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

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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..18.

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.

Example

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A097214, A097416, A281993.

Cf. A002235. - Farideh Firoozbakht, May 03 2009

Sequence in context: A220994 A097416 A076373 * A281993 A126397 A164607

Adjacent sequences: A097212 A097213 A097214 * A097216 A097217 A097218

Keyword

nonn

Author

Matthew Vandermast, Aug 12 2004

Extensions

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

More terms from Jinyuan Wang, Feb 11 2020

Status

approved