A064402 -id:A064402 - cp's OEIS frontend

A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.

0, 1, 1, 0, 0, 2, 1, 3, 1, 2, 2, 5, 2, 3, 2, 4, 3, 5, 2, 3, 2, 4, 1, 8, 4, 4, 4, 3, 2, 5, 2, 4, 4, 4, 4, 8, 2, 4, 2, 5, 4, 6, 3, 2, 6, 6, 3, 11, 6, 10, 4, 8, 2, 11, 4, 7, 4, 7, 2, 12, 4, 6, 2, 6, 3, 8, 3, 5, 8, 12, 6, 12, 4, 15, 8, 11, 5, 12, 2, 11

Offset: 1

Zhi-Wei Sun, Mar 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 2, 3, 7, 9, 23. Moreover, for any r = 1,-1 and n > 5*(2+r) there is a positive integer k < n such that 2*k+r and prime(k*n)+k*n are both prime.
(ii) If n > 1 is not equal to 13, then prime(k*n) - k*n is prime for some k = 1, ..., n.
This conjecture implies that there are infinitely many positive integers m with prime(m) + m (or prime(m) - m) prime.

			a(7) = 1 since 11 and prime(7*(11+1)/2) + 7*(11+1)/2 = prime(42) + 42 = 181 + 42 = 223 are both prime.
a(23) = 1 since 7 and prime(23*(7+1)/2) + 23*(7+1)/2 = prime(92) + 92 = 479 + 92 = 571 are both prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169--187. (See Conjecture 3.21(i) and note that the typo 2k+1 there should be 2k-1.)

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A064269, A064402, A238573, A238576, A238878.

PQ[n_]:=PrimeQ[Prime[n]+n]
p[k_,n_]:=PQ[(Prime[k]+1)/2*n]
a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]

a(n) = {my(nb = 0); forprime(p=3, 2*n, if (isprime(prime(n*(p+1)/2) + n*(p+1)/2), nb++);); nb;} \\ Michel Marcus, Sep 21 2015

A231326 Primes p such that p - 2*k is also prime, where p is k-th prime.

Original entry on oeis.org

17, 19, 23, 37, 47, 67, 71, 73, 83, 89, 97, 113, 131, 137, 139, 149, 151, 157, 167, 179, 181, 197, 199, 223, 233, 263, 307, 331, 353, 379, 397, 419, 421, 439, 443, 457, 461, 463, 503, 557, 587, 613, 631, 641, 643, 659, 661, 677, 701, 719, 743, 761, 773, 839, 863

Offset: 1

K. D. Bajpai, Nov 07 2013

nonn

			a(2)= 19 which is 8th prime. prime(8)-2*8= 19-16= 3 which is also prime.
a(6)= 67 which is 19th prime. prime(19)-2*19= 67-38= 29 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5200

Cf. A061068 (primes: prime(m) plus its subscript).
Cf. A064402 (numbers n: prime(n)+n is prime).
Cf. A227420 (primes: p - pi(p) is also prime).
Cf. A231232 (primes: prime(k)+2*k is also prime).

KD := proc() local a,b; a:= ithprime(n); b := a-2*n; if isprime(b) then RETURN (a); fi;end: seq(KD(),n=1..500);

TK = Select[Table[{Prime[n], Prime[n] - 2*n}, {n, 200}], PrimeQ[#[[2]]] &]; Transpose[TK][[1]]

A231506 Primes p such that p + 3k and p - 3k, both are primes, where p is k-th prime.

Original entry on oeis.org

7, 13, 19, 53, 71, 101, 107, 139, 173, 199, 223, 229, 281, 293, 397, 463, 557, 569, 673, 787, 809, 839, 953, 1013, 1283, 1451, 1559, 1657, 1861, 1871, 1877, 1949, 1987, 1997, 2213, 2311, 2347, 2357, 2377, 2503, 2543, 2551, 2593, 2633, 2837, 2851, 2939, 2999, 3041

Offset: 1

K. D. Bajpai, Nov 09 2013

nonn

			a(7)= 107 which is 28th prime. prime(28)-3*28= 107-84= 23: prime(28)+3*28= 107+84= 191: 23 and 191 both are primes.
a(9)= 173 which is 40th prime. prime(40)-3*40= 173-120= 53: prime(40)+3*40= 173+120= 293: 53 and 293 both are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..5200

Cf. A061068 (primes: prime(m) plus its subscript).
Cf. A064402 (numbers n: prime(n)+n is prime).
Cf. A231232 (primes p : p+2*k is also primes).
Cf. A231383 (primes p : p+3*k is also primes).

