A318175 -id:A318175 - cp's OEIS frontend

A189000 Bi-unitary multiperfect numbers.

1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240, 342720, 523776, 1028160, 1528800, 6168960, 7856640, 7983360, 14443520, 22932000, 23569920, 43330560, 44553600, 51979200, 57657600, 68796000, 133660800, 172972800, 779688000, 1476304896, 2339064000, 6840038400

Offset: 1

R. J. Mathar, Apr 15 2011

nonn

All entries greater than 1 are even [Hagis].
14443520 is the first (only?) composite term not divisible by 3.  Excluding the factor p=3, all composite terms <= 172972800 have nonincreasing exponents in the factorization (sorted by primes). - D. S. McNeil, Apr 15 2011
Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - Tomohiro Yamada, Apr 15 2017
McNeil's observation about exponents does not hold in general. Indeed, a(41) = 2^8 * 3^5 * 5^2 * 7 * 11 * 13^2 * 17. - Giovanni Resta, Apr 15 2017
a(43) > 4.66*10^12. - Giovanni Resta, Sep 07 2018
We include 1 here, although this is not "multi"-perfect. - R. J. Mathar, Sep 08 2018

			n=120 divides A188999(120)=360.
n=90 divides A188999(90)=180.
n=672 divides A188999(672)=2016.

Giovanni Resta, Table of n, a(n) for n = 1..42
Peter Hagis, Bi-Unitary amicable and multiperfect numbers, Fib. Quart. 25 (2) (1987) 144-151
Pentti Haukkanen and V. Sitaramaiah, Bi-unitary multiperfect numbers, I, Notes Number Theory Discrete Math. 26 (1) (2020) 93-171.
Michel Marcus, Unexhaustive list of terms
C. R. Wall, Bi-unitary perfect numbers, Proc. Amer. Math. Soc. 33 (1) (1972) 39-42.
Tomohiro Yamada, Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024.

Cf. A007691 (analog for sigma).

Cf. A188999 (bi-unitary sigma), A318175, A318781 (the k coefficients).

bsig[n_] := If[n == 1, 1, Block[{p, e}, Product[{p, e} = pe; (p^(e + 1) - 1)/(p - 1) - If[EvenQ[e], p^(e/2), 0], {pe, FactorInteger[n]}]]]; Select[Range[10^5], Mod[bsig[#], #] == 0 &] (* Giovanni Resta, Apr 15 2017 *)

a188999(n) = {my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = ! frac(a188999(n)/n); \\ Michel Marcus, Sep 03 2018

{n | A188999(n)}.

a(18)-a(27) by D. S. McNeil, Apr 15 2011
a(28)-a(31) from Giovanni Resta, Apr 15 2017
a(1)=1 inserted by Giovanni Resta, Sep 07 2018

A318182 Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 15, 18, 24, 30, 60, 720, 1020, 4080, 8925, 14688, 14976, 16728, 17850, 35700, 36720, 37440, 66912, 71400, 285600, 308448, 381888, 428400, 602208, 636480, 763776, 856800, 1321920, 1505520, 3011040, 3084480, 21679488, 22276800, 30844800

Offset: 1

Michel Marcus, Aug 20 2018

nonn

a(86) > 3*10^11. All the prime factors of the first 85 terms belong to the set {2, 3, 5, 7, 11, 13, 17, 41, 43, 257}. - Giovanni Resta, Aug 25 2018
Like in A019278, here there are many instances where if x is a term, then A049417(x) is also a term.
Additionally, there exist longer chains of 3 or 4 elements like:
- 8 (3), 15 (4), 24 (5), 60 (6);
- 9 (2), 10 (3), 18 (4), 30 (5);
- 31615920 (4), 50585472 (5), 126463680 (6), 252927360 (12);
- 963407296051200 (16), 3134896756992000 (17), 15414516736819200 (18);
- 3541951043592192 (5), 8854877608980480 (6), 17709755217960960 (12), 53129265653882880 (20);
- 4829933241262080 (11), 17709755217960960 (12), 53129265653882880 (20);
  7871002319093760 (9), 26564632826941440 (10), 70839020871843840 (13), 265646328269414400 (14).

Giovanni Resta, Table of n, a(n) for n = 1..85 (first 58 terms from Tomohiro Yamada)
Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, 47 (2017) pp. 211-218. See Table 1.
Michel Marcus, Unexhaustive list of terms

Cf. A049417 (infinitary sigma).

Cf. A019278 (analog for sigma), A318175 (analog for bi-unitary sigma).

a049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)));}
isok(n) = frac(a049417(a049417(n))/n) == 0;

More terms from Giovanni Resta, Aug 25 2018

A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 18, 24, 27, 30, 54, 165, 238, 288, 512, 656, 660, 864, 952, 1536, 1968, 2464, 2880, 4608, 4680, 13824, 14448, 14976, 16728, 19008, 19992, 23040, 29376, 60928, 152064, 155520, 172368, 279552, 474936, 746928, 1070592, 1114560, 1524096, 1703520

Offset: 1

Tomohiro Yamada, Jan 16 2024

nonn

			A188999(18) = 4 * 10 = 40 and A034448(40) = 9 * 6 = 54 = 3 * 18, so 18 is a term with k = 3.

Amiram Eldar, Table of n, a(n) for n = 1..80

Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).

Cf. A369205 (analog for A188999(A034448(m))).

a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
a188999(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=if(e%2,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2));f[i,2]=1);factorback(f)};
isok(n) = (a034448(a188999(n))%n) == 0;

A369205 Numbers m such that A188999(A034448(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 9, 10, 15, 18, 21, 30, 40, 42, 60, 120, 288, 567, 630, 720, 756, 1023, 1134, 1428, 2046, 2160, 2268, 2520, 3024, 3276, 3570, 4092, 6048, 8184, 8925, 9240, 11424, 11550, 15345, 17850, 18144, 30690, 35700, 46200, 57120, 85680, 147312, 285600, 491040, 556920

Offset: 1

Tomohiro Yamada, Jan 16 2024

nonn

			A034448(18) = 4 * 10 = 40 and A188999(40) = 15 * 6 = 90 = 5 * 18, so 18 is a term with k = 5.

Amiram Eldar, Table of n, a(n) for n = 1..105

Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).

Cf. A369204 (analog for A034448(A188999(m))).

a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
a188999(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=if(e%2,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2));f[i,2]=1);factorback(f)};
isok(n) = (a188999(a034448(n))%n) == 0;

A318242 a(n) is the least k such that A188999(A188999(k)) = n*k, where A188999 is the bi-unitary sigma function, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 8, 15, 24, 42, 240, 648, 168, 480, 321408, 4320, 57120, 103680, 1827840, 23591520, 898128000, 374250240

Offset: 1

Michel Marcus, Aug 22 2018

It is also known that a(20) = 11975040.
Then for higher indices n, we have:
  a(19) <= 5235707393280;
  a(21) <= 49110437376000;
  a(22) <= 106780561395056640;
  a(24) <= 1099525819392000;
  a(25) <= 41252767395840;
  a(26) <= 202768780032000.

Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017. See Table 1.
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). See Table 1.

Cf. A188999, A318175.

Cf. A272930 (analog for sigma), A318272 (analog for infinitary sigma).

cp's OEIS Frontend

A189000 Bi-unitary multiperfect numbers.

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A318182 Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

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A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

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A369205 Numbers m such that A188999(A034448(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

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A318242 a(n) is the least k such that A188999(A188999(k)) = n*k, where A188999 is the bi-unitary sigma function, or 0 if no such k exists.

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