A069482 -id:A069482

A267896

a(n) = (Prime(n+1)^2 - Prime(n)^2) / 8.

+10
1

2, 3, 9, 6, 15, 9, 21, 39, 15, 51, 39, 21, 45, 75, 84, 30, 96, 69, 36, 114, 81, 129, 186, 99, 51, 105, 54, 111, 420, 129, 201, 69, 360, 75, 231, 240, 165, 255, 264, 90, 465, 96, 195, 99, 615, 651, 225, 114, 231, 354, 120, 615, 381, 390, 399, 135, 411, 279

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Robert Israel, Table of n, a(n) for n = 2..10000

FORMULA

a(n) = A024675(n) * A028334(n) / 2.

a(n) = (A000040(n+1)^2 - A000040(n)^2) / 8.

a(n) = A069482(n) / 8.

MAPLE

seq((ithprime(n+1)^2 - ithprime(n)^2)/8, n=2..100); # Robert Israel, Jan 22 2016

MATHEMATICA

Rest[Array[(Prime[# + 1]^2 - Prime[#]^2) / 8 &, 60]] (* Vincenzo Librandi, Jan 23 2016 *)

(#[[2]]-#[[1]])/8&/@Partition[Prime[Range[2, 60]]^2, 2, 1] (* Harvey P. Dale, Aug 01 2022 *)

PROG

(PARI) a(n) = (prime(n+1)^2 - prime(n)^2)/8; \\ Michel Marcus, Jan 22 2016

(Magma) [(NthPrime(n+1)^2-NthPrime(n)^2) div 8: n in [2..60]]; // Vincenzo Librandi, Jan 23 2016

CROSSREFS

Cf. A069482, A028334, A024675, A000040.

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Jan 22 2016

STATUS

approved

A276963

a(n) = prime(n+1)^4 - prime(n)^4.

+10
1

65, 544, 1776, 12240, 13920, 54960, 46800, 149520, 427440, 216240, 950640, 951600, 593040, 1460880, 3010800, 4226880, 1728480, 6305280, 5260560, 2986560, 10551840, 8508240, 15283920, 25787040, 15531120, 8490480, 18528720, 10078560, 21889200, 97097280, 34355280, 57775440

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..32.

EXAMPLE

a(1) = 3^4 - 2^4 = 65.

a(2) = 5^4 - 3^4 = 544.

CROSSREFS

Cf. A069482, A129701.

KEYWORD

nonn

AUTHOR

Bhushan Bade, Sep 22 2016

STATUS

approved

A295706

Primes p for which the difference between p^2 and the square of the next prime is both 1 more and 1 less than a prime.

+10
1

7, 17, 23, 37, 47, 59, 83, 89, 107, 113, 127, 131, 149, 163, 173, 257, 353, 433, 439, 457, 467, 521, 563, 761, 773, 839, 881, 953, 1009, 1031, 1213, 1307, 1319, 1321, 1697, 1733, 1759, 1811, 1861, 1871, 1913, 1979, 2153, 2221, 2281, 2287, 2309, 2393, 2593, 2767, 2789

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

I.e., primes p for which the difference between p^2 and the square of the next prime is the average of a twin prime pair.

LINKS

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

EXAMPLE

The primes 7 and 11 are consecutive and their squares are 49 and 121. The difference is 72, and both 71 and 73 are prime.

Likewise, the difference between the square of 563 and the next prime (569) is 6792, and 6791 and 6793 are twin primes.

