A072055 -id:A072055 - cp's OEIS frontend

A367573 Long legs of the only primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

12, 24, 60, 112, 264, 364, 612, 760, 1104, 1740, 1984, 2812, 3444, 3784, 4512, 5724, 7080, 7564, 9112, 10224, 10804, 12640, 13944, 16020, 19012, 20604, 21424, 23112, 23980, 25764, 32512, 34584, 37812, 38920, 44700, 45904, 49612, 53464, 56112, 60204, 64440

Offset: 1

Miguel-Ángel Pérez García-Ortega, Nov 23 2023

See Ejercicio 2.7. of García-Ortega.

			Triangles begin
   5,  12,  13;
   7,  24,  25;
  11,  60,  61;
  15, 112, 113;
  23, 264, 265;
  ...
Row n = (a, b, c) = (2*p + 1, 2*p^2 + 2*p, 2*p^2 + 2*p + 1), where p is the n-th prime number.
This sequence is the middle column.

Miguel-Ángel Pérez García-Ortega, Capítulo 2. Inradio, El Libro de las Ternas Pitagóricas.

Cf. A072055 (short leg).

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

A072056 Number of divisors of 2*prime(n)+1.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 4, 4, 2, 2, 6, 6, 2, 4, 4, 2, 4, 4, 8, 4, 6, 4, 2, 2, 8, 4, 6, 4, 4, 2, 8, 2, 6, 6, 4, 4, 12, 4, 4, 2, 2, 6, 2, 6, 4, 8, 6, 4, 8, 8, 2, 2, 8, 2, 4, 4, 6, 4, 8, 2, 10, 2, 8, 4, 8, 4, 8, 12, 4, 4, 4, 2, 12, 6, 8, 4, 4, 8, 4, 12, 2, 4, 2, 6, 4, 2, 4, 8, 4, 6, 8, 4, 12

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

			Divisors of A072055(8) = 2*A000040(8)+1 = 2*19+1=39: {1,3,13,39} with size 4, therefore a(8) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A072055, A072057.

Table[DivisorSigma[0,2Prime[n]+1],{n,100}] (* Harvey P. Dale, Apr 12 2021 *)

a(n) = numdiv(2*prime(n)+1); \\ Amiram Eldar, Apr 26 2024

lista(pmax) = forprime(p = 2, pmax, print1(numdiv(2*p+1), ", ")); \\ Amiram Eldar, Apr 26 2024

a(n) = A000005(A072055(n)).

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

Kritsada Moomuang, Jan 28 2019

nonn

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

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

Robert Israel, Table of n, a(n) for n = 1..10000

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.

map(p -> p^2-p-1, [seq(ithprime(i),i=1..100)]); # Robert Israel, Mar 11 2019

Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)

a(n) = {p=prime(n);p^2-p-1;} \\ Jinyuan Wang, Feb 02 2019

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

A023591 Greatest exponent in prime-power factorization of 2*prime(n)+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1

Offset: 1

Clark Kimberling

nonn

Cf. A051903, A072055.

A023591 := proc(n)
    A051903(2*ithprime(n)+1) ;
end proc: # R. J. Mathar, Jul 08 2015

a[n_] := Max[FactorInteger[2*Prime[n] + 1][[;;, 2]]]; Array[a, 100] (* Amiram Eldar, Sep 09 2024 *)

a(n) = vecmax(factor(2*prime(n)+1)[,2]); \\ Michel Marcus, Apr 20 2021

a(n) = A051903(A072055(n)). - Amiram Eldar, Sep 09 2024

A023593 Exponent of least prime factor of 2*prime(n)+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Clark Kimberling

nonn

			a(6) = 3 because 2*prime(6)+1 = 27 = 3^3.

Cf. A067029, A072055.

A023593 := proc(n)
    A067029(2*ithprime(n)+1) ;
end proc: # R. J. Mathar, Jul 08 2015

Table[FactorInteger[2*Prime[n] + 1][[1]][[2]], {n, 100}] (* Clark Kimberling, Oct 01 2013 *)

Corrected by Clark Kimberling, Oct 01 2013

A072058 Squarefree kernel of 2*prime(n)+1.

Original entry on oeis.org

5, 7, 11, 15, 23, 3, 35, 39, 47, 59, 21, 15, 83, 87, 95, 107, 119, 123, 15, 143, 21, 159, 167, 179, 195, 203, 69, 215, 219, 227, 255, 263, 55, 93, 299, 303, 105, 327, 335, 347, 359, 33, 383, 129, 395, 399, 141, 447, 455, 51, 467, 479, 483, 503

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

			n = 12: 2*A000040(12)+1 = 2*37+1 = 75 and as A007947(75) = A007947(7*5*5) = 15: a(12) = 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A007947, A072055.

rad[n_] := Times @@ FactorInteger[n][[;;,1]]; rad /@ (2 * Select[Range[250], PrimeQ] + 1) (* Amiram Eldar, Sep 07 2020 *)

a(n) = A007947(A072055(n)).

