A091515 -id:A091515 - cp's OEIS frontend

A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

42, 1316, 131080256

Offset: 1

G. L. Honaker, Jr., Jan 26 2020

The number 37778715690312487141376 is also in the sequence. - Daniel Suteu, Jan 27 2020
The first 3 terms have the form (2^p-1)*(2^(p-1))*((2^p-1)^2-2), i.e., a Perfect number times a Carol prime. - G. L. Honaker, Jr., Jan 27 2020
In other words, the values of p are given by the intersection of A091515 and A000043. Currently, only four such values of p are known: {2, 3, 7, 19}. - Daniel Suteu, Jan 27 2020
From Bernard Schott, Jan 28 2020: (Start)
Proposition: If a number N_p is of the form Q_p * C_p where Q_p = (2^(p-1)) * (2^p - 1) is a perfect number and C_p = (2^p-1)^2-2 is a Carol prime then, the sum of the nonprime proper divisors of N_p called S_p(N_p) is equal to N_p.
Proof:
The sum of the nonprime proper divisors of N_p is:
S_p(N_p) = (2* Q_p - 2 - (2^p-1)) + ((Q_p - 1) * C_p).
In the first parenthesis, there is the sum of the nonprime proper divisors of N_p coming only from the perfect number Q_p, then in the second parenthesis, there is the sum of the nonprime proper divisors of N_p coming from C_p.
Then, this sum of the nonprime proper divisors of N_p, S_p(N_p) is indeed equal to N_p = (2^(p-1)) * (2^p-1) * ((2^p-1)^2-2).
Hence, (2^19-1)*(2^(19-1))*((2^19-1)^2-2) = 37778715690312487141376 is a term. (End)
10^13 < a(4) <= 72872313094554244192 = 2^5 * 109 * 151 * 65837 * 2101546957. - Giovanni Resta, Jan 28 2020

			42 is a term because 42 = 1 + 6 + 14 + 21.
1316 is a term because 1316 = 1 + 4 + 14 + 28 + 94 + 188 + 329 + 658.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 42
Wikipedia, 42 (number)

Cf. A001065, A018252, A023890, A331858.

Cf. A000043, A091515, A091516 (Carol primes).

fun[p_, e_] := (p^(e+1) - 1)/(p - 1); npsigma[n_] := Times @@ fun @@@ (f = FactorInteger[n]) - Plus @@ First /@ f;; Select[Range[2, 1500], npsigma[#] == 2# &] (* Amiram Eldar, Jan 26 2020 *)

isok(n) = sigma(n) - n - vecsum(factor(n)[,1]) == n; \\ Daniel Suteu, Jan 27 2020

a(2) from Chuck Gaydos

a(3) from Amiram Eldar, Jan 26 2020

A098939 a(n) = (2^n - 1)^11 - 2.

Original entry on oeis.org

-2, -1, 177145, 1977326741, 8649755859373, 25408476896404829, 62050608388552823485, 138624799340320978519421, 296443535898840969287109373, 620340165367069806017523226109, 1284197945649659948122178573052925

Offset: 0

Parthasarathy Nambi, Oct 21 2004

sign

			a(2) = (2^2 - 1)^11 - 2 = 177145.

These are similar to Carol's numbers (A091515).

Table[(2^n - 1)^11 - 2, {n, 0, 10}] (* Stefan Steinerberger, Mar 06 2006 *)
(2^Range[0,10]-1)^11-2 (* Harvey P. Dale, Apr 29 2014 *)

first(m)=vector(m,i,i--;(2^i - 1)^11 - 2) \\ Anders Hellström, Aug 26 2015

More terms from Stefan Steinerberger, Mar 06 2006

A098940 a(n) = (2^n - 1)^7 - 2.

Original entry on oeis.org

-2, -1, 2185, 823541, 170859373, 27512614109, 3938980639165, 532875860165501, 70110209207109373, 9098007718612700669, 1172544775637859048445, 150599974607076225726461, 19309780749956356667109373, 2473765232992079297629970429

Offset: 0

Parthasarathy Nambi, Oct 21 2004

sign

			a(2) = (2^2 - 1)^7 - 2 = 2185.

These are similar to Carol's numbers (A091515).

Table[(2^n - 1)^7 - 2, {n, 0, 15}] (* Stefan Steinerberger, Mar 06 2006 *)

More terms from Stefan Steinerberger, Mar 06 2006

A099360 a(n) = (2^n + 1)^4 - 2.

