A034785 -id:A034785 - cp's OEIS frontend

A137389 a(n) = 2^prime(n) + 2^prime(n+1).

12, 40, 160, 2176, 10240, 139264, 655360, 8912896, 545259520, 2684354560, 139586437120, 2336462209024, 10995116277760, 149533581377536, 9147936743096320, 585467951558164480, 2882303761517117440, 149879795598890106880, 2508757194024499019776, 11805916207174113034240

Offset: 1

Max Sills, Apr 10 2008

			a(3) = 160 because 2^5 + 2^7 = 160.

Cf. A000040, A001043, A137781.

A137389:=n->2^ithprime(n)+2^ithprime(n+1): seq(A137389(n), n=1..20); # Wesley Ivan Hurt, Mar 27 2015

f[n_] := 2^Prime[n] + 2^Prime[n + 1]; Array[f, 18] (* Robert G. Wilson v *)
Table[2^Prime[n] + 2^Prime[n + 1], {n, 18}] (* Stefan Steinerberger, Apr 12 2008 *)

a(n) = A034785(n) + A034785(n+1). [R. J. Mathar, Jun 15 2009]

a(n) = 4*A137781(n). - Wesley Ivan Hurt, Mar 27 2015

More terms from N. J. A. Sloane, Apr 10 2008

A137781 a(n) = (2^prime(n) + 2^prime(n+1)) / 4.

Original entry on oeis.org

3, 10, 40, 544, 2560, 34816, 163840, 2228224, 136314880, 671088640, 34896609280, 584115552256, 2748779069440, 37383395344384, 2286984185774080, 146366987889541120, 720575940379279360, 37469948899722526720, 627189298506124754944

Offset: 1

Alexander R. Povolotsky, Apr 28 2008

Note that 4 is 2^prime(1).

G. C. Greubel, Table of n, a(n) for n = 1..460

Cf. A034785, A137389.

[(2^NthPrime(n) + 2^NthPrime(n+1)) / 4: n in [1..20]]; // Vincenzo Librandi, Mar 28 2015

A137781:=n->(2^ithprime(n)+2^ithprime(n+1))/4: seq(A137781(n), n=1..20); # Wesley Ivan Hurt, Mar 27 2015

Table[(2^Prime[n] + 2^Prime[n + 1])/4, {n, 20}] (* Wesley Ivan Hurt, Mar 27 2015 *)
(2^#[[1]]+2^#[[2]])/4&/@Partition[Prime[Range[20]],2,1] (* Harvey P. Dale, Dec 21 2019 *)

vector(15,n,(2^prime(n)+2^prime(n+1))/4) \\ Derek Orr, Mar 29 2015

From Wesley Ivan Hurt, Mar 27 2015: (Start)
a(n) = A137389(n)/4.
a(n) = (A034785(n) + A034785(n+1))/4. (End)

A157413 Decimal expansion of sum_{p = primes = A000040} 1/(p*2^p).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 28 2009

			0.174087071760979362471993... = 1/(2*2^2)+1/(3*2^3)+1/(5*2^5)+1/(7*2^7)+... = sum_{i>=1} 1/(A000040(i)*A034785(i)).

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

A002162 = sum_{n>=1} 1/(n*2^n) = 1/2 + this_constant_here + A157414 + equivalent terms of higher order k-almost primes.

A247938 Sum of divisors of 2^prime(n)-1.

Original entry on oeis.org

4, 8, 32, 128, 2160, 8192, 131072, 524288, 8567136, 539922240, 2147483648, 138055271872, 2199187780272, 8817412930560, 140828559963840, 9008745449302368, 576463955735383776, 2305843009213693952, 147573953351708377936, 2361193635521975063040

Offset: 1

Vincenzo Librandi, Sep 27 2014

nonn

b-file computed with factorizations in Wagstaff link. a(167) corresponding to 2^991-1 is currently the first unknown term. - Jens Kruse Andersen, Sep 28 2014

Conjecture: a(n)/2^prime(n) reaches its maximum value 135/128 at n = 5. - Jianing Song, Dec 31 2022

R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys, R.P. Macédoine, Vol. 5 (1954), pp. 5-15.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 96.

