A157468 -id:A157468 - cp's OEIS frontend

A238086 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k is the increasing list of all primes p such that (p+k)^2+k is prime but (p+j)^2+j is not prime for all 0

3, 7, 5, 11, 31, 13, 29, 47, 37, 19, 193, 41, 59, 43, 23, 139, 331, 113, 61, 79, 53, 107, 523, 409, 163, 67, 97, 73, 181, 293, 563, 457, 173, 71, 103, 83, 101, 277, 359, 769, 487, 199, 127, 241, 89, 17, 191, 541, 389, 853, 787, 211, 131, 271, 109

Offset: 1

Alois P. Heinz, Feb 17 2014

			Column k=3 contains prime 47 because (47+3)^2+3 = 2503 is prime and (47+2)^2+2 = 2403 = 3^3*89 and (47+1)^2+1 = 2305 = 5*461 are composite.
Square array A(n,k) begins:
:   3,   7,  11,  29, 193,  139, 107,  181, ...
:   5,  31,  47,  41, 331,  523, 293,  277, ...
:  13,  37,  59, 113, 409,  563, 359,  541, ...
:  19,  43,  61, 163, 457,  769, 389,  937, ...
:  23,  79,  67, 173, 487,  853, 397, 1381, ...
:  53,  97,  71, 199, 787, 1019, 401, 1741, ...
:  73, 103, 127, 211, 829, 1489, 433, 2551, ...
:  83, 241, 131, 251, 991, 1553, 461, 2617, ...

Alois P. Heinz, Antidiagonals n = 1..100, flattened

Column k=1-10 give: A157468, A238664, A238665, A238666, A238667, A238668, A238669, A238670, A238671, A238672.
Rows n=1-10 give: A238673, A238674, A238675, A238676, A238677, A238678, A238679, A238680, A238681, A238682.
Main diagonal gives A238663.
Cf. A238048.

A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

nmax = 12;
col[k_] := col[k] = Reap[For[cnt = 0; p = 2, cnt < nmax, p = NextPrime[p], If[PrimeQ[(p+k)^2+k] && AllTrue[Range[k-1], !PrimeQ[(p+#)^2+#]&], cnt++; Sow[p]]]][[2, 1]];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 03 2019 *)

A246748 Numbers n such that A242719(n) = (prime(n))^2+1 and A242720(n) - A242719(n) = 2*(prime(n)+1).

Original entry on oeis.org

3, 52, 104, 209, 343, 373, 398, 473, 628, 2633, 3273, 7538, 8060, 8813, 9025, 10847, 12493, 13768, 14196, 15486, 16865, 17486, 18362, 18613, 18842, 21175, 23522, 31825, 33537, 34507, 38740, 39603, 41802, 41947, 43314, 45479, 47550, 47668, 47787, 50321, 50682

Offset: 1

Vladimir Shevelev, Sep 02 2014

nonn

If the sequence is infinite, then lim inf(A242719(k)/(prime(k))^2) = 1 and lim inf(A242720(k)/(prime(k))^2) = 1.
In connection with this, one can conjecture that A242719(k) ~ A242720(k) ~ (prime(k))^2, as k goes to infinity (cf. A246819, A246821).
n is in the sequence if and only if prime(n)>=5 and is in the intersection of A001359, A062326, A157468.
Proof. Firstly note that A242719(n) = prime(n)^2 + 1 if and only if prime(n)^2 - 2 is prime. Indeed, let prime(n)^2 + 1 be A242719(n). Then we have lpf(prime(n)^2 - 2) > lpf(prime(n)^2) = prime(n). It is possible only when prime(n)^2 - 2 is prime, i. e., prime(n) is in A062326. Add that prime(n)^2+1 is the smallest value of A242719(n).
Let A242720(n) = A242719(n) + 2*prime(n) + 2 = prime(n)^2 + 2*prime(n) + 3. Then, by the definition of A242720, we have lpf(prime(n)^2 + 2*prime(n) + 2) > lpf(prime(n)*(prime(n)+2)) >= prime(n). Thus prime(n) + 2 is prime, i.e., prime(n) is in A001359. Besides, lpf(prime(n)^2 + 2*prime(n) + 2) > prime(n), or lpf((prime(n)+1)^2 + 1) >= prime(n+1) = prime(n) + 2. So (prime(n)+1)^2+1 is prime, i.e., prime(n) is also in A157468.
Add that, for n>=3, N=prime(n)^2 + 2*prime(n) + 3 is the smallest possible value of A242720(n). Indeed, let prime(n)^2+1 <= N <= prime(n)^2 + 2*prime(n) + 2. Then prime(n)^2-2 <= N - 3 <= prime(n)^2 + 2*prime(n) - 1. Since it should be lpf(N-3) >= prime(n), then there are only two possibilities: N-3 = prime(n)^2 + prime(n) or N-3 = prime(n)^2. However, lpf(prime(n)^2 + prime(n)) = 2, while, although lpf(prime(n)^2) = prime(n), however, in this case, lpf(N-1) = lpf(prime(n)^2+2) = 3, n>=3, and, so the inequalities lpf(N-1) > lpf(N-3) >= prime(n) are impossible in the considered cases for n>=3. - Vladimir Shevelev, Sep 03 2014

Cf. A001359, A062326, A157468, A242719, A242720, A246819, A246821.

