A000396 -id:A000396 - cp's OEIS frontend

A138818 Concatenation of initial digit of n-th even superperfect number A061652(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n).

236, 472, 134, 618, 483, 618, 251, 122, 122, 361, 811, 811

Offset: 1

Omar E. Pol, Apr 05 2008

Also, concatenation of initial digit of n-th superperfect number A019279(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of A138124(n), A135613(n) and A135617(n).

L. C. Noll, Mersenne Prime Digits and Names.
J. M. Pedersen, Known Perfect Numbers.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A135613, A135617, A138124, A138816, A138817, A138819, A138841, A138842, A138843.

A138819 Concatenation of final digit of n-th even superperfect number A061652(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n).

Original entry on oeis.org

236, 478, 616, 478, 616, 616, 478, 478, 616, 616, 478, 478, 616, 478, 478, 478, 616, 616, 616, 478, 616, 616, 616, 616, 616, 616, 616, 478, 478, 616, 478, 478, 616, 478, 616, 616, 616, 616, 616

Offset: 1

Omar E. Pol, Apr 05 2008

Also, concatenation of final digit of n-th superperfect number A019279(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of A138125(n), A080172(n) and A094540(n).
For n>1 a(n) is equal to 478 or 616, only.
Note that, for n>1: if the final digit of n-th Mersenne prime A000668(n) is 1 then the final digit of n-th even superperfect number is 6 and the final digit of n-th perfect number also is 6, otherwise the final digit of n-th even superperfect number is 4 and the final digit of n-th perfect number is 8 (see example).

			===================================================================
.................. SHORT TABLE OF FINAL DIGITS ...................
===================================================================
... Final digit of even ..... Final digit of ..... Final digit of
... superperfect number ..... Mersenne prime ..... perfect number
........ A061652 ............... A000668 ............. A000396
===================================================================
n = 1 ..... (2) ................... (3) .................. (6)
n > 1 ..... (4) ................... (7) .................. (8)
n > 1 ..... (6) ................... (1) .................. (6)

L. C. Noll, Mersenne Prime Digits and Names.
J. M. Pedersen, Known Perfect Numbers.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138816, A138817, A138818, A138841, A138842, A138843.

A153800 Indices of perfect numbers (A000396) in the sequence of multiply-perfect numbers (A007691).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 25, 41, 110, 192, 229, 294

Offset: 1

Omar E. Pol, Jan 13 2009

The sequence contains no further entries between 294 and 1600. - R. J. Mathar, May 26 2011

			a(7) is 25 because the 25th multiply-perfect number A007691(25)=137438691328 is also the 7th perfect number A000396(7).

Cf. A000396, A007691.

A174292 Spoof-perfect numbers: Freestyle perfect numbers (A058007) which are not perfect numbers (A000396).

Original entry on oeis.org

60, 84, 90, 120, 336, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8190, 8280, 9240, 9828, 9900, 10080, 10530, 11088, 11400, 13680, 14040, 15120, 16380

Offset: 1

Daniel Forgues, Nov 27 2010

Spoof-perfect numbers are freestyle perfect numbers which are not perfect numbers.
Only one odd spoof perfect number is known: 198585576189, found by Descartes.
Assuming all integer factorizations were tried in the range [1..9900] in A058007, where I removed 6, 28, 496, 8128 from the list (I did not do the search for spoof perfect numbers myself, so the accuracy of my list depends on the accuracy of A058007's list.)
Roughly said, a spoof-perfect number is a number that would be perfect if some (one or more) of its composite factors were wrongly assumed to be prime, i.e., taken as a spoof prime.
Contribution from M. F. Hasler, Jan 13 2013: (Start)
I added "roughly said" to the above last phrase, since different interpretations of "would be perfect if some of its composite factors were wrongly assumed to be prime" are possible, and Descartes's example does not help to decide: (Notations are those from A058007, n = Sum (f_i)^(e_i).)
(a) If a spoof prime factor f_i is composite, may it have some of the smaller (spoof or true prime) f_j as factors or not? (In Descartes's example, this is not the case. And "assumed to be prime" could well imply that the answer is "no". But there is no such restriction in A058007.)
(b) If f_i is composite, is it required that e_i is the highest possible power, i.e., the (f_i)-valuation of n (or of n divided by all smaller f_j to the powers e_j)? (In Descartes's example this is the case. And if product(f_i^e_i) is to be a "prime factorization" of n, then it should be the case. But there is no such restriction in A058007. Note that this is not a consequence of (a), because the f_i could have common factors: e.g., even if f_1=21, f_2=35, f_3=45 are "wrongly assumed to be prime", then n=21*35*45 would have the (f_1)-valuation = 2, i.e., factorization n = f_1^2*75.)
(c) Is it reasonable to allow for even spoof primes f_i? (In Descartes's example this is not the case. And it seems somehow inconceivable that an even number be "wrongly assumed to be prime". But there is no such restriction in A058007.
Depending on the answer to each of these questions, "spoof-perfect numbers" as defined using "composite factors were wrongly assumed to be prime", could mean at least 8 different sequences. (End)

			n = 60 = (3^1)*(4^1)*(5^1), s = 120 = (3^2-1)/(3-1) * (4^2-1)/(4-1) * (5^2-1)/(5-1): s-n = 120-60 = n, so 60 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B1.

Giovanni Resta, Table of n, a(n) for n = 1..500
OEIS wiki, Spoof perfect numbers.
C. Rivera (Ed.), Prime Puzzle 111. Spoof odd Perfect numbers, on primepuzzles.net (before 2000)
Giovanni Resta, Spoof factorizations of a(1)-a(500)

Cf. A058007, A000396; A222263, A222264.

r[s_, n_, f_] := Catch[If[n==1, s==1, Block[{p,e}, Do[e=1; While[Mod[n, p^e] == 0, r[s*(p^(e+1)-1) / (p-1), n/p^e, p] && Throw@True; e++], {p, Select[Divisors@n, f < # &]}]]; False]]; spoofQ[n_] := r[1/2/n, n, 1] && DivisorSigma[-1, n] != 2; Select[Range[10^4], spoofQ] (* Giovanni Resta, Feb 28 2013 *)

a(37)-a(44) from Amiram Eldar, Dec 27 2018

A090945 Harmonic numbers (A001599) which are not perfect (A000396).

Original entry on oeis.org

1, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720

Offset: 1

N. J. A. Sloane, Feb 28 2004

nonn

			A001599(4) = 140, but 336 = sigma(140) <> 2*140 = 280. Thus, 140 is a harmonic number which is not perfect. - _Muniru A Asiru_, Nov 26 2018

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.

Amiram Eldar, Table of n, a(n) for n = 1..930 (terms below 10^14; terms 1..77 from Muniru A Asiru)
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Cf. A001599, A003601. Different from A007340.

For the associated harmonic means, see A102408.

Concatenation([1],Filtered([2,4..2000000],n->Sigma(n)<>2*n and IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018

Select[Range[2 10^7], IntegerQ[HarmonicMean[Divisors[#]]] && !DivisorSigma[1, #]==2 # &] (* Vincenzo Librandi, Nov 27 2018 *)

isok(n) = my(sn = sigma(n)); (frac(n*numdiv(n)/sn) == 0) && (sn != 2*n); \\ Michel Marcus, Nov 28 2018

A061193 Number of digits in n-th even perfect number (A000396).

Original entry on oeis.org

1, 2, 3, 4, 8, 10, 12, 19, 37, 54, 65, 77, 314, 366, 770, 1327, 1373, 1937, 2561, 2663, 5834, 5985, 6751, 12003, 13066, 13973, 26790, 51924, 66530, 79502, 130100, 455663, 517430, 757263, 841842, 1791864, 1819050, 4197919, 8107892, 12640858, 14471465, 15632458, 18304103, 19616714, 22370543

Offset: 1

N. J. A. Sloane

The next known values following a(48) are 44677235, 46498850, and 49724095, but these may not be the next terms. [Updated by M. F. Hasler, Nov 28 2017, Ivan Panchenko, Apr 07 2018, Apr 17 2018, Amiram Eldar, Oct 16 2024]

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
Martin Gardner, Mathematical Magic Show, Alfred A. Knopf, 1977, p. 165.
Paul Hoffman, Archimedes' Revenge, Penguin, 1988, p. 11.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 244-245.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
Donald D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co., 1995, p. 80.

Ivan Panchenko, Table of n, a(n) for n = 1..48 [Updated by _Ivan Panchenko_, Apr 17 2018, _Amiram Eldar_, Oct 16 2024]
A. Byerley, Relationship Between Mersenne Primes and Perfect Numbers.
Chris K. Caldwell, Table of Known Mersenne Primes.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
H. J. Smith, Mersenne Primes. [broken link]

Cf. A000043, A000396, A007524, A055642.

Table[n=MersennePrimeExponent@k;IntegerLength[2^(n-1)(2^n-1)],{k,45}] (* Giorgos Kalogeropoulos, Sep 03 2020 *)
Array[IntegerLength@*PerfectNumber, 18] (* Giorgos Kalogeropoulos, Sep 03 2020 *)

apply( p->(2*p-1)*log(2)\log(10)+1, A000043) \\ where A000043 is the vector of the known Mersenne primes. - M. F. Hasler, Nov 28 2017

a(n) = ceiling((2*A000043(n)-1)*A007524), with A000043 = Mersenne prime exponents, A007524 = log_10(2). - M. F. Hasler, Nov 28 2017

This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Lekraj Beedassy, May 30 2001
More terms from Harry J. Smith, Apr 16 2003
Entry revised by N. J. A. Sloane, Jun 10 2012
a(39) through a(45) from M. F. Hasler, Nov 28 2017

A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 248, 496, 127, 254, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336, 131071, 262142, 524284, 1048568, 2097136, 4193792

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the divisors of n-th perfect number that are not powers of 2.

First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

			Triangle begins:
  3, 6,
  7, 14, 28
  31, 62, 124, 248, 496
  127, 254, 508, 1016, 2032, 4064, 8128
  ...
==========================================================
Row .... First term ..... Last term ....... Row sum ......
n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .
==========================================================
1 ............ 3 .............. 6 ......... 3^2 = 9
2 ............ 7 ............. 28 ......... 7^2 = 49
3 ........... 31 ............ 496 ........ 31^2 = 961
4 .......... 127 ........... 8128 ....... 127^2 = 16129
5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481

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

Cf. A000043, A000396, A000668, A018254, A018487, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655.

A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).

Original entry on oeis.org

24, 224, 15872, 1040384, 274844352512, 1125891316908032, 72057456598974464, 4951760154835678090382802944, 6129982163463555430774932117031404988667342368173719552

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..14

Subsequence of A181595 and A181702.

Cf. A000043, A000396, A147538, A181596, A000396, A181701, A181702.

With[{p = MersennePrimeExponent[Range[10]]}, 2^(2*p - 1)*(2^p - 1)] (* Amiram Eldar, Apr 29 2024 *)

a(n) = A147538(A000043(n)). - Amiram Eldar, Apr 29 2024

a(6)-a(9) from Amiram Eldar, Apr 29 2024

A237286 Numbers n such that sigma(n) - n = perfect number (A000396).

Original entry on oeis.org

6, 25, 28, 496, 652, 8128, 10682, 10828, 33550336, 44655764, 8589869056, 8623554304, 137438691328

Offset: 1

Jaroslav Krizek, Feb 28 2014

Also numbers n such that sum of the proper divisors of n (A001065) = perfect number (A000396).
Supersequence of perfect numbers (A000396).
a(14) > 10^12. - Giovanni Resta, Mar 01 2014
Up to a(13), note that the terms that are not perfect are deficient. - Michel Marcus, Mar 12 2014

			Number 25 is a term because sigma(25) - 25 = 31 - 25 = 6 (perfect number).

Cf. A000203 (sigma(n)).

isok(n) = sigma(smn=sigma(n) - n) == 2*smn; \\ Michel Marcus, Mar 01 2014

a(10) from Michel Marcus, Mar 01 2014

a(11)-a(13) from Giovanni Resta, Mar 01 2014

A291901 Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).

Original entry on oeis.org

2, 4, 13, 16, 64, 4096, 65536, 262144, 3145341, 932181397, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864, 75603657215035519123837860069507929970384679

Offset: 1

Jaroslav Krizek, Nov 02 2017

nonn

Numbers n such that A229335(n) is in the sequence of perfect numbers, A000396.
Corresponding values of perfect numbers: 6, 28, 28, 496, 8128, 33550336, 8589869056, 137438691328, 33550336, ...
All even superperfect numbers A061652(n) are terms in this sequence.
Primes q of the form 2^(p-2) * (2^p - 1) - 1 where p is a Mersenne exponent (A000043) are terms: 2, 13, ...

			Divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of sums of elements of subsets = 1 + 2 + 4 + 3 + 5 + 6 + 7 = 28 (perfect number).
sigma(16) * 2^(tau(16) - 1) = 31 * 16 = 496 (perfect number).

Max Alekseyev, Table of n, a(n) for n = 1..71

Cf. A000043, A000396, A061652, A090748, A229335.

[n: n in [1..10^6]  | SumOfDivisors(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1))) eq 2*(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1)))];

isA000396 := proc(n)
    numtheory[sigma](n)=2*n ;
    simplify(%) ;
end proc:
for n from 1 do
    if isA000396(A229335(n)) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2017

Select[Range[2^20], DivisorSigma[1, DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)] == 2 (DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)) &] (* Michael De Vlieger, Nov 02 2017 *)

Terms a(10) onward added by Max Alekseyev, Sep 18 2024

cp's OEIS Frontend

A138818 Concatenation of initial digit of n-th even superperfect number A061652(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n).

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A138819 Concatenation of final digit of n-th even superperfect number A061652(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n).

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A153800 Indices of perfect numbers (A000396) in the sequence of multiply-perfect numbers (A007691).

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A174292 Spoof-perfect numbers: Freestyle perfect numbers (A058007) which are not perfect numbers (A000396).

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Mathematica

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A090945 Harmonic numbers (A001599) which are not perfect (A000396).

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A061193 Number of digits in n-th even perfect number (A000396).

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A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

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A181710 Near-perfect numbers (A181595) of the form m*2^p, where m = 2^(p-1)*(2^p-1) is a perfect number (A000396).

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A237286 Numbers n such that sigma(n) - n = perfect number (A000396).

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A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).