Jens Kruse Andersen and Amiram Eldar, Table of n, a(n) for n = 1..197 (terms 1..166 Jens Kruse Andersen)
Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 271, exercise 22.
Sam Wagstaff, The Cunningham Project.

Subsequence of A075708.

Cf. A000203, A001348, A034785.

[SumOfDivisors(2^p-1): p in PrimesUpTo(100)];

with(numtheory): A247938:=n->sigma(2^ithprime(n)-1): seq(A247938(n), n=1..20); # Wesley Ivan Hurt, Sep 27 2014

Table[DivisorSigma[1, 2^Prime[n]-1], {n, 30}]

vector(50,n,sigma(2^prime(n)-1)) \\ Derek Orr, Sep 27 2014

a(n) = A000203(A001348(n)). - Michel Marcus, Sep 27 2014

Limit_{n->oo} a(n)/A001348(n) = 1 (Bojanić, 1954). - Amiram Eldar, Mar 04 2021

A269327 a(n) = 7^prime(n).

Original entry on oeis.org

49, 343, 16807, 823543, 1977326743, 96889010407, 232630513987207, 11398895185373143, 27368747340080916343, 3219905755813179726837607, 157775382034845806615042743, 18562115921017574302453163671207, 44567640326363195900190045974568007

Offset: 1

Emre APARI, Feb 23 2016

			The second prime is 3, hence a(2) = 7^3 = 343.
The third prime is 5, hence a(3) = 7^5 = 16807.

Cf. A000040, A000420, A034785, A053810, A057901, A057902, A132822.

[7^p: p in PrimesUpTo(45)]; // Vincenzo Librandi, Feb 25 2016

A269327:=n->7^ithprime(n): seq(A269327(n), n=1..20); # Wesley Ivan Hurt, Mar 07 2016

7^Prime[Range[20]] (* Alonso del Arte, Feb 23 2016 *)

a(n) = 7^prime(n); \\ Altug Alkan, Mar 04 2016

a(n) = 7^A000040(n).

Sum_{n>=1} 1/a(n) = A132822. - Amiram Eldar, Aug 11 2020

A346173 Decimal expansion of Sum_{k>=1} prime(k)/2^prime(k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 08 2021

This constant is irrational (Hančl and Tijdeman, 2004).

Jaroslav Hančl and Robert Tijdeman, On the irrationality of Cantor series, Journal für die reine und angewandte Mathematik, Vol. 2004, No. 571 (2004), pp. 145-158.

Cf. A034785, A098990, A100127.

RealDigits[Sum[Prime[n]/2^Prime[n], {n, 1, 100}], 10, 100][[1]]

suminf(k=1, prime(k)/2^prime(k)) \\ Michel Marcus, Jul 09 2021

1.09306425770250716540258595269676368295547596540121...

A135173 a(n) = 5^p - 3^p - 2^p, where p = prime(n).

Original entry on oeis.org

12, 90, 2850, 75810, 48648930, 1219100610, 762810181890, 19072323542370, 11920834803510690, 186264446292181467330, 4656612255401848810530, 72759575691550215703252290, 45474735052173413319557911170, 1136868376887903321203168728290, 710542735733511371371429275935010

Offset: 1

Omar E. Pol, Nov 25 2007

nonn

			a(4) = 75810 because the 4th prime number is 7, 5^7 = 78125, 3^7 = 2187, 2^7 = 128 and 78125-2187-128 = 75810.

Ivan Panchenko, Table of n, a(n) for n = 1..200

Cf. A034785 (2^p), A057901 (3^p), A057902 (5^p).

