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 1-5 of 5 results.

A072055 a(n) = 2*prime(n)+1.

Original entry on oeis.org

5, 7, 11, 15, 23, 27, 35, 39, 47, 59, 63, 75, 83, 87, 95, 107, 119, 123, 135, 143, 147, 159, 167, 179, 195, 203, 207, 215, 219, 227, 255, 263, 275, 279, 299, 303, 315, 327, 335, 347, 359, 363, 383, 387, 395, 399, 423, 447, 455, 459, 467, 479
Offset: 1

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Author

Reinhard Zumkeller, Jun 11 2002

Keywords

Links

Crossrefs

Cf. A000040, A005385, A023589, A023590, A023591, A023592, A023593, A072059, A023595, A072060, A072056, A072057, A072058, A072192, A165286, A278230.
One less than A089241. After the initial term equal to A166496.
Row 4 of A286625, column 4 of A286623.

Programs

Formula

a(n) = A089241(n)-1.

A029933 Numerator of n * Product_{d|n} (1 + 1/d).

Original entry on oeis.org

2, 6, 8, 15, 12, 28, 16, 135, 80, 198, 24, 455, 28, 360, 256, 2295, 36, 2660, 40, 2079, 1408, 828, 48, 11375, 312, 1134, 2240, 6525, 60, 76384, 64, 75735, 1088, 1890, 3456, 1599325, 76, 2340, 4480, 767151, 84
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(6) = 28. The divisors of 6 are 1, 2, 3, 6. We multiply 2 * (1 + 1/2)(1 + 1/3)(1 + 1/6) to get 14/3, which multiplied by 6 is 28.
a(7) = 16, since 2 * 7 + 2 = 16.
a(8) = 135. The divisors of 8 are 1, 2, 4, 8. We multiply 2 * (1 + 1/2)(1 + 1/4)(1 + 1/8) to get 135/32, which multiplied by 8 is 135/4, the numerator of which is 135.

Links

Crossrefs

Cf. A029934 (denominators).

Programs

  • Maple
    with(numtheory): A029933 := proc(n) local i,j; j := n; for i in divisors(n) do j := j*(1+1/i); od; end;
  • Mathematica
    Numerator[Table[n * Times@@((1 + 1/#)&/@Divisors[n]), {n, 50}]] (* Harvey P. Dale, Dec 14 2014 *)
  • PARI
    a(n) = my(d=divisors(n)); numerator(n*prod(i=1, #d, (d[i]+1)/d[i])); \\ Michel Marcus, Mar 06 2017

Formula

Given p prime, a(p) = 2p + 2, since p(1 + 1/1)(1 + 1/p) = 2p + 2, see A089241. - Alonso del Arte, Mar 02 2017

A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

Original entry on oeis.org

1, 5, 19, 41, 109, 155, 271, 341, 505, 811, 929, 1331, 1639, 1805, 2161, 2755, 3421, 3659, 4421, 4969, 5255, 6161, 6805, 7831, 9311, 10099, 10505, 11341, 11771, 12655, 16001, 17029, 18631, 19181, 22051, 22649, 24491, 26405, 27721, 29755, 31861, 32579, 36289
Offset: 1

Views

Author

Kritsada Moomuang, Jan 28 2019

Keywords

Comments

Terms are divisible by 5 iff p is of the form 10*m + 3 (A030431).

Examples

			a(3) = 19 because 5^2 - 5 - 1 = 19.

Links

Crossrefs

Supersequence of A091568.
Subsequence of A028387 or A165900.
Second column of A378979.
Cf. A000040, A001248, A030431, A033879, A036689, A036690, A060800, A065479, A065488, A072055, A089241.
A039914 is an essentially identical sequence.

Programs

  • Maple
    map(p -> p^2-p-1, [seq(ithprime(i),i=1..100)]); # Robert Israel, Mar 11 2019
  • Mathematica
    Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)
  • PARI
    a(n) = {p=prime(n);p^2-p-1;} \\ Jinyuan Wang, Feb 02 2019

Formula

a(n) = A036689(n) - 1.
a(n) = A036690(n) - A072055(n).
a(n) = A060800(n) - A089241(n).
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065488.
Product_{n>=2} (1 - 1/a(n)) = A065479. (End)
a(n) = A033879(A001248(n)). [Deficiency of squares of primes] - Antti Karttunen, Dec 13 2024

A367335 Long leg of the only primitive Pythagorean triple whose inradius is the n-th odd prime and whose short leg is an even number.

Original entry on oeis.org

15, 35, 63, 143, 195, 323, 399, 575, 899, 1023, 1443, 1763, 1935, 2303, 2915, 3599, 3843, 4623, 5183, 5475, 6399, 7055, 8099, 9603, 10403, 10815, 11663, 12099, 12995, 16383, 17423, 19043, 19599, 22499, 23103, 24963, 26895, 28223, 30275, 32399, 33123, 36863
Offset: 1

Views

Author

Miguel-Ángel Pérez García-Ortega, Nov 23 2023

Keywords

Comments

See Ejercicio 2.7. of the reference file.

Examples

			Triangles begin
    8,  15,  17;
   12,  35,  37;
   16,  63,  65;
   24, 143, 145;
   28, 195, 197;
   ...
This sequence is the middle column.

Links

Crossrefs

Cf. A065091, A089241 (short leg).

Formula

a(n) = p^2 + 2*p where p = A065091(n).

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.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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