A281993 -id:A281993 - cp's OEIS frontend

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

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

A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

Original entry on oeis.org

34, 568, 147328, 603971584, 9663643648, 39582416502784, 696341272098017608537735168, 765635325572111542783369494684623699968, 3615610599582728119969414707766982030374842621310535527825408, 3791242500068058721125048996612134914443116117566314438843154038784

Offset: 1

Jinyuan Wang, Feb 12 2020

nonn

This is the case h = 8 of the h-perfect numbers as defined in the Harborth link.

			34 is a term since sigma(34) + sigma(8*34) = 612, that is 18*34.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A002236, A050524, A136539, A281993, A330413.

isok(m) = sigma(m) + sigma(8*m) == 18*m;

a(n) = 2^A002236(n) * A050524(n).

A330413 Integers m such that sigma(m) + sigma(16m) = 34m.

Original entry on oeis.org

268, 4336, 69568, 73014378496, 18691696623616, 80280230208715249156096, 5516815412193254337299253840314368, 22596875928343569838211798520159010816, 106710729501573572985208420194451100911225778218295042768896, 7689318425915528602346510723233181380859942271270135051778769275060995751936

Offset: 1

Jinyuan Wang, Feb 12 2020

nonn

This is the case h = 16 of the h-perfect numbers as defined in the Harborth link.

			268 is a term since sigma(268) + sigma(16*268) = 9112, that is 34*268.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A001774, A136539, A281993, A330412.

isok(m) = sigma(m) + sigma(16*m) == 34*m;

a(n) = 2^A001774(n) * (17*2^A001774(n) - 1).

A334128 Integers m such that sigma(m) + sigma(7m) = 16m.

Original entry on oeis.org

135, 1365, 42819777

Offset: 1

Jinyuan Wang, May 15 2020

This is the case h = 7 of the h-perfect numbers as defined in the Harborth link.
If gcd(m, 7) = 1 and m is even, then m must be of the form 2*q, where 27*sigma(q) = 32*q and gcd(q, 7) = 1.
a(4) > 3.5*10^12, if it exists. - Giovanni Resta, May 17 2020

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A281993, A330412, A330413.

isok(m) = sigma(m) + sigma(7*m) == 16*m;

cp's OEIS Frontend

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

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A330412 Integers m such that sigma(m) + sigma(8*m) = 18*m.

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A330413 Integers m such that sigma(m) + sigma(16*m) = 34*m.

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A334128 Integers m such that sigma(m) + sigma(7*m) = 16*m.

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A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

A330413 Integers m such that sigma(m) + sigma(16m) = 34m.

A334128 Integers m such that sigma(m) + sigma(7m) = 16m.