MAPLE

N:= 10^4: # to get all terms <= N

p:= 1: q:= 2: A:= NULL:

while p < N do

p:= q; q:= nextprime(p);

d:= q^2-p^2;

if isprime(d+1) and isprime(d-1) then A:= A, p fi

od:

A; # Robert Israel, Mar 02 2018

MATHEMATICA

For[p = 1, p < 10000, p++,

a = Prime[p];

b = Prime[p + 1];

c = b^2 - a^2;

d = (c + 1);

e = (c - 1);

If[And[PrimeQ[d] == True, PrimeQ[e] == True], Print[a]];

]

(* Second program: *)

Select[Partition[Prime@ Range@ 300, 2, 1], AllTrue[{# + 1, # - 1}, PrimeQ] &[#2^2 - #1^2] & @@ # &][[All, 1]] (* Michael De Vlieger, Dec 03 2017 *)

PROG

(PARI) lista(nn) = { my(pp=2); forprime(p=3, nn, my(d=p^2-pp^2); if(isprime(d+1) && isprime(d-1), print1(pp, ", ")); pp=p); } \\ Iain Fox, Dec 03 2017

CROSSREFS

Cf. A014574 (average of twin prime pairs), A069482 (difference between squares of consecutive primes).

KEYWORD

nonn

AUTHOR

Geoffrey Marnell, Nov 25 2017

STATUS

approved

A128926

Smaller member p of a pair of consecutive primes (p,q) such that either q^2-p^2+1 or q^2-p^2-1 is also prime.

+10
0

3, 5, 7, 11, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 73, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 241, 251, 257, 263, 281, 307, 311, 313, 331, 337, 353, 359, 367, 373, 379

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

3 and 5 are consecutive primes, 5^2-3^2 = 25-9 = 16. 17 is prime, hence 3 is in the sequence.

79 and 83 are consecutive primes, 83^2-79^2 = 6889-6241 = 648. 647 is prime, hence 79 is in the sequence.

89 and 97 are consecutive primes, 97^2-89^2 = 9409-7921 = 1488. 1487 (as well as 1489) is prime, hence 89 is in the sequence.

MAPLE

isA128926 := proc(n) local p, q ; p := ithprime(n) ; q := ithprime(n+1) ; isprime((p+q)*(q-p)+1) or isprime((p+q)*(q-p)-1) ; end:

for n from 1 to 100 do if isA128926(n) then printf("%d, ", ithprime(n)) ; fi ; od ; # R. J. Mathar, Apr 26 2007

MATHEMATICA

Prime@ Select[ Range@ 75, PrimeQ[ Prime[ # + 1]^2 - Prime@#^2 - 1] || PrimeQ[ Prime[ # + 1]^2 - Prime@#^2 + 1] &] (* Robert G. Wilson v *)

PROG

(Magma) [ p: p in PrimesUpTo(380) | IsPrime(q^2-p^2-1) or IsPrime(q^2-p^2+1) where q is NextPrime(p) ]; /* Klaus Brockhaus, May 05 2007 */

CROSSREFS

Cf. A069482.

KEYWORD

nonn

AUTHOR

J. M. Bergot, Apr 25 2007

EXTENSIONS

Corrected and extended by Robert G. Wilson v, R. J. Mathar and Klaus Brockhaus, Apr 26 2007

STATUS

approved

A157494

Primes in A014150.

+10
0

2, 1429, 32869, 3189059, 5157791, 62701339, 139181423, 296686879, 522304883, 5070516751, 6276844867, 7098350179, 8983996079, 9331926623, 21211375343, 31177858939, 34861039007, 38865340309, 39918757589, 62858815181

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..20.

MATHEMATICA

s0=s1=s2=0; lst={}; Do[p=Prime[n]; s0+=p; s1+=s0; s2+=s1; If[PrimeQ[s2], AppendTo[lst, s2]], {n, 7!}]; lst

CROSSREFS

Cf. A001223, A014148, A014150, A061789, A069482, A122382, A129701, A157492, A157493

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

STATUS

approved

A254860

Sorted integers m = (prime(n+1)^2 - prime(n)^2)/24, where prime(n) is A000040(n), with duplicates removed.

+10
0

1, 2, 3, 5, 7, 10, 12, 13, 15, 17, 18, 23, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 43, 45, 47, 52, 55, 58, 62, 65, 67, 70, 72, 75, 77, 80, 85, 87, 88, 93, 95, 100, 103, 105, 107, 110, 117, 118, 120, 127, 130, 133, 135, 137, 138, 140, 143, 147

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A069482 gives the values of (prime(n+1)^2 - prime(n)^2), in order, with duplicates.

