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 1-10 of 13 results. Next

A141130 a(n) = prime(prime(prime(prime(prime(A028815(n) - 1) - 1) - 1) - 1) - 1) - 1.

Original entry on oeis.org

1, 2, 150, 4210, 12496, 43206, 83046, 161092, 202966, 305068, 498936, 633160, 906426, 1125418, 1248412, 1460566, 1808478, 2264752, 2339136, 3026112, 3331266, 3501748, 4015168, 4529520, 5049852, 5806336, 6448536, 6792726, 7214610
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jul 31 2008

Keywords

Links

Crossrefs

Cf. A000040, A028815, A141132, A141133, A141136, A141138.

Programs

  • Magma
    p:=NthPrime;
A141130:= func< n | n eq 0 select 1 else p(p(p(p(p(p(n))-1)-1)-1)-1)-1 >;
[A141130(n): n in [0..50]]; // G. C. Greubel, Aug 05 2024
  • Mathematica
    A141130[n_]:= With[{p=Prime}, If[n==0, 1, p[p[p[p[p[p[n]]-1]-1]-1]-1]-1 ]];
Table[A141130[n], {n,0,60}] (* G. C. Greubel, Aug 05 2024 *)
  • SageMath
    p=nth_prime
def A141130(n): return 1 if n==0 else p(p(p(p(p(p(n))-1)-1)-1)-1)-1
[A141130(n) for n in range(51)] # G. C. Greubel, Aug 05 2024

Formula

a(n) = prime(prime(prime(prime(prime(prime(n)) - 1) - 1) - 1) - 1) - 1, with a(0) = 1. - G. C. Greubel, Aug 05 2024

Extensions

More terms from D. S. McNeil, Mar 21 2009
Offset changed by G. C. Greubel, Aug 05 2024

A141132 a(n) = prime(prime(prime(prime(A028815(n) - 1) - 1) - 1) - 1) - 1.

Original entry on oeis.org

1, 2, 36, 576, 1492, 4512, 8110, 14778, 18222, 26416, 41452, 51576, 71740, 87552, 96352, 111372, 135660, 167106, 172152, 218550, 238990, 250252, 284148, 317856, 351706, 400408, 441478, 463398, 490246, 505780, 527280, 643990, 679036, 718182
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jul 31 2008

Keywords

Links

Crossrefs

Cf. A000040, A028815, A141130, A141133, A141136, A141138.

Programs

  • Magma
    p:=NthPrime;
A141132:= func< n | n eq 0 select 1 else p(p(p(p(p(n))-1)-1)-1)-1 >;
[A141132(n): n in [0..50]]; // G. C. Greubel, Aug 05 2024
  • Mathematica
    A141132[n_]:= With[{p=Prime}, If[n==0, 1, p[p[p[p[p[n]]-1]-1]-1]-1 ]];
Table[A141132[n], {n, 0, 60}] (* G. C. Greubel, Aug 05 2024 *)
  • SageMath
    p=nth_prime
def A141132(n): return 1 if n==0 else p(p(p(p(p(n))-1)-1)-1)-1
[A141132(n) for n in range(51)] # G. C. Greubel, Aug 05 2024

Formula

a(n) = prime(prime(prime(prime(prime(n)) - 1) - 1) - 1) - 1, with a(0) = 1. - G. C. Greubel, Aug 05 2024

Extensions

Extended by D. S. McNeil, Mar 21 2009
Offset changed by G. C. Greubel, Aug 05 2024

A141133 a(n) = prime(prime(prime(A028815(n) - 1) - 1) - 1) - 1.

Original entry on oeis.org

1, 2, 12, 106, 238, 612, 1020, 1732, 2088, 2902, 4336, 5278, 7102, 8500, 9282, 10566, 12640, 15268, 15678, 19488, 21142, 22062, 24766, 27430, 30108, 33892, 37056, 38706, 40786, 41980, 43626, 52360, 54982, 57900, 60496, 66808, 68398, 72670
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jul 31 2008

Keywords

Links

Crossrefs

Cf. A000040, A028815, A141130, A141132, A141136, A141138.

Programs

  • Magma
    p:=NthPrime;
A141133:= func< n | n eq 0 select 1 else p(p(p(p(n))-1)-1) -1 >;
[A141133(n): n in [0..50]]; // G. C. Greubel, Aug 05 2024
  • Mathematica
    A141133[n_]:= With[{p=Prime}, If[n==0, 1, p[p[p[p[n]]-1]-1]-1 ]];
Table[A141133[n], {n,0,60}] (* G. C. Greubel, Aug 05 2024 *)
  • SageMath
    p=nth_prime
def A141133(n): return 1 if n==0 else p(p(p(p(n))-1)-1)-1
[A141133(n) for n in range(51)] # G. C. Greubel, Aug 05 2024

Formula

a(n) = prime(prime(prime(prime(n)) - 1) - 1) - 1, with a(0) = 1. - G. C. Greubel, Aug 05 2024

Extensions

Corrected and extended by D. S. McNeil, Mar 21 2009
Offset changed by G. C. Greubel, Aug 05 2024

A141136 a(n) = prime(prime(A028815(n) - 1) - 1) - 1.