Original entry on oeis.org

14, 79, 623, 6559, 83519, 1185919, 17850623, 276922879, 4362470399, 69257922559, 1103812890623, 17626570956799, 281749955297279, 4505799053311999, 72075187834650623, 1153062248537784319, 18447869999386460159

Offset: 0

Parthasarathy Nambi, Nov 16 2004

Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A091513, A091515.

Table[(2^n + 1)^4 - 2, {n, 0, 18}] (* Stefan Steinerberger, Mar 06 2006 *)

a(n)=(2^n+1)^4-2 \\ Charles R Greathouse IV, Oct 01 2012

More terms from Stefan Steinerberger, Mar 06 2006

A118558 a(n) = (2^n-1)^4 - 2.

Original entry on oeis.org

-2, -1, 79, 2399, 50623, 923519, 15752959, 260144639, 4228250623, 68184176639, 1095222947839, 17557851463679, 281200199450623, 4501401006735359, 72040003462430719, 1152780773560811519, 18445618199572250623, 295138898083176775679, 4722294425687923097599

Offset: 0

Jonathan Vos Post, May 03 2006

			a(0) = (2^0 - 1)^4 - 2 = 0^4 - 2 = -2.
a(1) = (2^1 - 1)^4 - 2 = 1^4 - 2 = -1.
a(2) = (2^2 - 1)^4 - 2 = 3^4 - 2 = 79.

Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Cf. A091516, A091515, A098878, A091514.

Cf. A093112, A098878.

(2^Range[0, 20] - 1)^4 - 2 (* Paolo Xausa, Apr 19 2024 *)

a(n)=(2^n-1)^4-2 \\ Charles R Greathouse IV, Feb 19 2016

a(n) = (2^n - 1)^4 - 2.

G.f.: (1984*x^4-2120*x^3+510*x^2-61*x+2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). - Colin Barker, Apr 30 2013

Offset changed to 0 by Paolo Xausa, Apr 19 2024

A173888 Exactly one of (2^n-1)^2-2 and (2^n+1)^2-2 is prime.

Original entry on oeis.org

0, 1, 4, 5, 6, 7, 8, 9, 10, 17, 19, 23, 25, 32, 51, 55, 65, 87, 129, 132, 159, 171, 175, 180, 242, 315, 324, 358, 393, 435, 467, 491, 501, 507, 555, 591, 680, 786, 800, 1070, 1459, 1650, 1707, 2813, 2923, 3281, 4217, 5153, 6287, 6365, 6462, 10088, 10367, 14289

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

The numbers which are in A091513 or A091515, but not in both sequences. - R. J. Mathar, Mar 09 2010

			a(1)=1 because (2^1-1)^2-2=-1 is nonprime and (2^1+1)^2-2=7 is prime.

Cf. A091513, A091515, A091514, A091516.

Corrected (0 inserted, 12, 16, 18, 21 removed) and extended by R. J. Mathar, Mar 09 2010

A173903 Numbers k such that both (2^k+1)^2-2 and (2^k-1)^2-2 are prime.

Original entry on oeis.org

2, 3, 12, 15, 18, 21, 27

Offset: 1

Vincenzo Librandi, Mar 08 2010

a(8) > 9394. - Max Z. Scialabba, Jan 21 2024

a(8) > 695631 using A091513 and A091515. - Michael S. Branicky, Oct 24 2024

Cf. A091513, A091515.

[n: n in [1..400] | IsPrime((2^n-1)^2-2) and IsPrime((2^n+1)^2-2)];

Select[Range[3000], PrimeQ[((2^# + 1)^2 - 2)]&&PrimeQ[((2^# - 1)^2 - 2)] &] (* Vincenzo Librandi, Aug 21 2014 *)

A091513 INTERSECT A091515. - R. J. Mathar, Jul 06 2010

Definition clarified by Jon E. Schoenfield, Jun 18 2010

cp's OEIS Frontend

A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

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A098939 a(n) = (2^n - 1)^11 - 2.

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A098940 a(n) = (2^n - 1)^7 - 2.

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A099360 a(n) = (2^n + 1)^4 - 2.

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A118558 a(n) = (2^n-1)^4 - 2.

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A173888 Exactly one of (2^n-1)^2-2 and (2^n+1)^2-2 is prime.

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Author

Keywords

Comments

Examples

Crossrefs

Extensions

A173903 Numbers k such that both (2^k+1)^2-2 and (2^k-1)^2-2 are prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions