A076821 -id:A076821 - cp's OEIS frontend

A038702 Prime(n)^2 mod prime(n-1).

1, 1, 4, 2, 4, 3, 4, 16, 13, 4, 5, 16, 4, 16, 36, 36, 4, 36, 16, 4, 36, 16, 36, 64, 16, 4, 16, 4, 16, 83, 16, 36, 4, 100, 4, 36, 36, 16, 36, 36, 4, 100, 4, 16, 4, 144, 144, 16, 4, 16, 36, 4, 100, 36, 36, 36, 4, 36, 16, 4, 100, 196, 16, 4

Offset: 2

Neil Fernandez, May 01 2000

a(n+1) = n-th prime gap squared mod the n-th prime = A076821(n) mod A000040(n). Probably a(n) = A076821(n+1) for n > 31. This holds up to 4 * 10^18. - Charles R Greathouse IV, Apr 17 2012

			To get a(4): square the fourth prime to get 7^2 = 49. The remainder when this is divided by the third prime, 5, is 4. So a(3) = 4.

Vincenzo Librandi, Table of n, a(n) for n = 2..10000

Cf. A038703.

A038702 := proc(n)
    modp( ithprime(n)^2,ithprime(n-1)) ;
end proc: # R. J. Mathar, Jan 09 2015

Table[Mod[Prime[n+1]^2,Prime[n]],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2010 *)
Table[PowerMod[Prime[n],2,Prime[n-1]],{n,2,70}] (* Harvey P. Dale, Mar 22 2012 *)

a(n)=prime(n)^2%prime(n-1) \\ Charles R Greathouse IV, Apr 17 2012

A167770 a(n) = prime(n)^2 modulo prime(n+1).

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 16, 7, 4, 36, 16, 4, 16, 36, 36, 4, 36, 16, 4, 36, 16, 36, 64, 16, 4, 16, 4, 16, 69, 16, 36, 4, 100, 4, 36, 36, 16, 36, 36, 4, 100, 4, 16, 4, 144, 144, 16, 4, 16, 36, 4, 100, 36, 36, 36, 4, 36, 16, 4, 100, 196, 16, 4, 16, 196, 36, 100, 4, 16, 36, 64, 36, 36, 16

Offset: 1

Zak Seidov, Nov 11 2009

nonn

Only for three cases n = 4,9,30, a(n) < (prime(n+1)-prime(n))^2 because only in these cases (prime(n+1)-prime(n))^2 > prime(n+1):
n = 4: a(4) = 5 < ((p(5)-p(4))^2 = (11-7)^2 = 16) and 16 > 11.
n = 9: a(9) = 7 < ((p(10)-p(9))^2 = (29-23)^2 = 36) and 36 > 29.
n = 30: a(30) = 69 < ((p(31)-p(30))^2 = (127-113)^2 = 196) and 196 > 127.
In all other cases, a(n) = A076821(n) = (prime(n+1)-prime(n))^2, is highly probable but not proved conjecture.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A076821 (squares of the differences between consecutive primes).
Cf. A001223 (modular square roots of this sequence).
Cf. A000040 (primes), A001248 (squares of primes).

A167770:=n->ithprime(n)^2 mod ithprime(n+1): seq(A167770(n), n=1..70); # Wesley Ivan Hurt, Oct 01 2014

Table[PowerMod[Prime[n], 2, Prime[n+1]], {n, 221265}]

a(n)=prime(n)^2%prime(n+1) \\ M. F. Hasler, Oct 04 2014

a(n) = prime(n)^2 modulo prime(n+1).

a(n) == A001223(n)^2 (mod A000040(n+1)). - L. Edson Jeffery, Oct 01 2014

A074741 Sum of squares of gaps between consecutive primes.

Original entry on oeis.org

1, 5, 9, 25, 29, 45, 49, 65, 101, 105, 141, 157, 161, 177, 213, 249, 253, 289, 305, 309, 345, 361, 397, 461, 477, 481, 497, 501, 517, 713, 729, 765, 769, 869, 873, 909, 945, 961, 997, 1033, 1037, 1137, 1141, 1157, 1161, 1305, 1449, 1465, 1469, 1485, 1521

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 05 2002

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York, Heidelberg, 1994, problem A8.

