A069482 -id:A069482 - cp's OEIS frontend

A267896 a(n) = (Prime(n+1)^2 - Prime(n)^2) / 8.

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

Offset: 2

Franklin T. Adams-Watters, Jan 22 2016

nonn

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

Cf. A069482, A028334, A024675, A000040.

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

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

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 *)

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

a(n) = A024675(n) * A028334(n) / 2.
a(n) = (A000040(n+1)^2 - A000040(n)^2) / 8.
a(n) = A069482(n) / 8.

A276963 a(n) = prime(n+1)^4 - prime(n)^4.

Original entry on oeis.org

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

Offset: 1

Bhushan Bade, Sep 22 2016

nonn

			a(1) = 3^4 - 2^4 = 65.
a(2) = 5^4 - 3^4 = 544.

Cf. A069482, A129701.

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.

Original entry on oeis.org

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

Offset: 1

Geoffrey Marnell, Nov 25 2017

nonn

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.

			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.

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

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

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

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 *)

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

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.

Original entry on oeis.org

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

Offset: 1

J. M. Bergot, Apr 25 2007

nonn

			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.

Cf. A069482.

[ 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 */

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

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

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

A157494 Primes in A014150.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

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

s0=s1=s2=0;lst={};Do[p=Prime[n];s0+=p;s1+=s0;s2+=s1;If[PrimeQ[s2],AppendTo[lst,s2]],{n,7!}];lst
Select[Nest[Accumulate[#]&,Prime[Range[700]],3],PrimeQ] (* Harvey P. Dale, Jul 11 2025 *)

A254860 Sorted integers m = (prime(n+1)^2 - prime(n)^2)/24, where prime(n) is A000040(n), with duplicates removed.

Original entry on oeis.org

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

Offset: 1

Richard R. Forberg, Feb 19 2015

nonn

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.

Cf. A069482, A000040.

DeleteDuplicates[Sort[Table[(Prime[n + 1]^2 - Prime[n]^2)/24, {n, 3, 300}]]]
Union[Differences[Prime[Range[3,300]]^2]/24] (* Harvey P. Dale, Jul 18 2025 *)

A261464 a(n) = prime(n+2)^3 - prime(n+1)^2 + prime(n).

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Aug 20 2015

nonn

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

Cf. A136612, A069482, A129701.

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

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

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

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

More terms from Vincenzo Librandi, Aug 20 2015

A261465 a(n) = prime(n+1)^2 - prime(n).

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Aug 20 2015

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

Cf. A036689, A001223, A069482, A129701.

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

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

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

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

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

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

cp's OEIS Frontend

A267896 a(n) = (Prime(n+1)^2 - Prime(n)^2) / 8.

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A276963 a(n) = prime(n+1)^4 - prime(n)^4.

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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.

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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.

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A157494 Primes in A014150.

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A254860 Sorted integers m = (prime(n+1)^2 - prime(n)^2)/24, where prime(n) is A000040(n), with duplicates removed.

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A261464 a(n) = prime(n+2)^3 - prime(n+1)^2 + prime(n).

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A261465 a(n) = prime(n+1)^2 - prime(n).

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