More terms from Peter J. C. Moses, Sep 02 2014

A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

Original entry on oeis.org

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840

Offset: 1

Labos Elemer, Apr 23 2002

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008
Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012
Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012
{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

			150 is a term since 149, 151 and 22501 are all primes.

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

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014

Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014

A246824 Numbers k for which A242720(k) = (prime(k)+1)^2 + 2.

Original entry on oeis.org

3, 35, 41, 52, 57, 81, 104, 209, 215, 343, 373, 398, 473, 477, 584, 628, 768, 774, 828, 872, 1117, 1145, 1189, 1287, 1324, 1435, 1615, 1634, 1653, 1704, 1886, 1925, 2070, 2075, 2123, 2171, 2193, 2425, 2449, 2605, 2633, 2934, 2948, 3019, 3194, 3273, 3533, 3552, 3685, 3758

Offset: 1

Vladimir Shevelev, Sep 04 2014

nonn

By a comment in A246748, A242720(k) >= (prime(k)+1)^2 + 2, and equality is attained in this sequence.

Prime(a(n)) >= 5 and is in the intersection of A001359 and A157468.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001359, A157468, A242719, A242720, A246748, A246819, A246821.

lpf[n_] := FactorInteger[n][[1, 1]]; aQ[n_] := Module[{k=6}, While[PrimeQ[k-3] && PrimeQ[k-1] || lpf[k-1]<=lpf[k-3] || lpf[k-3]Amiram Eldar, Dec 10 2018 *)

lpf(k) = factorint(k)[1, 1];
f(n) = my(k=6); while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)A242720
isok(n) = f(n) == (prime(n)+1)^2 + 2; \\ Michel Marcus, Dec 10 2018

from sympy import prime, isprime, factorint
A246824_list = [a for a, b in ((n, prime(n)+1) for n in range(3,10**3)) if (not (isprime(b**2-1) and isprime(b**2+1)) and (min(factorint(b**2+1)) > min(factorint(b**2-1)) >= b-1))] # Chai Wah Wu, Jun 03 2019

a(40)-a(50) from b-file by Robert Price, Sep 08 2019

A238048 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k is the increasing list of all primes p such that (p+k)^2+k is also prime.

Original entry on oeis.org

3, 7, 5, 5, 13, 13, 3, 7, 19, 19, 7, 11, 11, 31, 23, 5, 31, 13, 19, 37, 53, 3, 13, 43, 23, 47, 43, 73, 7, 5, 19, 67, 29, 59, 79, 83, 11, 13, 11, 29, 73, 31, 61, 97, 89, 3, 23, 43, 19, 59, 109, 41, 67, 103, 109, 13, 17, 29, 73, 23, 73, 157, 43, 71, 109, 149

Offset: 1

Alois P. Heinz, Feb 17 2014

Prime 2 is not contained in this array.

			Column k=3 contains prime 47 because (47+3)^2+3 = 2503 is prime.
Square array A(n,k) begins:
   3,  7,  5,  3,   7,   5,  3,   7, ...
   5, 13,  7, 11,  31,  13,  5,  13, ...
  13, 19, 11, 13,  43,  19, 11,  43, ...
  19, 31, 19, 23,  67,  29, 19,  73, ...
  23, 37, 47, 29,  73,  59, 23,  79, ...
  53, 43, 59, 31, 109,  73, 29, 103, ...
  73, 79, 61, 41, 157,  83, 31, 109, ...
  83, 97, 67, 43, 163, 103, 41, 127, ...

Alois P. Heinz, Antidiagonals n = 1..150, flattened

Column k=1 gives A157468.

Cf. A238086.

A:= proc(n, k) option remember; local p;
      p:= `if`(n=1, 1, A(n-1, k));
      do p:= nextprime(p);
         if isprime((p+k)^2+k) then return p fi
      od
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..11);

A[n_, k_] := A[n, k] = Module[{p}, For[p = If[n == 1, 1, A[n-1, k]] // NextPrime, True, p = NextPrime[p], If[PrimeQ[(p+k)^2+k], Return[p]]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 11}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

A343148 Numbers k such that A083266(k) is prime.

Original entry on oeis.org

2, 6, 10, 15, 21, 26, 28, 30, 35, 38, 39, 40, 42, 44, 45, 46, 51, 55, 60, 63, 68, 69, 70, 78, 84, 93, 95, 96, 102, 105, 106, 116, 123, 124, 126, 130, 135, 136, 138, 143, 146, 150, 153, 155, 166, 174, 176, 178, 201, 203, 205, 218, 219, 221, 222, 231, 232, 234, 236, 240, 244, 245, 246, 248, 249

Offset: 1

J. M. Bergot and Robert Israel, Apr 06 2021

nonn

Includes 2*p for p in A157468.

			a(3) = 10 is a term because A083266(10) = 37 is prime.

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

Cf. A083266, A157468, A343151.

f:= n -> numtheory:-sigma(n) + n*numtheory:-phi(n)/2 - 1:
select(t -> isprime(f(t)), [$2..1000]);

Select[Range[250], PrimeQ[DivisorSigma[1, #] + # EulerPhi[#]/2 - 1] &] (* Michael De Vlieger, Apr 07 2021 *)

A359185 Numbers k such that for any positive integers x,y, if x*y=k then (x+y)^2+1 is a prime number.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 23, 25, 39, 53, 55, 73, 83, 89, 109, 119, 133, 149, 155, 159, 169, 179, 203, 223, 229, 239, 263, 269, 283, 299, 305, 313, 339, 349, 383, 395, 419, 439, 443, 463, 469, 473, 543, 569, 593, 643, 653, 673, 689, 699, 703, 713, 739, 763, 859, 863, 889, 909

Offset: 1

Michel Lagneau, Dec 19 2022

nonn

Conjecture: if a term k is a perfect square > 1, then sqrt(k) is in the sequence A236068 (Primes p such that f(f(p)) is prime, where f(z) = z^2 + 1).
The conjecture is false.  A counterexample is 296147^2 = 87703045609 where 296147 = 47 * 6301. - Robert Israel, Mar 05 2024
The primes of the sequence are in A157468.
All terms except 1 are congruent to 3, 5 or 9 (mod 10). - Robert Israel, Mar 05 2024

			909 is in the sequence because 909 = 3^2*101 with 3 decompositions:
909 = 1*909 and (1+909)^2+1 = 910^2+1 = 828101 is prime;
909 = 3*303 and (3+303)^2+1 = 306^2+1 = 93637 is prime;
909 = 9*101 and (9+101)^2+1 = 110^2+1 = 12101 is prime.

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

Cf. A001358, A002496, A134406, A157468, A236068.

filter:= proc(n) local F;
  F:= select(t -> t^2 <= n, numtheory:-divisors(n));
  andmap(t -> isprime((t + n/t)^2+1), F)
end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Mar 05 2024

t={};Do[ds=Divisors[n];k=1;While[k<=(Length[ds]+1)/2&&(ok=PrimeQ[(ds[[k]]+ds[[-k]])^2+1]),k++];If[ok,AppendTo[t,n]],{n,1,2000}];t

isok(k) = fordiv(k, d, if ((d<=k/d) && !isprime((d+k/d)^2+1), return(0));); return(1); \\ Michel Marcus, Dec 19 2022

A157473 Primes p such that (p-2)^(1/3) -+ 2 are also primes.

Original entry on oeis.org

2, 127, 91127, 328511, 1157627, 2146691, 12326393, 125751503, 693154127, 751089431, 1033364333, 2102071043, 2222447627, 2893640627, 3314613773, 3951805943, 6591796877, 9063964127, 13464285941, 16406426423, 19880486831

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

			(127-2)^(1/3) - 2 = 3 and (127-2)^(1/3) + 2 = 7, so 127 is in the sequence.

Cf. A127435, A127436, A157467, A157468.

q=2;lst={};Do[p=Prime[n];r=(p-q)^(1/3)-q;u=(p-q)^(1/3)+q;If[PrimeQ[r]&&PrimeQ[u],AppendTo[lst,p]],{n,4*9!}];lst
lst = {}; p = 0; While[p < 2955, If[ PrimeQ[p - 2] && PrimeQ[p + 2] && PrimeQ[p^3 + 2], AppendTo[lst, p^3 + 2]]; p++ ]; lst (* Robert G. Wilson v, Mar 08 2009 *)
Select[Prime[Range[10^6]],AllTrue[Surd[#-2,3]+{2,-2},PrimeQ]&] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Aug 31 2024 *)

a(8)-a(21) from Robert G. Wilson v, Mar 08 2009

cp's OEIS Frontend

A238086 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k is the increasing list of all primes p such that (p+k)^2+k is prime but (p+j)^2+j is not prime for all 0

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A246748 Numbers n such that A242719(n) = (prime(n))^2+1 and A242720(n) - A242719(n) = 2*(prime(n)+1).

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A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

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A246824 Numbers k for which A242720(k) = (prime(k)+1)^2 + 2.

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A238048 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k is the increasing list of all primes p such that (p+k)^2+k is also prime.

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A343148 Numbers k such that A083266(k) is prime.

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A359185 Numbers k such that for any positive integers x,y, if x*y=k then (x+y)^2+1 is a prime number.

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A157473 Primes p such that (p-2)^(1/3) -+ 2 are also primes.

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