T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Erdös, Some problems on the distribution of prime numbers, C.I.M.E., Teoria dei numeri (1955).
D. R. Heath-Brown, Gaps between primes and the pair correlation of zeros of the zeta function, Acta Arithmetica, vol. XLI (1982), 85.
M. Wolf, Some conjectures on the gaps between consecutive primes, preprint IFTUWr 894//95, submitted to Asterisque.

Partial sums of A076821. - Michel Marcus, May 26 2018

Cf. A001223.

with(numtheory): a := proc(n) option remember: if (n=1) then RETURN(1) else RETURN(a(n-1)+(ithprime(n+1)-ithprime(n))^2) fi: end:

Rest[FoldList[Plus, 0, (#[[2]] - #[[1]])^2 & /@ Partition[Prime[Range[100]], 2, 1]]]
nn=60;With[{dsp=Differences[Prime[Range[nn+1]]]^2},Table[Total[Take[ dsp,n]],{n,nn}]] (* Harvey P. Dale, Nov 30 2011 *)
Accumulate[Differences[Prime[Range[60]]]^2] (* Harvey P. Dale, May 08 2015 *)

a(n) = sum(k=1, n, (prime(k+1) - prime(k))^2); \\ Michel Marcus, May 26 2018

from sympy import nextprime
from itertools import islice, accumulate
def gen():
    p, q = 2, 3
    while True:
        r = (q - p) ** 2
        yield r
        p, q = q, nextprime(q)
print(list(accumulate(islice(gen(), 51)))) # Adrienne Leonardo, Dec 18 2024

a(n) = Sum_{k=1..n} (prime(k+1)-prime(k))^2 = Sum_{k=1..n} A001223(k)^2.

Asymptotic expressions: D. R. Heath-Brown's conjecture: Sum_{prime(n)<=N} (prime(n)-prime(n-1))^2 ~ 2*N*log(N). Marek Wolf's conjecture: Sum_{prime(n)A000720(n).

A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.

Original entry on oeis.org

8, 13, 41, 53, 137, 173, 305, 397, 533, 877, 977, 1373, 1697, 1885, 2245, 2813, 3517, 3737, 4493, 5077, 5345, 6277, 6953, 7937, 9413, 10217, 10613, 11465, 12077, 12785, 16165, 17165, 18869, 19325, 22237, 22837, 24665, 26605, 27925, 29933, 32141, 32765, 36497, 37253, 38953, 39745

Offset: 1

Aaron T Cowan, Feb 20 2023

nonn

The value d is the square of the length of the diagonal of a trapezoid with a height and bases that are consecutive primes, respectively. The diagonal length is calculated using the Pythagorean theorem, but this distance is squared so that the value is an integer.

			        p(2)=3
        _ _ _ _
a(1):  |        \  d^2=2^2+(5-3)^2=8
p(1)=2 |_ _ _ _ _\
        p(3)=5
        p(3)=5
        _ _ _ _ _ _
a(2):  |           \    d^2=3^2 + (7-5)^2 = 9+4 = 13
p(2)=3 |            \
       |_ _ _ _ _ _ _\
        p(4)=7
a(3)= 5^2+(11-7)^2 = 25+16 = 41
a(7)= 17^2+(23-19)^2=305 = 5*61

Aaron T Cowan, Table of n, a(n) for n = 1..500

Cf. A001248, A076821.

Cf. A131019, A106171.

%shorter 1 line version
arrayfun(@(p) p^2+(nextprime(nextprime(p+1)+1)-nextprime(p+1))^2,[primes(10^6)])

Map[(#[[1]]^2 + (#[[3]] - #[[2]])^2) &, Partition[Prime[Range[50]], 3, 1]] (* Amiram Eldar, Feb 24 2023 *)

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2; \\ Michel Marcus, Feb 23 2023

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

a(n) = A001248(n) + A076821(n+1). - Michel Marcus, Feb 23 2023

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A038702 Prime(n)^2 mod prime(n-1).

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A167770 a(n) = prime(n)^2 modulo prime(n+1).

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A074741 Sum of squares of gaps between consecutive primes.

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A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.

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