A132797 -id:A132797 - cp's OEIS frontend

A057902 a(n) = 5^prime(n).

25, 125, 3125, 78125, 48828125, 1220703125, 762939453125, 19073486328125, 11920928955078125, 186264514923095703125, 4656612873077392578125, 72759576141834259033203125, 45474735088646411895751953125

Offset: 1

Henry Bottomley, Sep 29 2000

nonn

			a(4) = 5^7 = 78125.

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

Cf. A000040, A034785, A053810, A057902, A132797.

A057902:=n->5^ithprime(n); seq(A057902(n), n=1..30); # Wesley Ivan Hurt, Jun 14 2014

5^Prime@Range@30 (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

a(n) = 5^A000040(n).

Sum_{n>=1} 1/a(n) = A132797. - Amiram Eldar, Aug 11 2020

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

Original entry on oeis.org

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

A132806 Decimal expansion of Sum_{n >= 1} 1/4^prime(n).

Original entry on oeis.org

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

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-4 expansion. - M. F. Hasler, Jul 04 2017

			0.079162851037850149671771117962208184613038569751878...

Cf. A000720, A051006 (analog for base 2), A132800 (analog for base 3), A132797 (analog for base 5), A010051 (characteristic function of the primes), A000040 (the primes).

/* 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,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/4^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

Equals 3 * Sum_{k>=1} pi(k)/4^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 04 2017

A132817 Decimal expansion of Sum_{n >= 1} 1/6^prime(n).

Original entry on oeis.org

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

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-6 expansion. - M. F. Hasler, Jul 05 2017

			0.032539583308525544049260050781274181192986076617578098887664610099...

Vincenzo Librandi, Table of n, a(n) for n = 0..1110

Cf. A000720, A051006 (analog for base 2), A132800 (analog for base 3), A132806 (analog for base 4), A132797 (analog for base 5), A132822 (analog for base 7), A010051 (characteristic function of the primes), A000040 (the primes).

Join[{0}, RealDigits[FromDigits[{{Table[If[PrimeQ[n], 1, 0], {n, 370}]}, 0}, 6], 10, 111][[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,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/6^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

Equals 5 * Sum_{k>=1} pi(k)/6^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 05 2017

A132822 Decimal expansion of Sum_{n >= 1} 1/7^prime(n).

Original entry on oeis.org

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

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-7 expansion. - M. F. Hasler, Jul 05 2017

			0.023384328960353739909859822495912373489340935935944869619982884656523568...

Cf. A000720, A051006 (analog for base 2), A132800 (analog for base 3), A132806 (analog for base 4), A132797 (analog for base 5), A132817 (analog for base 6), A010051 (characteristic function of the primes), A132799 (base 8), A269327.

/* 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,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/7^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 05 2017

From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A269327(k).
Equals 6 * Sum_{k>=1} pi(k)/7^(k+1), where pi(k) = A000720(k). (End)

Offset corrected by R. J. Mathar, Feb 05 2009

Edited by M. F. Hasler, Jul 05 2017

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A057902 a(n) = 5^prime(n).

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

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A132806 Decimal expansion of Sum_{n >= 1} 1/4^prime(n).

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A132817 Decimal expansion of Sum_{n >= 1} 1/6^prime(n).

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A132822 Decimal expansion of Sum_{n >= 1} 1/7^prime(n).

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