A099440 -id:A099440 - cp's OEIS frontend

A099439 Numbers k such that A000295(k) = 2^k-k-1 is prime.

4, 10, 14, 16, 26, 50, 56, 70, 116, 2072, 6250, 13670, 14216, 14626, 396128

Offset: 1

Hugo Pfoertner, Oct 18 2004

The next term is > 400000.
Equals A063791 + 1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

			a(1) = 4 because 2^4 - 4 - 1 = 11 is prime.

Cf. A000295 (2^n-n-1), A099440 (primes in A000295), A099441 (2^n-n-1 is a semiprime), A099442 (semiprimes in A000295).

a(15) (a PRP) from Karsten Bonath, Jun 07 2018

A099442 Semiprimes of the form 2^k-k-1.

Original entry on oeis.org

4, 26, 57, 247, 502, 4083, 1073741793, 4294967263, 8589934558, 70368744177617, 4503599627370443, 4611686018427387841, 18889465931478580854709, 75557863725914323419059, 77371252455336267181195177, 316912650057057350374175801245

Offset: 1

Hugo Pfoertner, Oct 18 2004

nonn

Semiprimes in A000295.

			a(4) = 247 because 247 = 13*19 = 2^8-8-1 = 2^A099441(4)-A099441(4)-1.

Hugo Pfoertner, Table of n, a(n) for n = 1..31 (first 19 terms from Vincenzo Librandi)

Cf. A099439, A099440, A099441.

Select[Table[2^n - n - 1, {n, 300}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)

from sympy.ntheory.factor_ import primeomega
def ok(n): return primeomega(2**n-n-1) == 2
print([2**m-m-1 for m in range(2, 100) if ok(m)]) # Michael S. Branicky, Apr 26 2021

A099441 Numbers n such that A000295(n) = 2^n-n-1 is a semiprime.

Original entry on oeis.org

3, 5, 6, 8, 9, 12, 30, 32, 33, 46, 52, 62, 74, 76, 86, 98, 130, 134, 154, 228, 230, 242, 256, 266, 346, 352, 382, 412, 428, 474, 488, 634, 650, 662, 688, 704, 722, 772, 896, 986, 1108, 1222, 1246, 1326

Offset: 1

Hugo Pfoertner, Oct 18 2004

A candidate for the next term is a(45) = 1736. 2^1736-1737 is composite with 523 decimal digits and unknown factorization. - Tyler Busby, Jan 05 2025

			a(3) = 6 because 2^6-6-1 = 57 = 3*19 is a semiprime.

Dario Alpern, Factorization using the Elliptic Curve Method.
Status of 2^1736-1737 in factordb.com.

Cf. A000295 (2^n-n-1 (column 2 of the Eulerian numbers)), A099439 (2^n-n-1 is prime), A099440 (primes in A000295), A099442 (semiprimes in A000295).

More terms from Hugo Pfoertner, Aug 13 2007
a(24) corrected by Hugo Pfoertner, Sep 07 2017
a(32)-a(44) from Tyler Busby, Jan 05 2025

A206028 a(n) is the sum of distinct values of sigma(d) where d runs over the divisors of n and sigma = A000203.

Original entry on oeis.org

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 248

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A007429: a(66) = 248, A007429(66) = 260. Number 66 is the smallest number with at least two divisors d with the same sigma(d); see A206030.
In A007429 all values of sigma(d) of the divisors d of n are included in the sum with repetitions allowed. In this sequence only the distinct values of sigma(d) of the divisors d of n are included in the sum.
If a term is a prime p when n = 2^j then p = 2^(j+2)-(j+3) is also a term of A099440 (primes of the form 2^n-n-1). Greater of twin primes are terms. - Metin Sariyar, Apr 03 2020

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Sum of k = 1+3+4+12 = 20.
For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Sum of k = 1+3+4+12+36+48+144 = 248 (note that only one twelve is added.).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000203, A000217, A007429, A184387, A206029, A206030.

Table[Total[Union[DivisorSigma[1, Divisors[n]]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)={vecsum(Set(apply(sigma, divisors(n))))} \\ Andrew Howroyd, Aug 01 2018

a(p) = p+2, a(pq) = (p+2)*(q+2) for p, q = distinct primes.
a(n) = A184387(n) - A206029(n) = A000217(A000203(n)) - A206029(n).
a(2^n) = 2^(n+2) - (n+3). - Metin Sariyar, Apr 09 2020

Name clarified by David A. Corneth, Aug 01 2018

a(62)-a(66) from Andrew Howroyd, Aug 01 2018

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A099439 Numbers k such that A000295(k) = 2^k-k-1 is prime.

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A099442 Semiprimes of the form 2^k-k-1.

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A099441 Numbers n such that A000295(n) = 2^n-n-1 is a semiprime.

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A206028 a(n) is the sum of distinct values of sigma(d) where d runs over the divisors of n and sigma = A000203.

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