A084920 -id:A084920 - cp's OEIS frontend

A341655 a(n) is the number of divisors of prime(n)^2 - 1.

2, 4, 8, 10, 16, 16, 18, 24, 20, 32, 28, 24, 40, 32, 24, 32, 32, 32, 32, 60, 30, 48, 32, 60, 42, 48, 40, 32, 64, 48, 54, 64, 40, 64, 48, 60, 32, 40, 40, 32, 48, 96, 64, 32, 72, 90, 64, 56, 32, 64, 60, 96, 72, 96, 40, 40, 64, 96, 32, 80, 32, 48, 96, 80, 40, 32

Offset: 1

Jon E. Schoenfield, Feb 25 2021

nonn

a(n) >= A309906(2) = 32 for n > 21.

			        p =                factorization
   n  prime(n)  p^2 - 1      of p^2 - 1      a(n)
  --  --------  -------  ------------------  ----
   1      2         3    3                     2
   2      3         8    2^3                   4
   3      5        24    2^3 * 3               8
   4      7        48    2^4 * 3              10
   5     11       120    2^3 * 3 * 5          16
   6     13       168    2^3 * 3 * 7          16
   7     17       288    2^5 * 3^2            18
   8     19       360    2^3 * 3^2 * 5        24
   9     23       528    2^4 * 3 * 11         20
  10     29       840    2^3 * 3 * 5 * 7      32
  11     31       960    2^6 * 3 * 5          28
  12     37      1368    2^3 * 3^2 * 19       24
  13     41      1680    2^4 * 3 * 5 * 7      40
  14     43      1848    2^3 * 3 * 7 * 11     32
  15     47      2208    2^5 * 3 * 23         24
  16     53      2808    2^3 * 3^3 * 13       32
  17     59      3480    2^3 * 3 * 5 * 29     32
  18     61      3720    2^3 * 3 * 5 * 31     32
  19     67      4488    2^3 * 3 * 11 * 17    32
  20     71      5040    2^4 * 3^2 * 5 * 7    60
  21     73      5328    2^4 * 3^2 * 37       30
  22     79      6240    2^5 * 3 * 5 * 13     48
  23     83      6888    2^3 * 3 * 7 * 41     32
  24     89      7920    2^4 * 3^2 * 5 * 11   60

Cf. A000005, A000040, A084920, A309906.

Table[DivisorSigma[0,Prime[n]^2-1],{n,66}] (* Stefano Spezia, Feb 25 2021 *)

a(n) = numdiv(prime(n)^2-1); \\ Michel Marcus, Feb 25 2021

a(n) = A000005(A000040(n)^2 - 1) = A000005(A084920(n)).

A084922 a(n) = (prime(n)-1)*(prime(n)+1)/6.

Original entry on oeis.org

4, 8, 20, 28, 48, 60, 88, 140, 160, 228, 280, 308, 368, 468, 580, 620, 748, 840, 888, 1040, 1148, 1320, 1568, 1700, 1768, 1908, 1980, 2128, 2688, 2860, 3128, 3220, 3700, 3800, 4108, 4428, 4648, 4988, 5340, 5460, 6080, 6208, 6468, 6600, 7420

Offset: 3

Reinhard Zumkeller, Jun 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 3..840

Cf. A000040, A006093, A008864, A084920, A084921.

[(p^2-1)/6: p in PrimesInInterval(4, 250)]; // Vincenzo Librandi, Apr 11 2013

Select[Range[0, 7000], PrimeQ[Sqrt[6 # + 1]]&] (* Vincenzo Librandi, Apr 11 2013 *)
(Prime[Range[3,60]]^2 -1)/6 (* G. C. Greubel, May 02 2024 *)

a(n) = (prime(n)^2-1)/6; \\ Michel Marcus, Mar 22 2016

[(n^2-1)//6 for n in prime_range(4,301)] # G. C. Greubel, May 02 2024

a(n) = A084920(n)/6.

a(n) = A084921(n)/3.

A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

Original entry on oeis.org

4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380, 18240, 18624, 19404, 19800

Offset: 2

Richard R. Forberg, May 28 2013

Subsequence of A055523 restricted to the case of the other (shorter) leg of the triangle equal to a prime.
There is only one value of a(n) for each prime(n). (This is not necessarily true if the shorter leg is not a prime.)
Note that a(1) is nonexistent since there is no solution with prime = 2.
All terms are divisible by 4.
The values of m (the length of the hypotenuse) always equals a(n) + 1.
a(n) = (prime(n)^2 - 1)/2 for all n > 1.
This follows algebraically given m = a(n) + 1 (or vice versa).
The same two relationships apply when the shorter leg is an odd nonprime, but for only those results corresponding to the longest possible leg of the triangle.

			24^2 + 7^2 = 625 = 25^2 = (24 +1)^2  and a(4) = (prime(4)^2 -1)/2 = (49 - 1)/2 = 24.

Vincenzo Librandi, Table of n, a(n) for n = 2..4000

Subset of A055523.
Equals 4*A061066.
Equals A084921 excluding its first term.

[(NthPrime(n)^2 - 1)/2: n in [2..50]]; // G. C. Greubel, Dec 14 2018

A216244:=n->(ithprime(n)^2-1)/2: seq(A216244(n), n=2..100); # Wesley Ivan Hurt, May 03 2017

Table[(Prime[n]^2 - 1)/2, {n, 2, 100}] (* Vincenzo Librandi, Jun 15 2014 *)

vector(50, n, n++; (prime(n)^2 -1)/2) \\ G. C. Greubel, Dec 14 2018

[(nth_prime(n)^2 -1)/2 for n in (2..50)] # G. C. Greubel, Dec 14 2018

a(n) = (prime(n)^2 - 1)/2 for n >= 2.
a(n) = 4*A061066(n) = A084920(n)/2.
a(n) = A084921(n) for n > 1.
a(n) = (prime(n)-1)*(prime(n)+1)/2 = lcm(prime(n)+1, prime(n)-1) for n > 1 because one of prime(n)+1 or prime(n)-1 is even and the other is divisible by 4. Say prime(n)-1 is divisible by 4; then (prime(n)+1)/2 and (prime(n)-1)/4 must be coprime. - Frank M Jackson, Dec 11 2018
Product_{n>=2} (1 + 1/a(n)) = 3/2. - Amiram Eldar, Jun 03 2022

New name (taken from Formula entry) from Jon E. Schoenfield, Jul 11 2021

A166010 a(n) = prime(n)^2-4.

Original entry on oeis.org

0, 5, 21, 45, 117, 165, 285, 357, 525, 837, 957, 1365, 1677, 1845, 2205, 2805, 3477, 3717, 4485, 5037, 5325, 6237, 6885, 7917, 9405, 10197, 10605, 11445, 11877, 12765, 16125, 17157, 18765, 19317, 22197, 22797, 24645, 26565, 27885, 29925, 32037

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

Least common multiple of prime(n)-2 and prime(n)+2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A066830, A084921, A166011.

Cf. A049001, A084920, A182200; A182174.

[NthPrime(n)^2-4: n in [1..41]]; // Bruno Berselli, Apr 17 2012

f[n_]:=LCM[n-2,n+2]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
Prime[Range[5!]]^2 - 4 (* Zak Seidov, Apr 17 2012 *)

a(n)=prime(n)^2-4 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = A001248(n)-4 = A040976(n)*A052147(n). [Bruno Berselli, Apr 17 2012]

Definition rewritten by Bruno Berselli, Apr 17 2012

A024700 a(n) = (prime(n+2)^2 - 1)/3.

Original entry on oeis.org

8, 16, 40, 56, 96, 120, 176, 280, 320, 456, 560, 616, 736, 936, 1160, 1240, 1496, 1680, 1776, 2080, 2296, 2640, 3136, 3400, 3536, 3816, 3960, 4256, 5376, 5720, 6256, 6440, 7400, 7600, 8216, 8856, 9296, 9976, 10680, 10920, 12160, 12416, 12936, 13200, 14840, 16576, 17176

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

Numbers of the form 4*h*(3*h +- 1). - Vincenzo Librandi, May 21 2013

This sequence is also: Numbers n such that k is prime and its square is of the form 3*n + 1 (i.e., k^2 = 3*n + 1). For this case, the sequence is to be prepended with a(0) = 1. - G. C. Greubel, Sep 18 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040 (prime(n)), A001248, A024701, A024702, A061066, A081115, A084920, A084921, A084922.

[(NthPrime(n+2)^2-1)/3: n in [1..50]]; // Bruno Berselli, May 22 2013

Select[Range[2,10000], PrimeQ[Sqrt[3*#+1]] &] (* G. C. Greubel, Sep 18 2016 *)
(Prime[Range[3,50]]^2-1)/3 (* Harvey P. Dale, May 05 2022 *)

a(n) = (prime(n+2)^2-1)/3; \\ Altug Alkan, Sep 18 2016

[(n^2 -1)/3 for n in prime_range(4,301)] # G. C. Greubel, May 02 2024

a(n) = (A001248(n+2) - 1)/3. - Elmo R. Oliveira, Jan 20 2023

a(n) = 8*A024702(n+2) = 4*A081115(n+2) = 2*A084922(n+2) = (2/3)*A084921(n) = (4/3)*A024701(n+1) = (8/3)*A061066(n+2). - Alois P. Heinz, Jan 20 2023

A141767 A positive integer k is included if (p-1)*(p+1) divides k for every prime p that divides k.

