A146768 -id:A146768 - cp's OEIS frontend

A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812

Offset: 1

Jason Earls, Aug 09 2005

All admirable numbers are abundant.
If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - Farideh Firoozbakht, Jan 28 2006
In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - Charles R Greathouse IV, Jan 30 2014
Subsequence of A083207. - Ivan N. Ianakiev, Mar 20 2017
The concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers. - Amiram Eldar, Aug 22 2018
Odd terms are listed in A109729. For abundant nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020

			12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Giovanni Resta, admirable numbers
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers

Subsequence of A005101 (abundant numbers).
Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.
Cf. A109729 (odd admirable numbers).

with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); # Walter Kehowski, Aug 12 2005

fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (* Robert G. Wilson v, Aug 13 2005 *)
Select[Range[812],MemberQ[Most[Divisors[#]],(DivisorSigma[1,#]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *)

for(n=1,10^3,ap=sigma(n)-2*n;if(ap>0 && (ap%2)==0,d=ap/2;if(d!=n && (n%d)==0, print1(n",")))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

is(n)=if(issquare(n)||issquare(n/2),0,my(d=sigma(n)/2-n); d>0 && d!=n && n%d==0) \\ Charles R Greathouse IV, Jun 21 2011

Better definition from Walter Kehowski, Aug 12 2005

A119591 Least k such that 2*n^k - 1 is prime.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 1, 10, 1, 1, 6, 1, 2, 6, 1, 2, 136, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 2, 4, 4, 1, 2, 1, 1, 44, 1, 1, 2, 1, 3, 2, 5, 3, 2, 2, 1, 4, 1, 768, 4, 1, 1, 52, 34, 2, 132, 1, 1, 14, 7, 1, 2, 2, 1, 8, 1, 2, 10, 1, 24, 60, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1

Offset: 2

Pierre CAMI, Jun 01 2006

From Eric Chen, Jun 01 2015: (Start)
Conjecture: a(n) is defined for all n.
a(303) > 10000, a(304)..a(360) = {1, 2, 11, 1, 990, 1, 1, 2, 2, 4, 74, 5, 1, 10, 6, 6, 4, 1, 1, 2, 1, 9, 12, 1, 80, 2, 1, 1, 2, 14, 3, 2, 3, 1, 12, 1, 60, 36, 1, 8, 4, 34, 1, 522, 3, 15, 14, 1, 6, 2, 3, 1, 4, 5, 4, 10, 1}.
a(n) = 1 if and only if n is in A006254. (End)
From Eric Chen, Sep 16 2021: (Start)
Now a(303) is known to be 40174, also other terms > 10000: a(383) = 20956, a(515) = 58466, a(522) = 62288, a(578) = 129468, a(581) > 400000, a(590) = 15526, a(647) = 21576, a(662) = 16590, a(698) = 127558, a(704) = 62034, see the a-file and the references.
a(n) = 2 if and only if n is in A066049 but not in A006254.
a(n) = 3 if and only if n is in A214289 but not in A006254 or A066049. (End)

Eric Chen, Table of n, a(n) for n = 2..580
Gary Barnes, Riesel conjectures and proofs
Eric Chen, Table of n, a(n) for n = 2..2050 status
Prime Wiki, Riesel prime small bases least n

Cf. A119624, A253178.

Numbers r such that 2*k^r-1 is prime: A090748 (k=2), A003307 (k=3), A146768 (k=4), A120375 (k=5), A057472 (k=6), A002959 (k=7), ... (k=8), ... (k=9), A002957 (k=10), A120378 (k=11), ... (k=12), A174153 (k=13), A273517 (k=14), ... (k=15), ... (k=16), A193177 (k=17), A002958 (k=25).

f[n_] := Block[{k = 0}, While[ ! PrimeQ[2*n^k - 1], k++ ]; k ]; Table[f[n], {n, 2, 106}] (* Ray Chandler, Jun 08 2006 *)

a(n) = for(k=1, 2^24, if(ispseudoprime(2*n^k-1), return(k))) \\ Eric Chen, Jun 01 2015

From Eric Chen, Sep 16 2021: (Start)
a(6*n) = A098873(n).
a(2^n) = A279095(n).
a(A006254(n)) = 1.
a(A066049(n)) <= 2.
a(A214289(n)) <= 3. (End)

Corrected and extended by Ray Chandler, Jun 08 2006

A138576 Numbers k such that 2^(2*k - 1) - 1 is prime.

Original entry on oeis.org

2, 3, 4, 7, 9, 10, 16, 31, 45, 54, 64, 261, 304, 640, 1102, 1141, 1609, 2127, 2212, 4845, 4971, 5607, 9969, 10851, 11605, 22249, 43122, 55252, 66025, 108046, 378420, 429717, 628894, 699135, 1488111, 1510689, 3486297, 6733459, 10498006, 12018292, 12982476, 15201229, 16291329, 18578334

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 13 2008

nonn

			2^(2*2 - 1) - 1 = 7;
2^(2*3 - 1) - 1 = 31;
2^(2*4 - 1) - 1 = 127.

Mario Fernando Garcia Rivera, Table of n, a(n) for n = 1..47 (derived from A000043)

Cf. A000043, A146768.

(MersennePrimeExponent[Range[2, 48]] + 1)/2 (* Amiram Eldar, Oct 22 2024 *)

is(n)=isprime(2^(2*n-1)-1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A000043(n+1) + 1)/2. - Charles R Greathouse IV, Aug 30 2010

a(n) = A146768(n) + 1. - César Aguilera, May 27 2020

More terms from Charles R Greathouse IV, Aug 30 2010

More terms from Mario Fernando Garcia Rivera, Jul 18 2022

A139481 a(n) = A139480(n)/2.

Original entry on oeis.org

0, 1, 2, 5, 7, 8, 14, 29, 43, 52, 62, 259, 302, 638, 1100, 1139, 1607, 2125, 2210, 4843, 4969, 5605, 9967, 10849, 11603, 22247, 43120, 55250, 66023, 108044, 378418, 429715, 628892, 699133, 1488109, 1510687, 3486295, 6733457, 10498004, 12018290

Offset: 2

Artur Jasinski, Apr 22 2008

Amiram Eldar, Table of n, a(n) for n = 2..48

Cf. A000043, A000668, A124477, A139479, A139480, A146768.

(MersennePrimeExponent[Range[2, 48]] - 3)/2 (* Amiram Eldar, Oct 17 2024 *)

a(n) = A146768(n-1)-1. - R. J. Mathar, Mar 30 2011

Edited by N. J. A. Sloane, May 23 2008

cp's OEIS Frontend

A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Extensions

A119591 Least k such that 2*n^k - 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A138576 Numbers k such that 2^(2*k - 1) - 1 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A139481 a(n) = A139480(n)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions