A165778 -id:A165778 - cp's OEIS frontend

A101260 Numbers n whose abundance is 56.

84, 140, 224, 308, 364, 476, 532, 644, 812, 868, 1036, 1148, 1204, 1316, 1372, 1484, 1652, 1708, 1876, 1988, 2044, 2212, 2324, 2492, 2716, 2828, 2884, 2996, 3052, 3164, 3556, 3668, 3836, 3892, 4172, 4228, 4396, 4544, 4564, 4676, 4844, 5012, 5068, 5348

Offset: 1

Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 17 2004

If n is of the form p*28, where p is a prime distinct from 2 or 7 then n is in this sequence, note that 28 is a perfect number. The terms in the sequence but not divisible by 28 are 4544, 9272, 14552, 25472, 74992, 495104... - Enrique Pérez Herrero, Apr 15 2012

If p=2^k-57 is prime (cf. A165778), then 2^(k-1)*p is in the sequence: For the first such k=6,7,8,10,16,19,22,28,..., this yields 224, 4544, 25472, 495104, 2145615872, 137424011264, 8795973484544, 36028789368553472, ... - M. F. Hasler, Apr 15 2012

			84 is a term of the sequence because 2*2*3*7 = 84 and 84 - 42 - 28 - 21 - 14 - 12 - 7 - 6 - 4 - 3 - 2 = g(84) = -55.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..3000
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A005101, A033880.

Cf. A141545, A141546.

[n: n in [1..10^4] |DivisorSigma(1,n) eq 2*n+56]; // Vincenzo Librandi, Jul 30 2015

Select[ Range[5500], DivisorSigma[1, # ] == 2# + 56 &] (* Robert G. Wilson v, Dec 22 2004 *)

A165779 Numbers k such that |2^k-993| is prime.

Original entry on oeis.org

1, 4, 6, 10, 14, 17, 26, 29, 54, 62, 77, 121, 344, 476, 1012, 1717, 1954, 2929, 2993, 3014, 3304, 4704, 8882, 24042, 43572, 45722, 54913, 57893, 72566, 74473, 82092, 117302

Offset: 1

M. F. Hasler, Oct 11 2009

If p = 2^k-993 is prime, then 2^(k-1)*p is a solution to sigma(x)-2x = 992 = 2^5*(2^5-1) = 2*A000396(3).

			a(4) = 10 since 2^10-993 = 31 is prime.
For exponents a(1) = 1, a(2) = 4 and a(3) = 6, we get 2^a(n)-993 = -991, -977 and -929 which are negative, but which are prime in absolute value.

Cf. A000396, A096818, A165778, A165780.

[n: n in [1..1100] |IsPrime(2^n-993)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 993]}, {n,0,100}], PrimeQ[#[[2]]] &][[All, 1]] (* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-993)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

from sympy import isprime, nextprime
def afind(limit):
    k, pow2 = 1, 2
    for k in range(1, limit+1):
        if isprime(abs(pow2-993)):
            print(k, end=", ")
        k += 1
        pow2 *= 2
afind(2000) # Michael S. Branicky, Dec 26 2021

a(23) from Altug Alkan, Apr 08 2016
a(24) from Michael S. Branicky, Dec 26 2021
a(25)-a(26) from Michael S. Branicky, Apr 06 2023
a(27)-a(32) from Michael S. Branicky, Sep 25 2024

A165780 Numbers n such that |2^n-16257| is prime.

Original entry on oeis.org

2, 3, 6, 8, 10, 12, 14, 16, 20, 22, 26, 30, 34, 36, 38, 43, 44, 50, 58, 64, 68, 80, 116, 142, 146, 254, 296, 298, 306, 396, 456, 730, 876, 1004, 1006, 1051, 1094, 1776, 1896, 1908, 2502, 2876, 3824, 3882, 4796, 4818, 5006, 5704, 6722, 8467, 9676

Offset: 1

M. F. Hasler, Oct 11 2009

nonn

If p=2^n-16257 is prime, then 2^(n-1)*p is a solution to sigma(x)-2x = 16256 = 2^7*(2^7-1) = 2*A000396(4).

			a(7)=14 since 2^14-16257 = 127 is prime.
For exponents a(1)=2 through a(6)=12, we get negative values for 2^a(k)-16257, which are prime in absolute value.

Cf. A096818, A165778, A165779.

[n: n in [1..1100] |IsPrime(2^n-16257)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 16257]},{n,0,100}], PrimeQ[#[[2]]] &][[All, 1]](* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-16257)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

More terms from Altug Alkan, Apr 08 2016

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A101260 Numbers n whose abundance is 56.

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A165779 Numbers k such that |2^k-993| is prime.

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A165780 Numbers n such that |2^n-16257| is prime.

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