[5^p-3^p-2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Dec 14 2010

a:= n-> (p-> 5^p-3^p-2^p)(ithprime(n)):
seq(a(n), n=1..15);  # Alois P. Heinz, May 30 2025

5^# - 3^# - 2^# &/@Prime[Range[20]] (* G. C. Greubel, Sep 30 2016 *)

a(n,p=prime(n))=5^p - 3^p - 2^p \\ Charles R Greathouse IV, Sep 30 2016

a(n) = 5^p - 3^p - 2^p, with p = A000040(n).

a(n) = A057902(n) - A057901(n) - A034785(n). - Michel Marcus, Jun 14 2014

More terms from Vincenzo Librandi, Dec 14 2010

A135174 a(n) = 5^prime(n) - 3^prime(n) + 2^prime(n).

Original entry on oeis.org

20, 106, 2914, 76066, 48653026, 1219116994, 762810444034, 19072324590946, 11920834820287906, 186264446293255209154, 4656612255406143777826, 72759575691550490581159234, 45474735052173417717604422274, 1136868376887903338795354772706, 710542735733511371652904252645666

Offset: 1

Omar E. Pol, Nov 25 2007

			a(4)=76066 because the 4th prime number is 7, 5^7=78125, 3^7=2187, 2^7=128 and 78125-2187+128=76066.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040.

Cf. 2^p: A034785; 3^p: A057901; 5^p: A057902.

[5^p-3^p+2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Dec 14 2010

Table[5^p-3^p+2^p,{p,Prime[Range[20]]}] (* Harvey P. Dale, Dec 12 2013 *)

a(n)= 5^A000040(n) - 3^A000040(n) + 2^A000040(n).

More terms from Vincenzo Librandi, Dec 14 2010

A135176 5^p + 3^p + 2^p, where p = prime(n).

Original entry on oeis.org

38, 160, 3400, 80440, 49007320, 1222305640, 763068724360, 19074649113880, 11921023106645560, 186264583554009938920, 4656613490752936345720, 72759576592118302363153960, 45474735125119410471945995080, 1136868377544417273584428927960, 710542735786689000370819259221240

Offset: 1

Omar E. Pol, Nov 25 2007

			a(4)=80440 because the 4th prime number is 7, 5^7=78125, 3^7=2187, 2^7=128 and 78125+2187+128=80440.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. 2^p: A034785. 3^p: A057901. 2^5: A057902.

[5^p+3^p+2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi Dec 14 2010

Table[5^p + 3^p + 2^p, {p, Prime[Range[20]]}] (* Vincenzo Librandi, May 24 2014 *)

p=A000040(n): a(n)= 5^p + 3^p + 2^p.

More terms from Vincenzo Librandi, Dec 14 2010

A166743 a(n) = (2^p - p^2 - 1)/6 where p = prime(n).

Original entry on oeis.org

1, 13, 321, 1337, 21797, 87321, 1398013, 89478345, 357913781, 22906492017, 366503875645, 1466015503393, 23456248058853, 1501199875789697, 96076792050570001, 384307168202281705, 24595658764946068073

Offset: 3

Tanin (Mirza Sabbir Hossain Beg) (mirzasabbirhossainbeg(AT)yahoo.com), Oct 21 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 3..468

Cf. A034785, A066872.

A166743 := proc(n) p := ithprime(n) ; (2^p-p^2-1)/6 ; end: seq(A166743(n),n=3..20) ; # R. J. Mathar, Oct 25 2009

Table[(2^p-p^2-1)/6,{p,Prime[Range[3,20]]}] (* Harvey P. Dale, May 16 2020 *)

a(n) = (A034785(n) - A066872(n))/6. - R. J. Mathar, Oct 25 2009

Missing exponentiation signs inserted in the definition by R. J. Mathar, Oct 25 2009

Corrected by D. S. McNeil, Aug 20 2010

cp's OEIS Frontend

A137389 a(n) = 2^prime(n) + 2^prime(n+1).

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A137781 a(n) = (2^prime(n) + 2^prime(n+1)) / 4.

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A157413 Decimal expansion of sum_{p = primes = A000040} 1/(p*2^p).

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A247938 Sum of divisors of 2^prime(n)-1.

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A269327 a(n) = 7^prime(n).

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A346173 Decimal expansion of Sum_{k>=1} prime(k)/2^prime(k).

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A135173 a(n) = 5^p - 3^p - 2^p, where p = prime(n).

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