A072059 Smallest prime p such that 2*p+1 has n distinct prime factors.

Original entry on oeis.org

2, 7, 97, 577, 7507, 217717, 5232727, 75172597, 1617423307, 59844662377, 2750790860317, 109455887488447, 4621264673452927, 218071376383127767, 10914293640945722527, 662082573402158125717, 41249727342503299116997

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

Note that for each n=1,...,8, the product of the smallest n-1 distinct prime factors of 2*a(n)+1 is p(n)#/2, where p(n)# is the primorial (A002110) of the n-th prime - and the n-th distinct prime factor >= p(n+1). - Rick L. Shepherd, Jul 06 2002

			a(4)=577=A000040(106): 2*577+1 = 1155 = 11*7*5*3, 4 distinct factors.

Cf. A001221, A023589, A072055, A072060.

for (n=1,8, p=1; until(isprime(p) && omega(2*p+1)==n, p++); print1(p,","))

More terms from Rick L. Shepherd, Jul 06 2002

More terms from Don Reble, Apr 15 2003

A289108 Triangle read by rows: T(n,k) = (k + 1)*prime(n) + k for n > 0, 0 <= k <= n, and with T(0,0) = 1.

Original entry on oeis.org

1, 2, 5, 3, 7, 11, 5, 11, 17, 23, 7, 15, 23, 31, 39, 11, 23, 35, 47, 59, 71, 13, 27, 41, 55, 69, 83, 97, 17, 35, 53, 71, 89, 107, 125, 143, 19, 39, 59, 79, 99, 119, 139, 159, 179, 23, 47, 71, 95, 119, 143, 167, 191, 215, 239, 29, 59, 89, 119, 149, 179, 209, 239, 269, 299, 329

Offset: 0

Vincenzo Librandi, Sep 02 2017

			Triangle begins:
   1;
   2,   5;
   3,   7,  11;
   5,  11,  17,  23;
   7,  15,  23,  31,  39;
  11,  23,  35,  47,  59,  71;
  13,  27,  41,  55,  69,  83,  97;
  17,  35,  53,  71,  89, 107, 125, 143;
  19,  39,  59,  79,  99, 119, 139, 159, 179;
  23,  47,  71,  95, 119, 143, 167, 191, 215, 239;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000040, A008578, A072055.

/* As triangle (here NthPrime(0)=1) */ [[(k+1)*NthPrime(n)+k: k in [0..n]]: n in [0.. 15]];

Join[{1}, T[n_,k_] := (k + 1) Prime[n] + k; Table[T[n, k], {n, 10}, {k, 0, n}]//Flatten]

def A289108(n,k): return 1 if n==0 else (k+1)*nth_prime(n) +k
flatten([[A289108(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 04 2024

Definition corrected by Bruno Berselli, Sep 06 2017

A166496 Prime plus the next composite.

Original entry on oeis.org

6, 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, 483, 503, 515, 527

Offset: 1

Zak Seidov, Oct 15 2009

nonn

Differs from A072055 in the first term.

			a(1)=2+4=6,
a(2)=3+4=7,
a(3)=5+6=11.

Cf. A072055.

Mathematica
```
Prepend[2*Prime[Range[2,100]]+1,6]
```

a(n)=prime(n)+smallest composite > prime(n).

A363700 a(n) = phi(2*prime(n)+1).

Original entry on oeis.org

4, 6, 10, 8, 22, 18, 24, 24, 46, 58, 36, 40, 82, 56, 72, 106, 96, 80, 72, 120, 84, 104, 166, 178, 96, 168, 132, 168, 144, 226, 128, 262, 200, 180, 264, 200, 144, 216, 264, 346, 358, 220, 382, 252, 312, 216, 276, 296, 288, 288, 466, 478, 264, 502, 408, 480, 420, 360, 288, 562

Offset: 1

Alain Rocchelli, Jun 16 2023

nonn

2*prime(n)+1 is prime iff a(n) = 2*prime(n).

Cf. A000010, A000040, A008331, A037225, A072055.

a[n_] := EulerPhi[2*Prime[n] + 1]; Array[a, 100] (* Amiram Eldar, Jun 16 2023 *)

PARI
```
a(n)=eulerphi(2*prime(n)+1)
```

a(n) = A000010(A072055(n)).

a(n) = A037225(A000040(n)).

cp's OEIS Frontend

A367573 Long legs of the only primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

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A072056 Number of divisors of 2*prime(n)+1.

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

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A023591 Greatest exponent in prime-power factorization of 2*prime(n)+1.

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A023593 Exponent of least prime factor of 2*prime(n)+1.

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Extensions

A072058 Squarefree kernel of 2*prime(n)+1.

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A072059 Smallest prime p such that 2*p+1 has n distinct prime factors.

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PARI

Extensions

A289108 Triangle read by rows: T(n,k) = (k + 1)*prime(n) + k for n > 0, 0 <= k <= n, and with T(0,0) = 1.

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