A157413 -id:A157413 - cp's OEIS frontend

A100042 a(n) = prime(n)*2^prime(n).

8, 24, 160, 896, 22528, 106496, 2228224, 9961472, 192937984, 15569256448, 66571993088, 5085241278464, 90159953477632, 378231999954944, 6614661952700416, 477381560501272576, 34011184385901985792

Offset: 1

Jorge Coveiro, Dec 26 2004

nonn

A lower bound for the 2-color van der Waerden number A005346(prime(n)); see Berlekamp reference. - Charles R Greathouse IV, Jul 13 2008

E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canadian Mathematical Bulletin, Vol. 11, No. 3 (1968), pp. 409-414.

Cf. A005346, A034785, A157413.

Maple
```
seq(ithprime(n)*2^ithprime(n),n=1..20);
```

#*2^#&/@Prime[Range[20]] (* Harvey P. Dale, Sep 18 2020 *)

a(n) = prime(n)*2^prime(n); \\ Michel Marcus, May 14 2017

Sum_{n>=1} 1/a(n) = A157413. - Amiram Eldar, Nov 17 2020

A157414 Decimal expansion of Sum_{q = semiprimes = A001358} 1/(q*2^q).

Original entry on oeis.org

0, 1, 8, 5, 5, 0, 2, 6, 6, 2, 7, 9, 9, 4, 9, 7, 0, 6, 5, 8, 9, 2, 6, 5, 4, 8, 5, 2, 8, 8, 2, 0, 4, 7, 7, 7, 4, 3, 0, 1, 6, 8, 9, 3, 1, 8, 6, 9, 2, 7, 5, 1, 2, 7, 0, 3, 2, 8, 2, 8, 9, 3, 0, 0, 3, 5, 0, 1, 5, 8, 8, 4, 7, 7, 6, 3, 7, 1, 6, 5, 7, 3, 8, 8, 0, 1, 5, 8, 5, 4, 6, 3, 6, 6, 7, 7, 0, 3, 8, 1, 7, 4, 1, 8, 3

Offset: 0

R. J. Mathar, Feb 28 2009

			0.0185502662799497065892654852882047774... = 1/(4*2^4)+1/(6*2^6)+1/(9*2^9)+1/(10*2^10)+... = Sum_{i>=1} 1/(A001358(i)*2^A001358(i)).

R. J. Mathar, Series of reciprocal powers of k-almost primes, arxiv:0803.0900, eq. (66).

Cf. A001358.

A002162 = Sum_{n>=1} 1/(n*2^n) = 1/2 + A157413 + this_constant_here + equivalent terms of higher order k-almost primes.

cp's OEIS Frontend

A100042 a(n) = prime(n)*2^prime(n).

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