Original entry on oeis.org

1, 2, 6, 28, 52, 112, 172, 270, 316, 420, 592, 700, 910, 1060, 1150, 1290, 1510, 1782, 1830, 2212, 2376, 2472, 2740, 2998, 3256, 3630, 3930, 4078, 4270, 4390, 4546, 5350, 5590, 5866, 6100, 6658, 6802, 7186, 7602, 7828, 8218, 8520, 8712, 9310, 9438, 9732
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jul 31 2008

Keywords

Links

Crossrefs

Cf. A000040, A028815, A141130, A141132, A141133, A141138.

Programs

  • Magma
    p:=NthPrime;
A141136:= func< n | n eq 0 select 1 else p(p(p(n))-1) -1 >;
[A141136(n): n in [0..50]]; // G. C. Greubel, Aug 05 2024
  • Mathematica
    A141136[n_]:= With[{p=Prime}, If[n==0, 1, p[p[p[n]]-1]-1]];
Table[A141136[n], {n,0,60}] (* G. C. Greubel, Aug 05 2024 *)
  • SageMath
    p=nth_prime
def A141136(n): return 1 if n==0 else p(p(p(n))-1)-1
[A141136(n) for n in range(51)] # G. C. Greubel, Aug 05 2024

Formula

a(n) = A000040(A000040(A000040(n)) - 1) - 1, with a(0) = 1. - G. C. Greubel, Aug 05 2024

Extensions

Corrected and extended by D. S. McNeil, Mar 21 2009
Offset changed by G. C. Greubel, Aug 05 2024

A141138 a(n) = prime(A028815(n) - 1) - 1.

Original entry on oeis.org

1, 2, 4, 10, 16, 30, 40, 58, 66, 82, 108, 126, 156, 178, 190, 210, 240, 276, 282, 330, 352, 366, 400, 430, 460, 508, 546, 562, 586, 598, 616, 708, 738, 772, 796, 858, 876, 918, 966, 990, 1030, 1062, 1086, 1152, 1170, 1200, 1216, 1296, 1408, 1432, 1446, 1470
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jul 31 2008

Keywords

Links

Crossrefs

Cf. A028815, A141130, A141132, A141133, A141136.

Programs

Formula

a(n) = A000040(A000040(n)) - 1, with a(0) = 1. - G. C. Greubel, Aug 05 2024

Extensions

a(14), a(28) corrected by D. S. McNeil, Mar 21 2009
Offset changed by G. C. Greubel, Aug 05 2024

A289055 Triangle read by rows: T(n,k) = (k+1)*A028815(n) for 0 <= k <= n.

Original entry on oeis.org

2, 3, 6, 4, 8, 12, 6, 12, 18, 24, 8, 16, 24, 32, 40, 12, 24, 36, 48, 60, 72, 14, 28, 42, 56, 70, 84, 98, 18, 36, 54, 72, 90, 108, 126, 144, 20, 40, 60, 80, 100, 120, 140, 160, 180, 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330
Offset: 0

Views

Author

Vincenzo Librandi, Sep 02 2017

Keywords

Examples

			Triangle begins:
   2;
   3,   6;
   4,   8,  12;
   6,  12,  18,  24;
   8,  16,  24,  32,  40;
  12,  24,  36,  48,  60,  72;
  14,  28,  42,  56,  70,  84,  98;
  18,  36,  54,  72,  90, 108, 126, 144;
  20,  40,  60,  80, 100, 120, 140, 160, 180;
  ...

Links

Crossrefs

Cf. A289108.
Columns k: A028815 (k=0), A089241 (k=1), A247159 (k=2), A273801 (k=3).

Programs

  • Magma
    /* As triangle (here NthPrime(0)=1) */ [[(k+1)*(NthPrime(n)+1): k in [0..n]]: n in [0.. 15]];
  • Mathematica
    Join[{2}, t[n_, k_] := (k + 1) (Prime[n] + 1); Table[t[n, k], {n, 10}, {k, 0, n}] //Flatten]
  • SageMath
    def A289055(n,k): return 2 if n==0 else (k+1)*(nth_prime(n) +1)
flatten([[A289055(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 05 2024

Formula

a(n) = A289108(n) + 1.

A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

Original entry on oeis.org

1, 3, 12, 72, 576, 6912, 96768, 1741824, 34836480, 836075520, 25082265600, 802632499200, 30500034969600, 1281001468723200, 56364064623820800, 2705475101943398400, 146095655504943513600, 8765739330296610816000, 543475838478389870592000, 36956357016530511200256000
Offset: 0

Views

Author

Labos Elemer, May 15 2000

Keywords

Links

Crossrefs

Cf. A000203, A002110, A005867, A008864, A028815, A051674, A054641, A070826, A073004, A074107.

