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 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Comments

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

Links

Crossrefs

Cf. A120875 (a subsequence).

Programs

  • Magma
    [ NthPrime(n-1)*NthPrime(n)-1: n in [1..50] ]; // Vincenzo Librandi, Dec 23 2010; simplified by Klaus Brockhaus, Dec 23 2010
  • Maple
    1,seq(ithprime(n)*ithprime(n-1)-1,n=2..40); # Muniru A Asiru, Apr 27 2019
  • Mathematica
    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 *)
  • PARI
    a(n) = if(n==1, 1, prime(n)*prime(n-1)-1) \\ Altug Alkan, Nov 10 2015

Formula

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)

A234093 Integers of the form (p*q - 1)/2, where p and q are distinct primes.

Original entry on oeis.org

7, 10, 16, 17, 19, 25, 27, 28, 32, 34, 38, 42, 43, 45, 46, 47, 55, 57, 59, 61, 64, 66, 70, 71, 72, 77, 79, 80, 88, 91, 92, 93, 100, 101, 102, 104, 106, 107, 108, 109, 110, 117, 118, 123, 124, 126, 129, 132, 133, 143, 145, 147, 149, 150, 151, 152, 154, 159
Offset: 1

Views

Author

Clark Kimberling, Dec 27 2013

Keywords

Comments

A102770 is subsequence. - Zak Seidov, Feb 21 2014

Examples

			(3*5 - 1)/2 = 7; (3*7 - 1)/2 = 10.

Links

Crossrefs

Cf. A102770, A233561, A234095, A234096.

Programs

  • Mathematica
    t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t - 1)/2, 120] (* A234093 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233561 *)
(w - 1)/2 (* A234095 *) (* Peter J. C. Moses, Dec 23 2013 *)
With[{nn=60},Take[Union[Select[(Times@@#-1)/2&/@Subsets[Prime[ Range[ nn]],{2}],IntegerQ]],nn]] (* Harvey P. Dale, Mar 08 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

Views

Author

Lekraj Beedassy, Jul 09 2006

Keywords

Comments

This sequence is a subsequence of A102770.

Links

Crossrefs

Cf. The subsequence A086870.

Programs

  • Mathematica
    (Times@@#-1)/2&/@Select[Partition[Prime[Range[200]], 2,1],Last[#]- First[#]== 2&] (* Harvey P. Dale, Jun 26 2011 *)
  • PARI
    for(n=1, 200, if(prime(n+1)-prime(n)==2, print1((prime(n)*prime(n+1)-1)/2", "))) \\ Altug Alkan, Oct 23 2015
  • PARI
    p=2; forprime(q=3, 1e3, if(q-p==2, print1(p*q\2", ")); p=q) \\ Charles R Greathouse IV, Apr 01 2016

Formula

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

Extensions

Corrected by T. D. Noe, Oct 25 2006
Edited by Jason Kimberley, Oct 23 2015

A370763 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that a = prime(n)*prime(n+1) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

15, 112, 113, 35, 612, 613, 77, 2964, 2965, 143, 10224, 10225, 221, 24420, 24421, 323, 52164, 52165, 437, 95484, 95485, 667, 222444, 222445, 899, 404100, 404101, 1147, 657804, 657805, 1517, 1150644, 1150645, 1763, 1554084, 1554085, 2021, 2042220, 2042221, 2491, 3102540, 3102541
Offset: 2

Views

Author

Miguel-Ángel Pérez García-Ortega, Mar 01 2024

Keywords

Comments

The pair of natural numbers (d,e) is said to be a pair of primitive twin divisors of a natural number m when d*e = m and gcd(d,e) = 1.
Given two prime numbers p and q (p

Examples

			Table begins:
  n=2:   15,   112,   113;
  n=3:   35,   612,   613;
  n=4:   77,  2964,  2965;
  n=5:  143, 10224, 10225;
  n=6:  221, 24420, 24421;
  ...

References

  • Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

Crossrefs

Cf. A000040, A006094 (short leg), A102770 (inradius).

Programs

  • Mathematica
    Apply[Join,Map[{#,(#^2-1)/2,(#^2+1)/2} &,Prime[Range[2,31]]Prime[Range[3, 32]]]]

Formula

Row n = (a, b, c) = (p * q, (p^2 * q^2 - 1)/2, (p^2 * q^2 + 1)/2), where p = prime(n) and q = prime(n+1).
Showing 1-4 of 4 results.

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