A019279 -id:A019279 - cp's OEIS frontend

A138869 Concatenation of first two digits and last two digits of n-th even superperfect number A061652(n).

22, 44, 1616, 6464, 4096, 6536, 2644, 1024, 1176, 3056, 8164, 8564, 3476, 2664, 5244, 7304, 2276, 1236, 9596, 1404, 2356, 1776, 1496, 2136, 2276, 2056, 4236, 2604, 2604, 2556, 3724, 8744, 6496, 2064, 4056, 3176, 6336, 2196, 4636, 6224, 1404, 6124, 1536, 6236, 1064, 8476, 1556

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of first two digits and last two digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Cf. A019279, A061652, A138863, A138867, A138868, A138875.

More terms from Jinyuan Wang, Mar 14 2020

A138871 Last 3 digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 16, 64, 96, 536, 144, 824, 976, 56, 64, 864, 576, 64, 544, 504, 176, 536, 496, 304, 56, 776, 96, 736, 376, 256, 336, 104, 504, 656, 224, 944, 296, 264, 856, 576, 136, 896, 536, 24, 704, 624, 936, 936, 464, 376, 256, 976

Offset: 1

Omar E. Pol, Apr 02 2008

Also, last 3 digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Cf. A019279, A061652, A138865, A138867, A138870, A138872.

a(n) = 2^(A000043(n)-1) mod 1000. - Max Alekseyev, Feb 16 2024

More terms from R. J. Mathar, Feb 05 2010
a(40)-a(41) from Max Alekseyev, Feb 11 2012
a(42)-a(47) from Jinyuan Wang, Mar 14 2020
a(48) from Max Alekseyev, Feb 16 2024

A138872 Concatenation of first 3 digits and last 3 digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

22, 44, 1616, 6464, 409096, 655536, 262144, 107824, 115976, 309056, 811064, 850864, 343576, 265064, 520544, 737504, 223176, 129536, 953496, 142304, 239056, 173776, 140096, 215736, 224376, 201256, 427336, 268104, 260504, 256656, 373224, 870944

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of first 3 digits and last 3 digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Cf. A019279, A061652, A138866, A138877, A138870, A138871.

More terms from Max Alekseyev, Feb 11 2012

A138882 Triangle read by rows: row n lists divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32, 64, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384

Offset: 1

Omar E. Pol, Apr 11 2008

The number of divisors of n-th even superperfect number is equal to A000043(n), then row n has A000043(n) terms.

The sum of divisors of n-th even superperfect number is equal to n-th Mersenne prime A000668(n), then n-th row sum is equal to A000668(n).

			Triangle begins:
  1, 2
  1, 2, 4
  1, 2, 4, 8, 16
  1, 2, 4, 8, 16, 32, 64
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096
  ...
==============================================================
..... Mersenne ..............................................
....... prime ...............................................
n ... A000668(n) = Sum of divisors of A061652(n) .............
==============================================================
1 ........ 3 ... = 1+2
2 ........ 7 ... = 1+2+4
3 ....... 31 ... = 1+2+4+8+16
4 ...... 127 ... = 1+2+4+8+16+32+64
5 ..... 8191 ... = 1+2+4+8+16+32+64+128+256+512+1024+2048+4096

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000005, A000043, A000203, A000668, A019279, A061652, A133031.

Flatten[Divisors[2^(MersennePrimeExponent[Range[7]]-1)]] (* Harvey P. Dale, Apr 28 2022 *)

A147648 Number of distinct even superperfect numbers dividing n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2

Offset: 1

Omar E. Pol, Nov 09 2008

Also, numbers of distinct superperfect numbers dividing n, if there are no odd superperfect numbers.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A019279, A061652, A080225, A147645.

Array[DivisorSum[#, 1 &, And[EvenQ@ #, Nest[DivisorSigma[1, #] &, #, 2] == 2 #] &] &, 105] (* Michael De Vlieger, Nov 06 2018 *)

A147648(n) = sumdiv(n,d,(!(d%2)&&(sigma(sigma(d))==(2*d)))); \\ Antti Karttunen, Nov 06 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A061652(n) = 0.828388215042... . - Amiram Eldar, Jan 01 2024

A153476 Sum of first n even superperfect numbers A061652, divided by 2.

Original entry on oeis.org

1, 3, 11, 43, 2091, 34859, 165931, 537036843, 576460752840460331, 154742505487133287202850859, 40564973949808827981181705422891, 42535336430091257741749807110676449323

Offset: 1

Omar E. Pol, Dec 27 2008

nonn

Also, sum of first n superperfect numbers A019279, divided by 2, if there are no odd superperfect numbers.