Programs

  • Magma
    [1/2*&*[(1+NthPrime(k)): k in [0..n-1]]: n in [1..19]]; // Vincenzo Librandi, May 08 2017
  • Maple
    a:= n-> mul(1+ithprime(j), j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Aug 24 2008
  • Mathematica
    Table[Product[1 + Prime[i], {i,n-1}], {n,100}] (* Geoffrey Critzer, Dec 01 2014 *)
  • PARI
    a(n)=prod(i=1,n,prime(i)+1) \\ Charles R Greathouse IV, Feb 13 2013
  • SageMath
    def A054640(n): return product(nth_prime(j)+1 for j in range(1,n+1))
[A054640(n) for n in range(41)] # G. C. Greubel, Aug 05 2024

Formula

a(n+1) = a(n)*(prime(n) + 1) = a(n)*A028815(n) (quotient=n-th prime+1 starting with 2).
a(n) ~ (6/Pi^2) * exp(gamma) * A002110(n) * log(prime(n)) + O(A002110(n)) (Jakimczuk, 2017). - Amiram Eldar, Feb 17 2021
a(n) = a(n-1) * A008864(n). - Flávio V. Fernandes, Mar 20 2021
a(n) = A002110(n) + A074107(n), a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n). - Antti Karttunen, Nov 19 2024

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 01 2021

A175216 The first nonprimes after the primes.

Original entry on oeis.org

4, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272
Offset: 1

Views

Author

Jaroslav Krizek, Mar 06 2010

Keywords

Comments

Essentially the same as A135731, A055670, A028815 and A008864. [R. J. Mathar, Mar 13 2010]

Links

Crossrefs

Cf. A000040, A008864, A028815, A055670, A135731.

Programs

  • Magma
    [n eq 1 select 4 else NthPrime(n) +1: n in [1..100]]; // G. C. Greubel, Aug 06 2024
  • Mathematica
    Table[Prime[n] +1 +Boole[n==1], {n,100}] (* G. C. Greubel, Aug 06 2024 *)
  • SageMath
    def A175216(n): return nth_prime(n) +1 +int(n==1)
[A175216(n) for n in range(1,101)] # G. C. Greubel, Aug 06 2024

Formula

a(1) = 4, for n >= 2, a(n) = A008864(n) = A000040(n) + 1.

A057024 Largest odd factor of (n-th prime+1); k when n-th prime is written as k*2^m-1 [with k odd].

Original entry on oeis.org

3, 1, 3, 1, 3, 7, 9, 5, 3, 15, 1, 19, 21, 11, 3, 27, 15, 31, 17, 9, 37, 5, 21, 45, 49, 51, 13, 27, 55, 57, 1, 33, 69, 35, 75, 19, 79, 41, 21, 87, 45, 91, 3, 97, 99, 25, 53, 7, 57, 115, 117, 15, 121, 63, 129, 33, 135, 17, 139, 141, 71, 147, 77, 39, 157, 159, 83, 169, 87
Offset: 1

Views

Author

Henry Bottomley, Jul 24 2000

Keywords

Comments

a(n) = 1 if and only if prime(n) is a Mersenne prime. - Ely Golden, Feb 06 2017

Examples

			a(5)=3 because 5th prime is 11 and 11=3*2^2-1.

Links

Crossrefs

Cf. A000040, A000265, A007814, A023512, A028815, A057023.

Programs

  • Magma
    A057024:= func< n | (NthPrime(n)+1)/2^Valuation(NthPrime(n)+1, 2) >;
[A057024(n): n in [1..100]]; // G. C. Greubel, Aug 06 2024
  • Mathematica
    Table[Max[Select[Divisors[Prime[n]+1],OddQ]],{n,100}] (* Daniel Jolly, Nov 15 2014 *)
  • PARI
    a(n) = (prime(n)+1)/2^valuation(prime(n)+1, 2); \\ Michel Marcus, Feb 05 2017
  • Sage
    def a(n):
    x=nth_prime(n)+1
    return x/2**((int(x)&int(-x)).bit_length()-1)
index=1
while(index<=10000):
    print(str(index)+" "+str(a(index)))
    index+=1
# Ely Golden, Feb 06 2017

Formula

a(n) = A000265(A000040(n) + 1) = A000265(A028815(n)).
a(n) = (A000040(n) + 1)/A007814(A000040(n) + 1).
a(n) = A028815(n)/A023512(n).

A054641 GCD of divisor-sum of primorials and primorials itself: a(n) = gcd(A002110(n), A000203(A002110(n))).

Original entry on oeis.org

1, 6, 6, 6, 6, 42, 42, 210, 210, 210, 210, 3990, 3990, 43890, 43890, 43890, 43890, 1360590, 23130030, 23130030, 855811110, 855811110, 855811110, 855811110, 855811110, 855811110, 11125544430, 11125544430, 11125544430, 11125544430
Offset: 1

Views

Author

Labos Elemer, May 15 2000

Keywords

Comments

Values are repeated for several arguments: observed from 1 to 8 times below n=100. Sites of jump seems not so regular.
From a sufficiently high value of n, A002110(n) divides the terms. E.g., from n=27, A002110(8) divides the values of this sequences.

Links

Crossrefs

Cf. A002110, A000203, A028815, A054640, A054642.

Programs

Showing 1-10 of 13 results. Next

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