A081115 -id:A081115 - cp's OEIS frontend

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

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

A231589 a(n) = sum_{k=1..(n-1)/2} (k^2 mod n).

Original entry on oeis.org

0, 0, 1, 1, 5, 5, 7, 6, 12, 20, 22, 19, 39, 35, 35, 28, 68, 60, 76, 65, 91, 99, 92, 74, 125, 156, 144, 147, 203, 175, 186, 152, 242, 272, 245, 201, 333, 323, 286, 270, 410, 392, 430, 363, 420, 437, 423, 340, 490, 550, 578, 585, 689, 639, 605, 546, 760, 812

Offset: 1

Michel Marcus, Nov 11 2013

nonn

This expression occurred to S. A. Shirali while demonstrating a result concerning A081115 and A228432. This led him to investigate integers n such that a(n) = n*(n-1)/4, a(n) = floor(n/4), or a(n) = n*(n-1)/4 - n.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Cf. A048152, A048153.

Table[Sum[PowerMod[k,2,n],{k,(n-1)/2}],{n,60}] (* Harvey P. Dale, Jan 30 2016 *)

PARI
```
a(n) = sum(k=1, (n-1)\2, k^2 % n);
```

A339527 Primes p for which p + k and p^2 + k are prime, where k = (p^2-1)/12.

Original entry on oeis.org

7, 17, 37, 43, 79, 97, 199, 241, 307, 331, 503, 727, 811, 829, 1297, 1303, 1423, 1879, 2017, 2179, 2593, 2617, 2663, 2953, 3121, 3229, 3761, 3779, 4327, 4357, 4391, 4409, 4663, 4861, 4951, 5021, 5147, 5167, 5237, 5669, 5939, 6569, 7129, 7829, 8269, 8731, 9649, 9781, 10159, 10459, 10531, 10663, 11789

Offset: 1

J. M. Bergot and Robert Israel, Dec 22 2020

nonn

			a(4) = 43 is a term because with k = (43^2-1)/12 = 154, 43, 43+154 = 197 and 43^2+154 = 2003 are all primes.

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

Cf. A081115.

select(t -> isprime(t) and isprime((13*t^2-1)/12) and isprime(t+(t^2-1)/12), [seq(seq(12*i+j,j=[1,5,7,11]),i=0..10000)]);

isok(p) = isprime(p) && iferr(isprime(p+(p^2-1)/12) && isprime(p^2+(p^2-1)/12), E,0); \\ Michel Marcus, Dec 23 2020

A228432 Sum_{i=1..floor(prime(n)/4)} floor(sqrt(i*prime(n))).

Original entry on oeis.org

0, 0, 2, 2, 7, 14, 24, 25, 37, 70, 71, 114, 140, 143, 170, 234, 274, 310, 357, 399, 444, 498, 552, 660, 784, 850, 856, 926, 990, 1064, 1310, 1395, 1564, 1574, 1850, 1859, 2054, 2173, 2277, 2494, 2623, 2730, 2986, 3104, 3234, 3246, 3656, 4085, 4235, 4370

Offset: 1

Michel Marcus, Nov 11 2013

nonn

If p = prime(n) in A002145 and n>3, or said differently, if n in A080148 and n>1, then a(n) = A081115(n).

			For n=7, p=17 and a(7) = floor(sqrt(17)) + floor(sqrt(34)) + floor(sqrt(51)) + floor(sqrt(68)) = 4+5+7+8 = 24.

S. A. Shirali, A family portrait of primes -- a case study in discrimination, Math. Mag., Vol. 70, No. 4 (Oct. 1997), pp. 263-272.

Cf. A002145, A080148, A081115.

Table[p = Prime[n]; Sum[Floor[Sqrt[i*p]], {i, Floor[p/4]}], {n, 100}] (* T. D. Noe, Nov 13 2013 *)

a(n) = p = prime(n); sum(i=1, p\4, sqrtint(i*p));

cp's OEIS Frontend

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

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A231589 a(n) = sum_{k=1..(n-1)/2} (k^2 mod n).

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A339527 Primes p for which p + k and p^2 + k are prime, where k = (p^2-1)/12.

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A228432 Sum_{i=1..floor(prime(n)/4)} floor(sqrt(i*prime(n))).

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