A038843 -id:A038843 - cp's OEIS frontend

A064012 usigma(usigma(n))=3*n where usigma(n) is the sum of unitary divisors of n (A034448).

10, 30, 288, 660, 720, 2146560

Offset: 1

Felice Russo, Sep 07 2001

nonn

Is this sequence finite?
18 is the only known integer usigma(usigma(n)) = kn for some integer k >= 4 (no other one <= 2^30). - Tomohiro Yamada, Apr 22 2017
a(7) > 2.85*10^11, if it exists. The same bound holds also for any n > 18 such usigma(usigma(n)) = k*n for some integer k >= 4. - Giovanni Resta, Apr 10 2019

			10 belongs to the sequence because usigma(usigma(10)) = 30 = 3*10.

V. Sitaramaiah and M. V. Subbarao, On the equation usigma(usigma(n)) = 2n, Util. Math. 53 (1998), 101--124.

Cf. A034448, A038843.

One more term from Naohiro Nomoto, Oct 21 2001

No other terms < 1070000000, Jud McCranie, Oct 28 2001

A328120 Exponential superperfect numbers (or e-superperfect numbers): numbers m such that esigma(esigma(m)) = 2m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

9, 12, 45, 60, 63, 84, 99, 117, 132, 153, 156, 171, 204, 207, 228, 261, 270, 276, 279, 315, 333, 348, 369, 372, 387, 420, 423, 444, 477, 492, 495, 516, 531, 549, 564, 585, 603, 636, 639, 657, 660, 693, 708, 711, 732, 747, 765, 780, 801, 804, 819, 852, 855, 873

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Hanumanthachari et al. proved that:
1) The only e-superperfect number of the form p^q with p and q primes is 9 = 3^2.
2) If p prime, m squarefree coprime to m with gcd(p+1, m) > 1 then p^2 * m is e-superperfect only if p = 2.
3) If k is squarefree coprime to esigma(m) then m*k is e-superperfect if and only if m is e-superperfect.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Hanumanthachari, V. V. Subrahmanya Sastri, and V. Srinivasan, On e-perfect numbers, Math. Student, Vol. 46, No. 1 (1978), pp. 71-80; entire issue.

The exponential version of A019279.

Cf. A038843, A051377.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 2n; Select[Range[1000], espQ]

esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
isok(k) = esigma(esigma(k)) == 2*k; \\ Amiram Eldar, Jan 09 2025

9 is in the sequence since esigma(9) = 12 and esigma(12) = 18 = 2*9.

A329884 Nonunitary superperfect numbers: numbers k such that nusigma(nusigma(k)) = k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

Original entry on oeis.org

24, 48, 56, 112, 192, 248, 252, 328, 448, 496, 768, 1016, 1792, 1984, 2032, 3240, 6462, 7936, 8128, 11616, 11808, 17412, 20538, 32512, 49152, 65528, 114688, 131056, 507904, 524224, 786432, 1048568, 1835008, 2080768, 2096896, 2097136, 3145728, 4194296, 7340032

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

Analogous to superperfect numbers (A019279) as nonunitary perfect numbers (A064591) is analogous to perfect numbers (A000396).

Amiram Eldar, Table of n, a(n) for n = 1..53 (terms below 10^10)

Cf. A000396, A019279, A034448, A038843, A048146, A064591, A329881.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; Select[Range[10^6], nusigma[nusigma[#]] == # &]

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;

A329881 Nonunitary doubly superperfect numbers: numbers k such that nusigma(nusigma(k)) = 2*k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

Original entry on oeis.org

4032, 13104, 58032, 69648, 237744, 278592, 365652, 1114368, 15333552, 71319552, 245364912, 981465264, 1141112832, 4564451328, 873139150710, 4020089387184

Offset: 1

Amiram Eldar, Nov 23 2019

Analogous to superperfect numbers (A019279) as nonunitary doubly perfect numbers (A064592) is analogous to perfect numbers (A000396).
If n = 2^k*3*1451 and nusigma(n) = 2^5*3*11^2*p, with p > 11 prime, then n is a term. This happens for k = 4, 6, 8, 14, 18, 20, 32, 62, 90, 108, 128, 522, 608, ... . Similarly, if p=2^k-1 is prime (A000043), then 2^4*3^2*13*p is a term for k > 2. - Giovanni Resta, Nov 23 2019
a(17) > 6*10^12. - Giovanni Resta, Nov 24 2019

Cf. A000396, A019279, A034448, A038843, A048146, A064592, A328120, A329884.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; Select[Range[3*10^5], nusigma[nusigma[#]] == 2*# &]

a(15)-a(16) from Giovanni Resta, Nov 24 2019

cp's OEIS Frontend

A064012 usigma(usigma(n))=3*n where usigma(n) is the sum of unitary divisors of n (A034448).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A328120 Exponential superperfect numbers (or e-superperfect numbers): numbers m such that esigma(esigma(m)) = 2m, where esigma(m) is the sum of exponential divisors of m (A051377).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A329884 Nonunitary superperfect numbers: numbers k such that nusigma(nusigma(k)) = k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A329881 Nonunitary doubly superperfect numbers: numbers k such that nusigma(nusigma(k)) = 2*k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions