A000396 -id:A000396 - cp's OEIS frontend

A023196 Nondeficient numbers: numbers k such that sigma(k) >= 2k; union of A000396 and A005101.

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252

Offset: 1

David W. Wilson

Also called the non-deficient numbers.
If k is a term, so is every positive multiple of k. The "primitive" terms form A006039.
The sequence of numbers k that give local minima for A004125, i.e., such that A004125(k-1) > A004125(k) and A004125(k) < A004125(k+1) coincides with this sequence for the first 1014 terms. Then there appears 4095 which is a term of A023196 but is not a local minimum. - Max Alekseyev, Jan 26 2005
Also, union of pseudoperfect and weird numbers. Cf. A005835, A006037. - Franklin T. Adams-Watters, Mar 29 2006
k is in the sequence if and only if A004125(k-1) > A004125(k). - Jaycob Coleman, Mar 31 2014
Like the abundant numbers, this sequence has density between 0.2474 and 0.2480, see A005101. - Charles R Greathouse IV, Nov 30 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B (2016) Vol. 3, 1-26.

Cf. A000203, A004125, A006039, A000396, A005101.

Filtered([1..260],n->Sigma(n)>=2*n); # Muniru A Asiru, Dec 04 2018

[n: n in [1..300] | not (2*n gt DivisorSigma(1,n))]; // Vincenzo Librandi, Dec 05 2018

A023196:=n->`if`(numtheory[sigma](n)>=2*n, n, NULL): seq(A023196(n), n=1..380); # Wesley Ivan Hurt, Apr 18 2017

Select[Range[300], DivisorSigma[1, #] >= 2# &] (* Harvey P. Dale, Sep 26 2014 *)

is(n)=sigma(n,-1)>=2 \\ Charles R Greathouse IV, Mar 09 2014

A139228 First differences of perfect numbers A000396.

Original entry on oeis.org

22, 468, 7632, 33542208, 8556318720, 128848822272, 2305842870701260800, 2658455991569831742348849607813890048, 191561942608236104636337386514471893476304705594327040

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

			a(1) = 22 because 6 and 28 are the first two perfect numbers, and their difference is 28 - 6 = 22.

Paolo Xausa, Table of n, a(n) for n = 1..14
Philippe Ellia, On the distance between perfect numbers, arXiv:1210.0450 [math.NT], 2012.
Florian Luca, Problem 10711, Amer. Math. Monthly, Vo. 106, No. 2 (1999) p. 166; Can Two Consecutive Numbers Both Be Perfect?, solution by Francis B. Coghlan, ibid., Vol. 108, No. 1 (2001), pp. 80-81.
Florian Luca and Carl Pomerance, On the radical of a perfect number, New York Journal of Math., Vol. 16 (2010), pp. 23-30; alternative link.
Florian Luca and Herman te Riele, phi and sigma: from Euler to Erdős, Nieuw Archief voor Wiskunde, Vol. 12, No. 5 (2011), pp. 31-36.

Cf. A000396, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

Differences[Select[Range[10000], DivisorSigma[1, #] == 2# &]] (* Alonso del Arte, Mar 05 2020 *)
Differences[PerfectNumber[Range[12]]] (* Paolo Xausa, Oct 20 2023 *)

a(n) = A000396(n+1) - A000396(n).
From Amiram Eldar, May 07 2021: (Start)
a(n) > 1 (Luca, 1999).
a(n) > 4 (Luca and te Riele, 2011). (End)

More terms from Omar E. Pol, Oct 02 2012

A139229 First differences of perfect numbers A000396, divided by 2.

Original entry on oeis.org

11, 234, 3816, 16771104, 4278159360, 64424411136, 1152921435350630400, 1329227995784915871174424803906945024, 95780971304118052318168693257235946738152352797163520

Offset: 1

Omar E. Pol, Apr 19 2008

nonn

Paolo Xausa, Table of n, a(n) for n = 1..14

Cf. A000396, A139228, A139230, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

Differences[Select[Range[10000], DivisorSigma[1, #] == 2# &]]/2 (* Alonso del Arte, Mar 05 2020 *)
Differences[PerfectNumber[Range[12]]]/2 (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139228(n)/2.

a(6)-a(9) from A139228(n)/2 by Jinyuan Wang, Mar 04 2020

A139230 Second differences of perfect numbers A000396.

Original entry on oeis.org

446, 7164, 33534576, 8522776512, 120292503552, 2305842741852438528, 2658455991569831740043006737112629248, 191561942608236101977881394944640151127455097780436992

Offset: 1

Omar E. Pol, Apr 19 2008

nonn

Paolo Xausa, Table of n, a(n) for n = 1..13

Cf. A000396, A139228, A139229, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

Differences[PerfectNumber[Range[12]],2] (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139228(n+1) - A139228(n). - Jinyuan Wang, Mar 04 2020

a(5)-a(8) from Jinyuan Wang, Mar 04 2020

A139233 Second differences of perfect numbers A000396, divided by 2.

Original entry on oeis.org

223, 3582, 16767288, 4261388256, 60146251776, 1152921370926219264, 1329227995784915870021503368556314624, 95780971304118050988940697472320075563727548890218496

Offset: 1

Omar E. Pol, Apr 19 2008

nonn

Paolo Xausa, Table of n, a(n) for n = 1..13

Cf. A000396, A139228, A139229, A139230, A139231, A139232, A139234, A139235, A139236, A139237.

Differences[PerfectNumber[Range[12]],2]/2 (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139230(n)/2.

a(5)-a(8) from Jinyuan Wang, Mar 04 2020

A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

Original entry on oeis.org

326, 748, 166, 748, 166, 166, 748, 748, 166, 166, 748, 748, 166, 748, 748, 748, 166, 166, 166, 748, 166, 166, 166, 166, 166, 166, 166, 748, 748, 166, 748, 748, 166, 748, 166, 166, 166, 166, 166

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th superperfect number A019279(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of n-th term of A080172, A138125(n) and A094540(n).
a(1)=326. For n>1 a(n) is equal to 166 or 748, only.

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138817.

A099057 A bisection of A000396.

Original entry on oeis.org

6, 496, 33550336, 137438691328, 2658455991569831744654692615953842176, 13164036458569648337239753460458722910223472318386943117783728128

Offset: 0

N. J. A. Sloane, Nov 15 2004

nonn

The next term is too large to include. - Joshua Zucker, May 12 2006

More terms from Joshua Zucker, May 12 2006

A133031 Triangle read by rows: row n lists divisors of n-th perfect number A000396(n).

Original entry on oeis.org

1, 2, 3, 6, 1, 2, 4, 7, 14, 28, 1, 2, 4, 8, 16, 31, 62, 124, 248, 496, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896

Offset: 1

Omar E. Pol, Oct 26 2007

Harvey P. Dale, Table of n, a(n) for n = 1..5000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A018254, A018487, A027750. Perfect numbers: A000396.

Divisors[PerfectNumber[Range[5]]]//Flatten (* Harvey P. Dale, Jul 29 2024 *)

A099058 A bisection of A000396.

Original entry on oeis.org

28, 8128, 8589869056, 2305843008139952128, 191561942608236107294793378084303638130997321548169216, 14474011154664524427946373126085988481573677491474835889066354349131199152128

Offset: 0

N. J. A. Sloane, Nov 15 2004

nonn

More terms from Joshua Zucker, May 12 2006

A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

Original entry on oeis.org

326, 742, 314, 168, 843, 168, 521, 212, 212, 631, 181, 181, 632, 521, 155

Offset: 1

Omar E. Pol, Apr 01 2008

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

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

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A135613, A135617, A138124, A138817.

a(13)-a(15) from Robert Price, Jun 16 2019

cp's OEIS Frontend

A023196 Nondeficient numbers: numbers k such that sigma(k) >= 2k; union of A000396 and A005101.

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A139228 First differences of perfect numbers A000396.

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A139229 First differences of perfect numbers A000396, divided by 2.

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A139230 Second differences of perfect numbers A000396.

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A139233 Second differences of perfect numbers A000396, divided by 2.

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A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

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A099057 A bisection of A000396.

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A133031 Triangle read by rows: row n lists divisors of n-th perfect number A000396(n).

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A099058 A bisection of A000396.

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