A097416 -id:A097416 - 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

A097215 Numbers m such that A076078(m) = m and bigomega(m) >= 2; or in other words, A097214, excluding powers of 2.

Original entry on oeis.org

10, 44, 184, 752, 12224, 49024, 61064, 981520, 12580864, 206158168064, 16492668126208, 1080863908958322688, 18374686467592175488, 885443715520878608384, 4703919738602662723328, 226673591177468092350464, 232113757366000005450563584, 3894222643901120685369075227951104

Offset: 1

Matthew Vandermast, Aug 12 2004

nonn

A076078(m) equals the number of sets of distinct positive integers with a least common multiple of m.
If 3*2^k - 1 is an odd prime then 2^k*(3*2^k-1) is in the sequence. - Farideh Firoozbakht, May 03 2009
For what seems to be an appearance of this sequence in a different context, see Harborth (2013). - N. J. A. Sloane, Jun 08 2013

			For example, there are 184 sets of distinct positive integers with a least common multiple of 184.

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A097214, A097416, A281993.

Cf. A002235. - Farideh Firoozbakht, May 03 2009

f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; t = Union[ Table[ f[n], {n, 28000000}]]; Select[t, f[ # ] == # && !IntegerQ[ Log[2, # ]] &] (* Robert G. Wilson v, Aug 17 2004 *)

A076078(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; }
lista(nn) = {my(w=List([]), m=1, q=2, g); for(k=1, logint(nn, 2)-1, q=nextprime(q+1); m=m*q; for(r=1, nn\2^k-1, g=factor(A076078(m*2^r))[, 2]; if(#g==k+1&&g[2]==1, listput(w, A076078(m*2^r))))); Set(w); } \\ Jinyuan Wang, Feb 11 2020

More terms from Robert G. Wilson v, Aug 18 2004

More terms from Jinyuan Wang, Feb 11 2020

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.

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