cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A076265 a(n) = Product_{i=1..n} prime(i)^prime(i).

Original entry on oeis.org

4, 108, 337500, 277945762500, 79301169838123235887500, 24018350267611933650627567399079537500, 19868946365457062696924774946056904675112420776003728137500
Offset: 1

Views

Author

Jeff Burch, Nov 23 2002

Keywords

Comments

Denominator of Sum_{i=1..n} 1/(p(i)^p(i)), where p(i) = i-th prime. The numerators are in A117579. E.g., 1/4, 31/108, 96983/337500, 79870008269/277945762500, ... - Jonathan Vos Post, Mar 29 2006
Equally, denominator of Sum_{k=1..n}(-1)^(k+1) * 1/p(k)^p(k), where p(k) = prime(k). - Alexander Adamchuk, Aug 22 2006
C = Sum_{k>=1} (-1)^(k+1)/(prime(k)^prime(k)) = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... A122147 is the decimal expansion of C = 0.213281748700785698255627... - Alexander Adamchuk, Aug 22 2006
Hyperprimorials, from primorials by analogy with hyperfactorials. See A006939. - Matthew Campbell, Jul 30 2015

Examples

			A122148(n)/a(n) begins 1/4, 23/108, 71983/337500, ... - _Alexander Adamchuk_, Aug 22 2006

Crossrefs

Cf. A051674, A122147, A122148, A094289, A117579, A076265, A000040.

Programs

  • Mathematica
    Table[Denominator[Sum[1/Prime[k]^Prime[k],{k,1,n}]],{n,1,10}] (* Alexander Adamchuk, Aug 22 2006 *)
Denominator[Accumulate[1/#^#&/@Prime[Range[10]]]] (* Harvey P. Dale, Jan 24 2013 *)
  • PARI
    a(n)=prod(i=1,n,prime(i)^prime(i)) \\ Charles R Greathouse IV, Aug 05 2015

Formula

log a(n) ~ (n^2 log^2 n)/2. - Charles R Greathouse IV, Sep 14 2015

Extensions

Entry revised by N. J. A. Sloane, Apr 10 2006
Edited by N. J. A. Sloane, Aug 04 2008 at the suggestion of R. J. Mathar

A118055 Numerator of Sum_{i=1..n} 1/(s(i)^s(i)) where s(i) = i-th semiprime.

Original entry on oeis.org

1, 733, 389546509, 15216660895232989, 165124648173861912289213141201, 516014525543318775927975356319557, 11473924061057077116469420939475877122177
Offset: 1

Views

Author

Jonathan Vos Post, Apr 11 2006

Keywords

Comments

Semiprime analog of A117579. Fractions are 1/256, 733/186624, 389546509/99179645184, 15216660895232989/3874204890000000000, 165124648173861912289213141201/42041202325478752505760000000000, 516014525543318775927975356319557/131378757267121101580500000000000000, 11473924061057077116469420939475877122177 / 2921293509192991260690562210500000000000000, 239106294995420151295311285049507497083520504633431021289373163777 / 6087713879404511830817263262876196035025072.

Examples

			a(2) = 733 because (1/semiprime(1)^semiprime(1)) + (1/semiprime(2)^semiprime(2))
= (1/256) + (1/46656) = 733/186624.

Crossrefs

Denominators = A118055. Cf. A001358, A051674, A114850, A117579.

Programs

  • Mathematica
    Numerator[Accumulate[1/#^#&/@Select[Range[25],PrimeOmega[#]==2&]]] (* Harvey P. Dale, Aug 09 2012 *)

Formula

a(n) = Numerator of Sum_{i=1..n} 1/(semiprime(i)^semiprime(i)).
a(n) = Numerator of Sum_{i=1..n} 1/(A001358(i)^A001358(i)).
a(n) = Numerator of Sum_{i=1..n} 1/A114850(n).

A118056 Denominator of Sum_{i=1..n} 1/(s(i)^s(i)) where s(i) = i-th semiprime.

Original entry on oeis.org

256, 186624, 99179645184, 3874204890000000000, 42041202325478752505760000000000, 131378757267121101580500000000000000, 2921293509192991260690562210500000000000000, 60877138794045118308172632628761960350250724033554048000000000000000
Offset: 1

Views

Author

Jonathan Vos Post, Apr 11 2006

Keywords

Comments

Semiprime analog of A076265. Fractions are 1/256, 733/186624, 389546509/99179645184, 15216660895232989/3874204890000000000, 165124648173861912289213141201/42041202325478752505760000000000, 516014525543318775927975356319557/131378757267121101580500000000000000, 11473924061057077116469420939475877122177 / 2921293509192991260690562210500000000000000, 239106294995420151295311285049507497083520504633431021289373163777 / 60877138794045118308172632628761960350250724033554048000000000000000.

Examples

			a(2) = 186624 because (1/semiprime(1)^semiprime(1)) + (1/semiprime(2)^semiprime(2))= (1/256) + (1/46656) = 733/186624.

Crossrefs

Numerators = A118055. Cf. A001358, A051674, A114850, A117579.

Programs

  • Mathematica
    Denominator[Accumulate[1/#^#&/@Select[Range[30],PrimeOmega[#]==2&]]] (* Harvey P. Dale, Feb 15 2012 *)

Formula

a(n) = Denominator of Sum_{i=1..n} 1/(semiprime(i)^semiprime(i)).
a(n) = Denominator of Sum_{i=1..n} 1/(A001358(i)^A001358(i)).
a(n) = Denominator of Sum_{i=1..n} 1/A114850(n).

Extensions

Corrected by Harvey P. Dale, Feb 15 2012

A122147 Decimal expansion of Sum[ (-1)^(k+1) * 1/p(k)^p(k) ], where p(k) = Prime[k].

Original entry on oeis.org

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

Views

Author

Alexander Adamchuk, Aug 22 2006

Keywords

Comments

C = Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,Infinity} ] = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... Partial sums are A122148[n] / A076265[n] = Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,n} ] = 1/4, 23/108, 71983/337500, ...

Examples

			C = 0.2132817487007856982556274813698484360277279725322464100714222201238395676003\
726900563712201186188234415559844581411471306301650311286030077813464608267160\
801494597797561591251174806253914566160177882...

Crossrefs

Cf. A051674, A122148, A076265, A094289, A117579, A076265, A000040.

A122148 Numerator of Sum[ (-1)^(k+1) * 1/p(k)^p(k), {k,1,n}], where p(k) = Prime[k].

Original entry on oeis.org

1, 23, 71983, 59280758269, 16913492177093188294859, 5122675745984257357873512804013239827, 4237683625666802603266159755806379107958975382128522814879
Offset: 1

Views

Author

Alexander Adamchuk, Aug 22 2006

Keywords

Comments

C = Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,Infinity} ] = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... A122147[n] is a decimal expansion of C = 0.213281748700785698255627...

Examples

			a[n] / A076265[n] begins 1/4, 23/108, 71983/337500, ...

Crossrefs

Cf. A051674, A122147, A094289, A117579, A076265, A000040.

Programs

  • Mathematica
    Table[Numerator[Sum[(-1)^(k+1)*1/Prime[k]^Prime[k],{k,1,n}]],{n,1,10}]

Formula

a(n) = Numerator[ Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,n} ] ].

A118062 Numerator of Sum_{i=1..n} 1/(t(i)^t(i)) where t(i) = i-th 3-almost prime.

Original entry on oeis.org

1, 265721, 75047458863267833, 938093235790847912650094635296999121, 2771420766426289313598405374054613260285749630619149892803, 83546357082134777747819786589906868700938637689935705237433756853637190925073724793683
Offset: 1

Views

Author

Jonathan Vos Post, Apr 11 2006

Keywords

Comments

3-almost prime analog of A117579. Semiprime analog of A117579 is A118056. Fractions are 1/16777216, 265721/4458050224128, 75047458863267833/1259085058409489202413568, 938093235790847912650094635296999121 / 15738563230118615030169600000000000000000000, 2771420766426289313598405374054613260285749630619149892803 / 46496637333593157266125580467610571799579852800000000000000000000.

Examples

			a(2) = 265721 because (1/A014612(1)^A014612(1)) + (1/A014612(2)^A014612(2))= (1/(8^8)) + (1/(12^12)) = (1/16777216) + (1/8916100448256) = 265721/4458050224128.

Crossrefs

Denominators = A118063. Cf. A001358, A014612, A051674, A114850, A114967, A117579, A118056.

Formula

a(n) = Numerator of Sum_{i=1..n} 1/(3almostprime(i)^3almostprime(i)).
a(n) = Numerator of Sum_{i=1..n} 1/(A014612(i)^A014612(i)).
a(n) = Numerator of Sum_{i=1..n} 1/A114967(n).

A118063 Denominator of Sum_{i=1..n} 1/(t(i)^t(i)) where t(i) = i-th 3-almost prime.

Original entry on oeis.org

16777216, 4458050224128, 1259085058409489202413568, 15738563230118615030169600000000000000000000, 46496637333593157266125580467610571799579852800000000000000000000
Offset: 1

Views

Author

Jonathan Vos Post, Apr 11 2006

Keywords

Comments

3-almost prime analog of A076265. (Semiprime analog of A076265 is A118056.) Fractions are 1/16777216, 265721/4458050224128, 75047458863267833/1259085058409489202413568, 938093235790847912650094635296999121 / 15738563230118615030169600000000000000000000, 2771420766426289313598405374054613260285749630619149892803 / 46496637333593157266125580467610571799579852800000000000000000000.

Examples

			a(2) = 4458050224128 because (1/A014612(1)^A014612(1)) + (1/A014612(2)^A014612(2))= (1/(8^8)) + (1/(12^12)) = (1/16777216) + (1/8916100448256) = 265721/4458050224128.

Crossrefs

Numerators = A118062. Cf. A001358, A014612, A051674, A114850, A114967, A117579, A118056.

Programs

  • Mathematica
    Accumulate[1/#^#&/@Select[Range[30],PrimeOmega[#]==3&]]//Denominator (* Harvey P. Dale, Apr 05 2020 *)

Formula

a(n) = Denominator of Sum_{i=1..n} 1/(3almostprime(i)^3almostprime(i)).
a(n) = Denominator of Sum_{i=1..n} 1/(A014612(i)^A014612(i)).
a(n) = Denominator of Sum_{i=1..n} 1/A114967(n).
Showing 1-7 of 7 results.

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