For n>=3 (prime(n+1)^2 - prime(n)^2)/24 is an integer.

The list here is sorted with duplicates removed to examine the nature and scope coverage over the integers of these ratios.

a(n) values have increasing differences on average, and approximately fit a curve for the n-th distinct value, given by (1/3)*n*log(n) + (3/10)*n*log(log(n))^3 for the first 10,000 values.

The differences between adjacent a(n) values, examined over the first 100,000 values, indicates all integers are covered (i.e., for any integer k there is at least one n where k = a(n+1) - a(n)).

Prime factorization of a(n) indicates every prime will appear as a factor for at least one a(n) value.

LINKS

Table of n, a(n) for n=1..58.

MATHEMATICA

DeleteDuplicates[Sort[Table[(Prime[n + 1]^2 - Prime[n]^2)/24, {n, 3, 300}]]]

CROSSREFS

Cf. A069482, A000040.

KEYWORD

nonn

AUTHOR

Richard R. Forberg, Feb 19 2015

STATUS

approved

A261464

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

+10
0

118, 321, 1287, 2083, 4755, 6583, 11823, 23879, 28973, 49721, 67583, 77863, 102015, 146711, 202617, 223553, 297101, 353483, 384043, 487781, 565619, 698159, 904835, 1020981, 1082623, 1214535, 1283683, 1431123, 2035723, 2232075, 2554319, 2666981, 3288765

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..33.

FORMULA

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

EXAMPLE

a(1) = 5^3 - 3^2 + 2 = 118.

MATHEMATICA

Table[Prime[n + 2]^3 - Prime[n + 1]^2 + Prime[n], {n, 60}] (* Vincenzo Librandi, Aug 20 2015 *)

PROG

(Magma) [NthPrime(n+2)^3-NthPrime(n+1)^2+NthPrime(n): n in [1.. 35]]; // Vincenzo Librandi, Aug 20 2015

(PARI) vector(40, n, prime(n+2)^3-prime(n+1)^2+prime(n)) \\ Michel Marcus, Aug 20 2015

CROSSREFS

Cf. A136612, A069482, A129701.

KEYWORD

nonn

AUTHOR

Altug Alkan, Aug 20 2015

EXTENSIONS

More terms from Vincenzo Librandi, Aug 20 2015

STATUS

approved

A261465

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

+10
0

7, 22, 44, 114, 158, 276, 344, 510, 818, 932, 1338, 1644, 1808, 2166, 2762, 3428, 3662, 4428, 4974, 5258, 6168, 6810, 7838, 9320, 10104, 10508, 11346, 11774, 12660, 16016, 17034, 18638, 19184, 22062, 22652, 24498, 26412, 27726, 29762

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..39.

FORMULA

a(n) = A036689(n+1) + A001223(n). - Michel Marcus, Aug 21 2015 [Corrected by Georg Fischer, Dec 12 2022]

a(n) ~ n^2 log^2 n. - Charles R Greathouse IV, Aug 22 2015

EXAMPLE

a(2) = 5^2 - 3 = 22.

MATHEMATICA

Table[Prime[n + 1]^2 - Prime[n], {n, 60}] (* Vincenzo Librandi, Aug 20 2015 *)

PROG

(PARI) first(m)=vector(m, i, prime(i+1)^2 - prime(i)) \\ Anders Hellström, Aug 20 2015

(PARI) a(n, p=prime(n)); nextprime(p+1)^2-p \\ Charles R Greathouse IV, Aug 22 2015

(PARI) first(n)=my(v=primes(n+1)); vector(n, i, v[i+1]^2-v[i]) \\ Charles R Greathouse IV, Aug 22 2015

(Magma) [NthPrime(n+1)^2-NthPrime(n): n in [1.. 70]]; // Vincenzo Librandi, Aug 20 2015

CROSSREFS

Cf. A036689, A001223, A069482, A129701.

KEYWORD

nonn,easy

AUTHOR

Altug Alkan, Aug 20 2015

STATUS

approved