Original entry on oeis.org

1, 24, 48, 72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 576, 600, 648, 672, 720, 768, 864, 960, 1008, 1080, 1152, 1200, 1296, 1320, 1344, 1440, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2160, 2304, 2352, 2400, 2592, 2640, 2688, 2880, 3000

Offset: 1

Leroy Quet, Jul 02 2008

nonn

For n>1, a(n) is a multiple of 24.

			120 has the prime factorization of 2^3 * 3^1 * 5^1. The distinct primes dividing 120 are therefore 2,3,5. (2-1)*(2+1)=3, (3-1)*(3+1)=8 and (5-1)*(5+1)=24 all divide 120. So 120 is included in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A140470, A141766, A124240.

Cf. A027748, A084920.

a141767 n = a141767_list !! (n-1)
a141767_list = filter f [1..] where
   f x = all (== 0) $
         map (mod x) $ zipWith (*) (map pred ps) (map succ ps)
         where ps = a027748_row x
-- Reinhard Zumkeller, Aug 27 2013

fQ[n_] := Block[{p = First /@ FactorInteger@ n}, Union@ Mod[n, (p - 1) (p + 1)] == {0}]; Select[ Range[2, 3000], fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

Added missing term 336 and a(14)-a(47) from Donovan Johnson, Sep 27 2008

a(1)=1 prepended by Max Alekseyev, Aug 27 2013

A182200 a(n) = prime(n)^2-3.

Original entry on oeis.org

1, 6, 22, 46, 118, 166, 286, 358, 526, 838, 958, 1366, 1678, 1846, 2206, 2806, 3478, 3718, 4486, 5038, 5326, 6238, 6886, 7918, 9406, 10198, 10606, 11446, 11878, 12766, 16126, 17158, 18766, 19318, 22198, 22798, 24646, 26566, 27886, 29926, 32038, 32758, 36478

Offset: 1

Bruno Berselli, Apr 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001248, A049001, A061725, A066872, A084920, A166010.

Magma
```
[NthPrime(n)^2-3: n in [1..43]];
```

A182200:=n->ithprime(n)^2-3; seq(A182200(k),k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Mathematica
```
Table[Prime[n]^2 - 3, {n, 43}]
```

a(n) = A061725(n)-5 = A066872(n)-4 = A001248(n)-3 = A084920(n)-2 = A049001(n)-1 = A166010(n)+1. [Formulas revised and extended by Bruno Berselli, Oct 15 2012]

A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

9, 44, 234, 664, 2628, 4354, 9774, 13660, 24264, 48690, 59488, 101194, 137718, 158884, 207504, 297594, 410580, 453778, 601324, 715608, 777814, 985840, 1143324, 1409670, 1825054, 2060298, 2185144, 2449764, 2589730, 2885454, 4096384, 4495788, 5142294

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), this sequence (m=4), A297492 (m=6), A297493 (m=8), A297494 (m=10).

Cf. A259825.

Let b(n) = 2*n^3 - 3*n - 1.

a(n) = b(prime(n)).

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

Original entry on oeis.org

33, 308, 2874, 11528, 72060, 141218, 414918, 648260, 1394328, 3528690, 4608800, 9358298, 14113470, 17077148, 24378288, 39426858, 60555180, 69195410, 100714868, 127012680, 141942878, 194693840, 237229188, 313639470, 442561238, 520209690, 562658408, 655294428

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), this sequence (m=6), A297493 (m=8), A297494 (m=10).

Cf. A259825.

b(n) = 5*n^4 - 9*n^2 - 5*n - 1;
a(n) = b(prime(n)); \\ Michel Marcus, Jan 01 2018

Let b(n) = 5*n^4 - 9*n^2 - 5*n - 1.

a(n) = b(prime(n)).

A297493 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^8H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

129, 2444, 39714, 224664, 2214948, 5133114, 19734534, 34465980, 89757384, 286456170, 399954528, 969369474, 1620023118, 2055854724, 3207878544, 5850511794, 10003119540, 11817917898, 18893239884, 25249088088, 29012002734, 43064859120, 55130420604

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), this sequence (m=8), A297494 (m=10).

Cf. A259825.

lista(nn) = forprime(p=2, nn, print1(14*p^5-28*p^3-20*p^2-7*p-1, ", ")); \\ Altug Alkan, Jan 01 2018

Let b(n) = 14*n^5 - 28*n^3 - 20*n^2 - 7*n - 1.

a(n) = b(prime(n)).

cp's OEIS Frontend

A341655 a(n) is the number of divisors of prime(n)^2 - 1.

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A084922 a(n) = (prime(n)-1)*(prime(n)+1)/6.

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A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

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A166010 a(n) = prime(n)^2-4.

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A024700 a(n) = (prime(n+2)^2 - 1)/3.

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A141767 A positive integer k is included if (p-1)*(p+1) divides k for every prime p that divides k.

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Haskell

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A182200 a(n) = prime(n)^2-3.

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A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

A297493 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^8H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.