A057901 -id:A057901 - cp's OEIS frontend

A132800 Decimal expansion of Sum_{n >= 1} 1/3^prime(n).

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

Offset: 0

Cino Hilliard, Nov 17 2007

Equivalently, the real number in (0,1) having the characteristic function of the primes, A010051, as its base-3 expansion. - M. F. Hasler, Jul 04 2017.

			0.15272690272572247152817541874425912430342364271463298508628837536732...

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A000720, A051006 (analog for base 2), A132797 (analog for base 5), A010051 (characteristic function of the primes), A057901, A132806 (base 4).

RealDigits[Sum[1/3^Prime[k], {k, 100}], 10, 100][[1]] (* Vincenzo Librandi, Jul 05 2017 *)

/* Sum of 1/m^p for primes p */ sumnp(n,m) = { local(s=0,a,j); for(x=1,n, s+=1./m^prime(x); ); a=Vec(Str(s)); for(j=3,100, print1(eval(a[j])",") ) }

suminf(n=1,1/3^prime(n)) \\ Then: digits(%\.1^default(realprecision))[1..-3] to remove the last 2 digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A057901(k).
Equals 2 * Sum_{k>=1} pi(k)/3^(k+1), where pi(k) = A000720(k). (End)

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 04 2017

A135171 3^p - 2^p, where p = prime(n).

Original entry on oeis.org

5, 19, 211, 2059, 175099, 1586131, 129009091, 1161737179, 94134790219, 68629840493971, 617671248800299, 450283768452043891, 36472994178147530851, 328256958598444055419, 26588814218220014932459, 19383245658672820642055731, 14130386091162273752461387579, 127173474823342767533669605651

Offset: 1

Omar E. Pol, Nov 25 2007

nonn

			a(4)=2059 because the 4th prime number is 7, 3^7=2187, 2^7=128 and 2187-128=2059.

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

Cf. A000040, A034785, A057901.

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

3^#-2^#&/@Prime[Range[20]] (* Harvey P. Dale, Jun 20 2011 *)

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

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

More terms from Vincenzo Librandi, Dec 14 2010

A135175 a(n) = 5^p + 3^p - 2^p, where p = prime(n).

Original entry on oeis.org

30, 144, 3336, 80184, 49003224, 1222289256, 763068462216, 19074648065304, 11921023089868344, 186264583552936197096, 4656613490748641378424, 72759576592118027485247016, 45474735125119406073899483976, 1136868377544417255992242883544, 710542735786689000089344282510584

Offset: 1

Omar E. Pol, Nov 25 2007

			a(4)=80184 because the 4th prime number is 7, 5^7=78125, 3^7=2187, 2^7=128 and 78125+2187-128=80184.

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

a:= n-> (p-> 5^p+3^p-2^p)(ithprime(n)):
seq(a(n), n=1..15);  # Alois P. Heinz, Jun 08 2025

5^#+3^#-2^#&/@Prime[Range[20]]  (* Harvey P. Dale, Apr 04 2011 *)
Table[5^p + 3^p - 2^p, {p, Prime[Range[20]]}] (* Vincenzo Librandi, May 24 2014 *)

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

More terms from Vincenzo Librandi, Dec 14 2010

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

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

A262782 a(n) = sum_{k=1..n} 3^prime(k).

Original entry on oeis.org

9, 36, 279, 2466, 179613, 1773936, 130914099, 1293175566, 95436354393, 68725813719276, 686399210003223, 450970305101000586, 36923966682271786989, 365180934076808864616, 26953995293034312152403, 19410199662973054208949126, 14149796291401707558973760193, 141323271117050318101857059796

Offset: 1

Emre APARI, Mar 24 2016

			a(2) = 3^prime(1) + 3^prime(2) = 3^2+3^3=36.

Cf. A057901.

Accumulate@ Array[3^Prime@ # &, {18}] (* Michael De Vlieger, Mar 24 2016 *)

a(n) = sum(k=1, n, 3^prime(k)); \\ Altug Alkan, Mar 24 2016

More terms from Michael De Vlieger, Mar 24 2016

cp's OEIS Frontend

A132800 Decimal expansion of Sum_{n >= 1} 1/3^prime(n).

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A135171 3^p - 2^p, where p = prime(n).

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A135175 a(n) = 5^p + 3^p - 2^p, where p = prime(n).

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

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

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Maple

Mathematica

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

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

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