A120875 -id:A120875 - cp's OEIS frontend

A023515 a(n) = prime(n)*prime(n-1) - 1.

1, 5, 14, 34, 76, 142, 220, 322, 436, 666, 898, 1146, 1516, 1762, 2020, 2490, 3126, 3598, 4086, 4756, 5182, 5766, 6556, 7386, 8632, 9796, 10402, 11020, 11662, 12316, 14350, 16636, 17946, 19042, 20710, 22498, 23706, 25590, 27220, 28890

Offset: 1

Clark Kimberling

nonn

a(1) = 1 assumes the not generally accepted convention prime(0) = 1. - Klaus Brockhaus, Dec 23 2010

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A120875 (a subsequence).

[ NthPrime(n-1)*NthPrime(n)-1: n in [1..50] ]; // Vincenzo Librandi, Dec 23 2010; simplified by Klaus Brockhaus, Dec 23 2010

1,seq(ithprime(n)*ithprime(n-1)-1,n=2..40); # Muniru A Asiru, Apr 27 2019

Prepend[Table[Prime@ n Prime[n - 1] - 1, {n, 2, 12}], 1] (* Michael De Vlieger, Nov 10 2015 *)
Join[{1},Times@@#-1&/@Partition[Prime[Range[40]],2,1]] (* Harvey P. Dale, Jul 06 2024 *)

a(n) = if(n==1, 1, prime(n)*prime(n-1)-1) \\ Altug Alkan, Nov 10 2015

From Jason Kimberley, Oct 23 2015: (Start)
a(n) = A006094(n-1) - 1 = A000040(n-1)*A000040(n)-1, for n>1.
a(n) = 2*A102770(n-2), for n>2.
(End)

A075369 Square associated with twin primes (p,p+2): p(p+2) + 1. Square of the average of twin primes.

Original entry on oeis.org

16, 36, 144, 324, 900, 1764, 3600, 5184, 10404, 11664, 19044, 22500, 32400, 36864, 39204, 51984, 57600, 72900, 79524, 97344, 121104, 176400, 186624, 213444, 272484, 324900, 360000, 381924, 412164, 435600, 656100, 675684, 685584, 736164

Offset: 1

Amarnath Murthy, Sep 20 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A014574, A037074, A120875.

a075369 = (^ 2) . a014574  -- Reinhard Zumkeller, Feb 10 2015

[a: n in [1..300] | IsSquare(a) where a is NthPrime(n)*NthPrime(n+1)+1]; // Vincenzo Librandi, Nov 19 2015

P:= select(isprime,{seq(2*i+1,i=1..1000)}):
T:= P intersect map(`+`,P,2):
sort(convert(map(t -> (t-1)^2, T), list)); # Robert Israel, Nov 18 2015

f[n_]:=Prime[n]*Prime[n+1]+1; lst={}; Do[If[IntegerQ[Sqrt[f[n]]],AppendTo[lst,f[n]]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

p=2; forprime(b=3, 1e3, if(b-p==2, print1(b*p+1", ")); p=b) \\ Altug Alkan, Nov 10 2015

a(n) = A037074(n) + 1. - Jon E. Schoenfield, Jan 13 2015
a(n) = A014574(n)^2. - Jon E. Schoenfield, Jan 14 2015
a(n) = A120875(n) + 2. - Jason Kimberley, Oct 22 2015

A120876 (Product of twin primes - 1)/2.

Original entry on oeis.org

7, 17, 71, 161, 449, 881, 1799, 2591, 5201, 5831, 9521, 11249, 16199, 18431, 19601, 25991, 28799, 36449, 39761, 48671, 60551, 88199, 93311, 106721, 136241, 162449, 179999, 190961, 206081, 217799, 328049, 337841, 342791, 368081, 388961, 520199, 532511, 551249, 563921

Offset: 1

Lekraj Beedassy, Jul 09 2006

nonn

This sequence is a subsequence of A102770.

Jason Kimberley, Table of n, a(n) for n = 1..5000

Cf. The subsequence A086870.

(Times@@#-1)/2&/@Select[Partition[Prime[Range[200]], 2,1],Last[#]- First[#]== 2&] (* Harvey P. Dale, Jun 26 2011 *)

for(n=1, 200, if(prime(n+1)-prime(n)==2, print1((prime(n)*prime(n+1)-1)/2", "))) \\ Altug Alkan, Oct 23 2015

p=2; forprime(q=3, 1e3, if(q-p==2, print1(p*q\2", ")); p=q) \\ Charles R Greathouse IV, Apr 01 2016

a(n) = A120875(n)/2 = A075369(n)/2-1 = A075369(n)^2/2-1.
a(n) = 18*A002822(n-1)^2-1, for n>1.
a(n) = A102770(A107770(n)). - Jason Kimberley, Nov 10 2015

Corrected by T. D. Noe, Oct 25 2006

Edited by Jason Kimberley, Oct 23 2015

A094949 Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

Original entry on oeis.org

192, 1152, 20160, 103680, 806400, 3104640, 12945600, 26853120, 108201600, 136002240, 362597760, 506160000, 1049630400, 1358807040, 1536796800, 2702128320, 3317529600, 5314118400, 6323748480, 9475464960, 14665694400

Offset: 1

Lekraj Beedassy, Jun 19 2004

nonn

If m=p*q for the twin prime pair (p, q), then the relation p^2 + q^2 = 2*(m+2) is evident from equations p*(p+2)=m=q*(q-2). Now phi(m)=(p-1)*(q-1)=p^2 - 1 and sigma(m)=(p+1)*(q+1)=q^2 - 1, so that phi(m)*sigma(m)=(p*q)^2 -(p^2 + q^2)+1=m^2-2*(m+2)+1=(m-3)*(m+1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001359, A006512, A037074.

EulerPhi[#]DivisorSigma[1,#]&/@Times@@@Select[Partition[Prime[ Range[ 200]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Apr 13 2017 *)

PARI
```
{m=400;p=1;while(p
				
```

a(n)=(m-3)*(m+1), where m=A037074(n).
a(n)=192*A002415(k), where k=A040040(n-1).
a(n) = (A120875(n))^2 - 4 = 4*((A120876(n))^2 - 1). - Lekraj Beedassy, Jul 09 2006

Corrected and extended by Jason Earls, Rick L. Shepherd, Vladeta Jovovic and Klaus Brockhaus, Jun 20 2004

cp's OEIS Frontend

A023515 a(n) = prime(n)*prime(n-1) - 1.

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A075369 Square associated with twin primes (p,p+2): p(p+2) + 1. Square of the average of twin primes.

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Haskell

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Maple

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A120876 (Product of twin primes - 1)/2.

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Programs

Mathematica

PARI

PARI

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Extensions

A094949 Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions