A074924 -id:A074924 - cp's OEIS frontend

A213739 Numbers n such that n and n^2 are sums of two successive primes.

12, 24, 42, 84, 90, 120, 204, 240, 372, 410, 456, 600, 630, 740, 762, 852, 882, 978, 1088, 1140, 1148, 1272, 1460, 1518, 1584, 1620, 1656, 1758, 1770, 1878, 1900, 1960, 2052, 2316, 2562, 2688, 2886, 2992, 3570, 3634, 3678, 3738, 3750, 3924, 4170, 4314, 4906

Offset: 1

Zak Seidov, Jun 19 2012

nonn

First terms not multiple of 6: 410, 740, 1088, 1148, 1460, 1900, 1960.

			12=5+7, 144=71+73;
24=11+13, 576=283+293;
42=19+23, 1764=881+883.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Zak Seidov, Table of n,p,q,n^2,r,s

Intersection of A001043 and A074924.

Reap[ Do[ If[ (p=NextPrime[n/2, -1]; p+NextPrime[p] == n) && (q=NextPrime[n^2/2, -1]; q+NextPrime[q] == n^2) , Sow[n]], {n, 2, 5000, 2}]][[2, 1]] (* Jean-François Alcover, Jul 17 2012 *)

p=2;forprime(q=3,1e3,n=p+q;if(precprime(n^2/2)+nextprime((n^2+1)/2)==n^2,print1(n", "));p=q) \\ Charles R Greathouse IV, Jun 21 2012

A213811 Numbers k such that k and k^3 are sums of two twin primes.

Original entry on oeis.org

1044, 1200, 2604, 2964, 4056, 4284, 4476, 7164, 7644, 9300, 9864, 10884, 14616, 15180, 20916, 24084, 40716, 51156, 55056, 65436, 66144, 70104, 74676, 92100, 99060, 104580, 105804, 163944, 164700, 165780, 209604, 218400, 219660, 222540, 226656, 257040, 281676

Offset: 1

Zak Seidov, Jun 20 2012

nonn

All terms are multiples of 12.

			1044=521+523, 1044^3=1137893184=568946591+568946593.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A001043, A074924, A213784, A213789.

sttpQ[n_]:=Module[{x=(n-2)/2},AllTrue[{x,x+2},PrimeQ]]; Select[Range[ 12,300000,12],sttpQ[#]&&sttpQ[#^3]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 25 2017 *)

isok(n) = !(n % 2) && isprime(n/2-1) && isprime(n/2+1) && isprime(n^3/2-1) && isprime(n^3/2+1); \\ Michel Marcus, Oct 19 2013

k/2 +/-1 and also (k^3)/2 +/- 1 are twin primes.

A225077 Smaller of the two consecutive primes whose sum is a triangular number.

Original entry on oeis.org

17, 37, 59, 103, 137, 149, 313, 467, 491, 883, 911, 1277, 1423, 1619, 1783, 2137, 2473, 2729, 4127, 4933, 5437, 5507, 6043, 6359, 10039, 10453, 11717, 13397, 15809, 17489, 20807, 21821, 23027, 27631, 28307, 28813, 29669, 33029, 36947, 39103, 44203, 48281

Offset: 1

Alex Ratushnyak, May 28 2013

nonn

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

Cf. A175132 (numbers n such that sum of two consecutive primes is triangular(n)).
Cf. A181902 and A154634 (average of two consecutive primes is a triangular number).
Cf. A075190 and A225195 (average of two consecutive primes is a square).
Cf. A074924 and A061275 (sum of two consecutive primes is a square).

f:= proc(n) local m,p,q;
  m:= n*(n+1)/2;
  p:= prevprime(ceil(m/2));
  q:= nextprime(p);
  if p+q=m then p fi
end proc:
map(f, [$3..500]); # Robert Israel, May 04 2020

tri[n_] := IntegerQ[Sqrt[1 + 8 n]]; t = {}; p1 = 2; While[Length[t] < 50, p2 = NextPrime[p1]; If[tri[p1 + p2], AppendTo[t, p1]]; p1 = p2]; t (* T. D. Noe, May 28 2013 *)

a(n) + nextprime(a(n)) = A000217(A175132(n)).

A251056 Numbers n such that n^2 is a sum of 8 consecutive primes.

Original entry on oeis.org

38, 414, 466, 514, 714, 844, 850, 1076, 1136, 1186, 1370, 1512, 1544, 1580, 1600, 1700, 1844, 1900, 1918, 2028, 2114, 2250, 2304, 2320, 2330, 2364, 2396, 2404, 2450, 2674, 2846, 2894, 3076, 3314, 3346, 3506, 3612, 3622, 3676, 3718, 3774, 3866, 3912, 3966, 4012, 4126, 4506, 4700

Offset: 1

Zak Seidov, Dec 14 2014

nonn

			38^2 = 1444 = prime(38) + ... + prime(45) = 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197,
414^2 = 171396 = prime(2401) + ... + prime(2408) = 21391 + 21397 + 21401 + 21407 + 21419 + 21433 + 21467 + 21481.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A074924, A051395, A252018, A252019, A252066.

Sqrt[#]&/@Select[Total/@Partition[Prime[Range[250000]],8,1], IntegerQ[ Sqrt[#]]&] (* Harvey P. Dale, Nov 28 2018 *)

A179485 Sums of two successive primes s such that s+-3 are primes.

Original entry on oeis.org

8, 100, 1120, 1220, 1300, 2240, 2380, 2414, 3536, 3634, 4906, 4940, 5566, 5740, 6706, 7240, 8864, 9224, 9394, 10136, 10850, 12040, 12476, 12586, 12920, 13180, 13334, 13754, 14630, 14720, 15134, 16270, 17710, 18430, 18800, 19916, 21014, 21320

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 16 2010

nonn

Intersection of A001043 and A087695. - Robert Israel, Oct 25 2017

			3+5=8,8-3=5(prime),8+3=11(prime),..

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

Cf. A074924, A074925, A167597, A176984, A176986, A075593, A075594, A116360, A076565

Cf. A001043, A087695.

q:= 2; p:= 3;
count:= 0:
while count < 100 do
  q:= p; p:= nextprime(p);
  s:= q+p;
  if isprime(s-3) and isprime(s+3) then
    count:= count+1; A[count]:= s;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Oct 25 2017

q=3;Select[Table[Prime[n]+Prime[n+1],{n,7!}],PrimeQ[ #-q]&&PrimeQ[ #+q]&]
Select[Total/@Partition[Prime[Range[1400]],2,1],AllTrue[#+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2018 *)

A212430 Numbers n such that n and n^4 are sums of two twin primes.

Original entry on oeis.org

384, 840, 8676, 33300, 34980, 37044, 39984, 42024, 50604, 53760, 55056, 61680, 64380, 71064, 83520, 88176, 97644, 103740, 120204, 129840, 133896, 148764, 154524, 160416, 168120, 173064, 184800, 188880, 199056, 207984, 234744, 266640, 292116, 307044, 356184

Offset: 1

Zak Seidov, Jun 21 2012

nonn

All terms are multiples of 12.

			384=191+193, 384^3=147456=73727+73729.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001043, A074924, A213784, A213789, A213811.

Select[12 Range[30000],AllTrue[{#/2+1,#/2-1,#^4/2+1,#^4/2-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 01 2017 *)

isok(n) = !(n % 2) && isprime(n/2-1) && isprime(n/2+1) && isprime(n^4/2-1) && isprime(n^4/2+1); \\ Michel Marcus, Oct 19 2013

n/2 +/-1 and (n^4)/2 +/- 1 are primes.

A226533 a(n) = smallest integer m such that m^n is a sum of two successive primes.

Original entry on oeis.org

5, 6, 2, 150, 22, 82, 2, 258, 70, 30, 42, 18, 2, 12, 262, 58, 460, 36, 552, 24, 318, 344, 450, 54, 274, 88, 36, 92, 90, 188, 554, 20, 404, 700, 240, 6, 136, 262, 578, 222, 2182, 276, 162, 60, 142, 326, 176, 198, 930, 1116

Offset: 1

Zak Seidov, Jun 09 2013

nonn

			5^1 = 5 = 2 + 3, 6^2 = 36  = 17 + 19, 2^3 = 8 = 3 + 5, 150^4 =506250000 = 253124999 + 253125001.

Zak Seidov, Table of n, a(n) for n = 1..150

a(2) = 6 = A074924(1), a(3) = 2 = A074925(1). Cf. A001043, A001597.

a[n_] := For[m = 2, True, m++, p = m^n/2 // NextPrime[#, -1]&; q = NextPrime[p]; If[p + q == m^n, Print["a(", n, ") = ", m]; Return[m]]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jun 10 2013 *)
tsp[n_]:=Module[{m=1,t},t=m^n;While[NextPrime[t/2]+NextPrime[t/2, -1]! = t,m++;t=m^n];m]; Array[tsp,50] (* Harvey P. Dale, Nov 10 2014 *)

a(n)=if(n==1,return(5)); my(m=1,M,p); while(1,M=m++^n;p=precprime(M/2); ispseudoprime(M-p) && M-p==nextprime(M/2) && return(m)) \\ Charles R Greathouse IV, Jun 10 2013

a(41)-a(50) from Jean-François Alcover, Jun 10 2013

A297620 Positive numbers n such that n^2 == p (mod q) and n^2 == q (mod p) for some consecutive primes p,q.

Original entry on oeis.org

6, 10, 12, 24, 42, 48, 62, 72, 76, 84, 90, 93, 108, 110, 120, 122, 128, 145, 146, 174, 187, 188, 194, 204, 208, 215, 220, 228, 232, 240, 241, 264, 297, 306, 310, 314, 317, 326, 329, 336, 349, 357, 366, 372, 386, 408, 410, 423, 426, 431, 444, 454, 456, 468, 470, 474, 518, 522, 535, 538, 546, 548

Offset: 1

Robert Israel and Thomas Ordowski, Jan 01 2018

nonn

Positive numbers n such that n^2 == p+q mod (p*q) for some consecutive primes p, q.
Each pair of consecutive primes p,q such that p is a quadratic residue mod q and p and q are not both == 3 (mod 4) contributes infinitely many members to the sequence.
Odd terms of this sequence are 93, 145, 187, 215, 241, 297, 317, 329, 349, 357, 423, 431, 535, ... - Altug Alkan, Jan 01 2018

			a(3) = 12 is in the sequence because 71 and 73 are consecutive primes with 12^2 == 73 (mod 71) and 12^2 == 71 (mod 73).

Robert Israel, Table of n, a(n) for n = 1..2305

Contains A074924.

N:= 1000: # to get all terms <= N
R:= {}:
q:= 3:
while q < N^2 do
  p:= q;
  q:= nextprime(q);
  if ((p mod 4 <> 3) or (q mod 4 <> 3)) and numtheory:-quadres(q,p) = 1 then
    xp:= numtheory:-msqrt(q,p); xq:= numtheory:-msqrt(p,q);
    for sp in [-1,1] do for sq in [-1,1] do
      v:= chrem([sp*xp,sq*xq],[p,q]);
      R:= R union {seq(v+k*p*q, k = 0..(N-v)/(p*q))}
    od od;
  fi;
od:
sort(convert(R,list));

A165744 Numbers k with property that 6^k is the sum of two consecutive primes.

Original entry on oeis.org

2, 3, 7, 36, 54, 143, 1102, 1678

Offset: 1

Zak Seidov, Sep 26 2009

			k=2: 6^2 = 36     = 17 + 19         = prime(7) + prime(8);
k=3: 6^3 = 216    = 107 + 109       = prime(28) + prime(29);
k=7: 6^7 = 279936 = 139967 + 139969 = prime(13005) + prime(13006).

Cf. A071087, A074924.

(* M6 *) Do[If[PreviousPrime[6^n/2]+NextPrime[6^n/2]==6^n,Print[n]],{n,1000}]

is(k) = my(t=6^k); precprime(t/2)+nextprime(1+t/2)==t; \\ Jinyuan Wang, Feb 18 2021

a(7) from Max Alekseyev, Dec 14 2011

a(8) from Amiram Eldar, Apr 06 2019

A175097 Primes in A139013.

Original entry on oeis.org

3, 5, 31, 61, 163, 193, 227, 383, 401, 521, 631, 653, 883, 1019, 1151, 1229, 1433, 1601, 1669, 1801, 1873, 2437, 2729, 3191, 3671, 4013, 4049, 4127, 4447, 5507, 5651, 5701, 5813, 6079, 6199, 6353, 6569, 6823, 6857, 7507, 7529, 7873, 7907, 8291, 8419, 8461

Offset: 1

Zak Seidov, Feb 03 2010

nonn

Cf. A062703, A074924, A139013.

cp's OEIS Frontend

A213739 Numbers n such that n and n^2 are sums of two successive primes.

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A213811 Numbers k such that k and k^3 are sums of two twin primes.

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A225077 Smaller of the two consecutive primes whose sum is a triangular number.

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A251056 Numbers n such that n^2 is a sum of 8 consecutive primes.

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A179485 Sums of two successive primes s such that s+-3 are primes.

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Maple

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A212430 Numbers n such that n and n^4 are sums of two twin primes.

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A226533 a(n) = smallest integer m such that m^n is a sum of two successive primes.

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A297620 Positive numbers n such that n^2 == p (mod q) and n^2 == q (mod p) for some consecutive primes p,q.

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