vandermast, Aug 13 2004

nonn

Odd members of A097216.

Cf. A097214, A097218.

A214547 Deficient numbers for which the (absolute value of) abundance is not a divisor.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106

Offset: 1

Jonathan Vos Post, Jul 20 2012

This is to A214408 as deficient numbers are to abundant numbers.
Differs from A097218, which does not contain 105, for example.
The deficient numbers which are *not* in the sequence are 2, 4, 8, 10, 16, 32, 44, 64, 128, 136, 152, 184, 256, 512, 752, 884, 1024, 2048, 2144, 2272, 2528, 4096, 8192, 8384, 12224, 16384, 17176, 18632, 18904, 32768, 32896, 33664, ... the union of powers of 2 and the terms of A060326. - M. F. Hasler, Jul 21 2012

			7 is in the sequence because 7 is deficient, and its abundance is -6, and |-6| = 6 does not divide 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005100, A033880, A060326, A097218, A214408.

filter:= proc(n) local t;
t:= 2*n-numtheory:-sigma(n);
t > 0 and n mod t <> 0
end proc:
select(filter, [$1..200]); # Robert Israel, Nov 13 2019

q[n_] := Module[{def = 2*n - DivisorSigma[1, n]}, def > 0 && !Divisible[n, def]]; Select[Range[120], q] (* Amiram Eldar, Apr 07 2024 *)

is_A214547(n)={sigma(n)<2*n & n%(2*n-sigma(n))} \\ M. F. Hasler, Jul 21 2012

Terms A005100(n) such that |A033880(A005100(n))| does not divide A005100(n).

Given terms double-checked with the PARI script by M. F. Hasler, Jul 21 2012

cp's OEIS Frontend

A076078 a(n) is the number of nonempty sets of distinct positive integers that have a least common multiple of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A097217 Odd numbers n such that A076078(n) > n, where A076078(n) equals the number of sets of distinct positive integers with a least common multiple of n.

Views

Author

Keywords

Comments

Crossrefs

A214547 Deficient numbers for which the (absolute value of) abundance is not a divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions