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 1-8 of 8 results.

A102770 (p*q - 1)/2 where p and q are consecutive odd primes.

Original entry on oeis.org

7, 17, 38, 71, 110, 161, 218, 333, 449, 573, 758, 881, 1010, 1245, 1563, 1799, 2043, 2378, 2591, 2883, 3278, 3693, 4316, 4898, 5201, 5510, 5831, 6158, 7175, 8318, 8973, 9521, 10355, 11249, 11853, 12795, 13610, 14445, 15483, 16199, 17285, 18431, 19010
Offset: 1

Views

Author

W. Neville Holmes, Feb 10 2005

Keywords

Comments

Primes in this sequence: 7, 17, 71, 449, 881, 2591, ... - Zak Seidov, Jan 14 2013

Examples

			a(1) = (3*5 - 1)/2 = 7.
a(2) = (5*7 - 1)/2 = 17.
a(3) = (7*11 - 1)/2 = 38.

Links

Crossrefs

Cf. A006094, A023515, A086870, A120876.

Programs

Formula

a(n) = (prime(n + 1)*prime(n + 2) - 1)/2.
a(n) ~ 0.5 n^2/log^2 n. - Charles R Greathouse IV, Jan 14 2013
a(n) = A023515(n+2)/2. - Jason Kimberley, Oct 23 2015

A120875 Product of twin primes minus 1.

Original entry on oeis.org

14, 34, 142, 322, 898, 1762, 3598, 5182, 10402, 11662, 19042, 22498, 32398, 36862, 39202, 51982, 57598, 72898, 79522, 97342, 121102, 176398, 186622, 213442, 272482, 324898, 359998, 381922, 412162, 435598, 656098, 675682, 685582, 736162
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2006

Keywords

Comments

This sequence is a subsequence of A023515.

Links

Crossrefs

Cf. A002822, A023515, A037074, A075369, A120876.

Programs

  • Mathematica
    Times[#, # + 2] - 1 & /@ Select[Prime@ Range@ 150, PrimeQ[# + 2] &] (* Michael De Vlieger, Oct 23 2015 *)
  • PARI
    for(n=1, 200, if(prime(n+1)-prime(n)==2, print1(prime(n)*prime(n+1)-1", "))) \\ Altug Alkan, Oct 23 2015

Formula

a(n) = A037074(n)-1 = (A014574(n))^2 -2 = A075369(n)-2.
a(n) = 2*A120876(n). - Jason Kimberley, Oct 23 2015
a(n) = 36*A002822(n-1)^2-2, for n>1. - Jason Kimberley, Oct 23 2015
a(n) = A023515(A107770(n)). - Jason Kimberley, Oct 23 2015

A023520 Exponent of 2 in prime factorization of prime(n)*prime(n-1) - 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 6, 2, 3
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

a(n) = 1 if and only if A080378(n-1) = 2. - Robert Israel, Feb 07 2018

Links

Crossrefs

Cf. A007814, A023515, A080378.

Programs

  • Maple
    seq(padic:-ordp(ithprime(n)*ithprime(n-1)-1,2),n=2..200); # Robert Israel, Feb 07 2018
  • PARI
    a(n) = valuation(prime(n)*prime(n-1) - 1, 2); \\ Michel Marcus, Sep 30 2013
  • Python
    from sympy import prime
def A023520(n): return (~(m:=prime(n)*prime(n-1)-1)& m-1).bit_length()    # Chai Wah Wu, Jul 07 2022

Formula

a(n) = A007814(A023515(n)). - Michel Marcus, Sep 30 2013

Extensions

Offset set to 2 and a(2) corrected by Michel Marcus, Sep 30 2013

A023516 Number of distinct prime divisors of prime(n)*prime(n-1) - 1.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 3, 4, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 2, 4, 2, 3, 2, 4, 3, 3, 4, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 3, 3, 4, 4, 4, 4, 4, 3, 4, 2, 3, 4, 2, 4, 4, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 2, 4
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

This is taking prime(0)=1 (see first comment in A023515). - Vincenzo Librandi, Apr 27 2019

Links

Crossrefs

Cf. A023515, A023524, A023568, A023575, A023582, A023589.

Programs

  • Magma
    [#PrimeDivisors(NthPrime(n)*(NthPrime(n-1))-1): n in [1..100]]; // Vincenzo Librandi, Apr 27 2019
  • Maple
    0,seq(nops(numtheory:-factorset(ithprime(n)*ithprime(n-1)-1)),n=2..120); # Muniru A Asiru, Apr 29 2019
  • Mathematica
    Prepend[Table[PrimeNu[Prime[n] Prime[n-1] - 1], {n, 2, 80}],0] (* Vincenzo Librandi, Apr 27 2019 *)
  • PARI
    a(n) = if (n==1, 0, omega(prime(n)*prime(n-1) - 1)); \\ Michel Marcus, Apr 30 2019

Formula

a(n) = A001221(A023515(n)).

A023517 Greatest prime divisor of prime(n)*prime(n-1) - 1.

Original entry on oeis.org

5, 7, 17, 19, 71, 11, 23, 109, 37, 449, 191, 379, 881, 101, 83, 521, 257, 227, 41, 2591, 31, 149, 1231, 83, 79, 743, 29, 17, 3079, 41, 4159, 997, 9521, 109, 1607, 439, 853, 1361, 107, 397, 167, 3457, 2633, 1901, 1153, 3499, 1307, 2531, 79, 13339, 9281, 929
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A006530, A023515, A023525.

Programs

  • PARI
    gpf(x) = {my (f=factor(x)); f[#f[,2],1]};
a023517(pmax) = {my(pp=2); forprime (p=3, pmax, print1(gpf(pp*p-1),", "); pp=p)};
a023517(300) \\ Hugo Pfoertner, Oct 09 2022

Extensions

a(1)=2 removed by Sean A. Irvine, Jun 05 2019

A023521 Sum of distinct prime divisors of prime(n)*prime(n-1) - 1.

Original entry on oeis.org

5, 9, 19, 21, 73, 18, 32, 111, 42, 451, 196, 381, 883, 108, 93, 526, 266, 232, 72, 2593, 36, 162, 1236, 98, 112, 752, 55, 26, 3081, 55, 4161, 1002, 9523, 135, 1616, 444, 863, 1368, 117, 415, 266, 3464, 2642, 1908, 1172, 3504, 1312, 2538, 135
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

If n-1 is in A103746, then a(n) = (prime(n)*prime(n-1)+3)/2. - Robert Israel, Jun 03 2020

Links

Crossrefs

Cf. A103746.

Programs

  • Maple
    p:= 2: R:= NULL:
for n from 2 to 50 do
  q:= p; p:= nextprime(p);
  R:= R, convert(numtheory:-factorset(p*q-1),`+`)
od:
R; # Robert Israel, Jun 03 2020
  • PARI
    sopf(n) = my(fac=factor(n)); sum(i=1, matsize(fac)[1], fac[i, 1]) ;
a(n) = sopf(prime(n)*prime(n-1) - 1); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A008472(A023515(n)). - Michel Marcus, Sep 30 2013

Extensions

Offset set to 2 and a(1) removed by Michel Marcus, Sep 30 2013

A023522 Sum of exponents in prime-power factorization of p(n)*p(n-1) - 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 3, 4, 2, 3, 3, 2, 4, 4, 3, 3, 4, 4, 2, 4, 4, 3, 5, 4, 3, 5, 5, 3, 5, 3, 4, 2, 4, 3, 5, 4, 4, 6, 4, 3, 3, 3, 4, 3, 4, 5, 4, 4, 3, 3, 3, 4, 4, 5, 4, 4, 3, 5, 2, 3, 5, 3, 4, 6, 3, 4, 3, 3, 4, 3, 5, 8, 3, 4, 6, 8, 3, 6
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Mathematica
    PrimeOmega[Times@@#-1]&/@Partition[Prime[Range[80]],2,1] (* Harvey P. Dale, Nov 22 2014 *)
  • PARI
    a(n) = bigomega(prime(n)*prime(n-1) - 1); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A001222(A023515(n)). - Michel Marcus, Sep 30 2013

Extensions

Offset set to 2 and a(1) removed by Michel Marcus, Sep 30 2013

A023519 Least odd prime divisor of prime(n)*prime(n-1) - 1, or 1 if prime(n)*prime(n-1) - 1 is a power of 2.

Original entry on oeis.org

1, 5, 7, 17, 19, 71, 5, 7, 109, 3, 449, 3, 379, 881, 5, 3, 3, 7, 3, 29, 2591, 3, 11, 3, 13, 31, 7, 5, 7, 3079, 5, 4159, 3, 9521, 5, 7, 3, 3, 5, 3, 3, 97, 5, 7, 5, 17, 3, 3, 5, 7, 13339, 3, 31, 5, 3, 3, 3, 7, 3, 11, 39761, 11, 5, 23869, 7, 5, 23, 3, 59, 151, 19, 3, 43, 3, 3, 11, 3, 19, 39799, 13
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Assumes the not generally accepted convention prime(0) = 1. - Michel Marcus, Jun 06 2019

Crossrefs

Cf. A023515, A078701.

Programs

  • PARI
    p(n) = if (n==0, 1, prime(n));
f(n) = my(v = select(x->((x%2)==1), factor(n)[, 1])); if (#v, vecmin(v), 1);
a(n) = f(p(n)*p(n-1) - 1); \\ Michel Marcus, Jun 06 2019

Formula

a(n) = A078701(A023515(n)). - Michel Marcus, Jun 06 2019

Extensions

Terms corrected by Sean A. Irvine, Jun 05 2019
Showing 1-8 of 8 results.

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