Cf. A153474, A153475.

a(n) = A153475(n)/2 [From Max Alekseyev, Jul 27 2009]

More terms from Max Alekseyev, Jul 27 2009

A324256 Larger of super amicable pair m < n defined by sigma(sigma(m)) = sigma(sigma(n)) = m + n.

Original entry on oeis.org

37, 28201, 34687, 65587, 2089951, 4091797, 8340613, 8161477, 10124833, 18927067, 37179433, 37393633, 25855567, 64346413, 107160373, 95150203, 159440893, 238973101, 257658061, 277743397, 322210813, 256268149, 349883707, 578403913, 814865497, 752724457, 704710543

Offset: 1

Amiram Eldar, Feb 19 2019

nonn

The terms are ordered according to the their lesser counterparts (A324255).

Analogous to A002046 as A019279 is analogous to A000396.

			(23, 37) are the first pair since sigma(sigma(23)) = sigma(sigma(37)) = 60 = 23 + 37.

Cf. A000203, A000396, A002046, A019279, A045614 (unitary analog), A051027, A324255.

seq={}; s[n_]:=DivisorSigma[1,DivisorSigma[1,n]]-n; Do[m=s[n];If[m>n && s[m]==n, AppendTo[seq, m]], {n,1,60000}]; seq

f(n) = sigma(sigma(n)) - n;
lista(nn) = {for (n=1, nn, my(fn = f(n)); if ((fn > n) && (f(fn) == n), print1(fn, ", ")););} \\ Michel Marcus, Feb 20 2019

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[#]] == # &]

A336613 Numbers m such that tau(sigma(m)) divides m, where tau(m) is the number of divisors function (A000005) and sigma(m) is the sum of divisors function (A000203).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 36, 48, 64, 72, 80, 81, 84, 100, 112, 120, 128, 140, 144, 156, 160, 162, 168, 192, 198, 200, 208, 210, 216, 240, 256, 270, 288, 300, 320, 324, 336, 357, 360, 368, 384, 390, 420, 432, 448, 464, 468, 480, 512, 560, 576, 592, 600, 624, 630

Offset: 1

Bernard Schott, Jul 29 2020

nonn

Two subsets of terms:
1) If 2^p - 1 is a Mersenne prime (p is in A000043 and 2^p-1 is in A000668), then m = 2^(p-1) is a term that belongs to A019279: the even superperfect numbers (2, 4, 16, 64, 4096, ...). Proof: sigma(m) = 1+2+...+2^(p-1) = 2^p - 1 that is a Mersenne prime so tau(2^p-1) = 2 that divides m = 2^(p-1); indeed, m/tau(sigma(m)) = 2^(p-2).
2) If m = 2^(p-1) is a term as above, then 3*m is another term (see example) with 3*m/tau(sigma(3*m)) = 2^(p-2).

			48 = 2^4 * 3, so, sigma(48) = sigma(2^4) * sigma(3) = (2^5 - 1) * (1+3) = 31 * 4 = 124; then, tau(2^2 * 31) = tau(4) * tau(31) = 3 * 2 = 6, and  48/6 = 8 = 2^3, hence 48 is a term.

Cf. A000005, A000203, A062068.

Cf. A019279 (subsequence), A336612 (sigma(tau(m)) divides m).

with(numtheory) filter:= m -> m/tau(sigma(m)) = floor(m/tau(sigma(m))) : select(filter, [$1..650]);

Select[Range[630], Divisible[#, DivisorSigma[0, DivisorSigma[1, #]]] &] (* Amiram Eldar, Jul 30 2020 *)

isok(m) = !(m % numdiv(sigma(m))); \\ Michel Marcus, Jul 30 2020

cp's OEIS Frontend

A138869 Concatenation of first two digits and last two digits of n-th even superperfect number A061652(n).

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A138871 Last 3 digits of n-th even superperfect number A061652(n).

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A138872 Concatenation of first 3 digits and last 3 digits of n-th even superperfect number A061652(n).

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A138882 Triangle read by rows: row n lists divisors of n-th even superperfect number A061652(n).

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A147648 Number of distinct even superperfect numbers dividing n.

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A153476 Sum of first n even superperfect numbers A061652, divided by 2.

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A324256 Larger of super amicable pair m < n defined by sigma(sigma(m)) = sigma(sigma(n)) = m + n.

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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).

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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).

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