KD := proc() local a,b,d;  a:= ithprime(n); b:= abs(a-3*n);d:=(a+3*n); if isprime(b) and  isprime(d) then RETURN (a); fi; end: seq(KD(), n=1..500);

A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 4, 2, 4, 4, 2, 4, 3, 5, 1, 4, 2, 3, 1, 2, 2, 2, 1, 1, 0, 0, 1, 4, 0, 1, 2, 0, 5, 2, 4, 4, 1, 3, 3, 3, 2, 3, 8, 2, 2, 3, 5, 5, 4, 3, 5, 3, 4, 3, 1, 3, 8, 4, 5, 4, 2, 6, 0, 12, 2, 4, 1, 5, 0, 4, 1, 4, 3, 3, 2, 5, 4, 7, 5, 3, 11, 1, 5, 4, 3, 4, 6, 2, 2, 5, 5, 6, 4, 4

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 794.
(ii) For any integer n > 59, there is a positive integer k < n such that m = phi(k) + phi(n-k)/4 is an integer with prime(m) - m and prime(m) + m both prime.
Clearly, part (i) of the conjecture implies that there are infinitely many positive integers m with m - 1, m + 1, prime(m) - m and prime(m) + m all prime.

			a(21) = 1 since phi(6) + phi(15)/2 = 6 with 6 - 1 = 5, 6 + 1 = 7, prime(6) - 6 = 7 and prime(6) + 6 = 19 all prime.
a(25) = 1 since phi(17) + phi(8)/2 = 18 with 18 - 1 = 17, 18 + 1 = 19, prime(18) - 18 = 43 and prime(18) + 18 = 79 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A014574, A064269, A064402, A232861, A235682.

q[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]-n]&&PrimeQ[Prime[n]+n]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A231432 Primes p such that abs(p - 3*k) is also prime, where p is the k-th prime.

Original entry on oeis.org

3, 7, 13, 19, 31, 41, 47, 53, 61, 71, 79, 89, 101, 107, 113, 139, 151, 173, 193, 199, 223, 229, 239, 251, 271, 281, 293, 349, 373, 397, 433, 457, 463, 521, 541, 557, 569, 593, 601, 613, 619, 641, 647, 673, 683, 743, 787, 809, 839, 911, 941, 953, 971, 1013, 1049

Offset: 1

K. D. Bajpai, Nov 09 2013

			The first prime, 2, is not a term since |2-3*1| = 1.
The second prime, 3, is a term, since |3-2*3| = 3 is a prime.
a(11) = 79 which is the 22nd prime, prime(22)-3*22 = 79-66 = 13 which is also prime.
a(15) = 113 which is the 30th prime, prime(30)-3*30 = 113-90 = 23 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..6400

Cf. A061068 (primes: prime(m) plus its subscript).
Cf. A064402 (numbers n: prime(n)+n is prime).
Cf. A231232 (primes p : p+2*k is also prime).
Cf. A231383 (primes p : p+3*k is also prime).

KD := proc() local a, b;  a:= ithprime(n); b:= abs(a-3*n); if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..500);

KD = Select[Table[{Prime[n], Prime[n] - 3*n}, {n, 200}], PrimeQ[#[[2]]] &]; Transpose[KD][[1]]
Select[Table[{k,Prime[k]},{k,200}],PrimeQ[Abs[#[[2]]-3#[[1]]]]&][[;;,2]] (* Harvey P. Dale, Jul 14 2024 *)

k=0;forprime(p=2,1e3,if(isprime(abs(p-k++*3)), print1(p", "))) \\ Charles R Greathouse IV, Mar 11 2014

A254867 Numbers n such that prime(n) + n and prime(n) + n^2 are prime.

Original entry on oeis.org

1, 2, 4, 22, 66, 96, 106, 144, 180, 222, 324, 378, 466, 492, 604, 742, 760, 778, 784, 960, 984, 990, 994, 1050, 1150, 1162, 1186, 1248, 1302, 1308, 1356, 1360, 1380, 1744, 1830, 1866, 1870, 1956, 2052, 2070, 2112, 2182, 2212, 2380, 2470, 2556, 2586, 2638, 2676, 2760, 2766

Offset: 1

Zak Seidov, Feb 09 2015

nonn

			a(4) = 22 = A064402(6): prime(22) = 79, 79 + {22, 22^2} = {101, 563} both prime.

Subsequence of A064402. Cf. A000040, A014688, A061067, A061068.

A254867:=n->`if`(isprime(ithprime(n)+n) and isprime(ithprime(n)+n^2), n, NULL): seq(A254867(n), n=1..10^4); # Wesley Ivan Hurt, Jan 16 2017

Select[Range[1000], PrimeQ[Prime[#] + #] && PrimeQ[Prime[#] + #^2] &] (* Alonso del Arte, Feb 09 2015 *)
Select[Range[3000],AllTrue[Prime[#]+{#,#^2},PrimeQ]&] (* Harvey P. Dale, Jan 17 2023 *)

A261832 Numbers n such that prime(n)^3 + n is prime.

Original entry on oeis.org

2, 4, 6, 24, 32, 34, 36, 84, 86, 88, 112, 172, 182, 200, 212, 240, 258, 290, 306, 320, 336, 360, 366, 396, 404, 406, 434, 480, 494, 504, 528, 536, 556, 558, 580, 612, 636, 718, 722, 732, 794, 906, 960, 966, 992, 994, 1008, 1020, 1116, 1132, 1176, 1184, 1186, 1212

Offset: 1

K. D. Bajpai, Sep 02 2015

nonn

			6 is in the list because prime(6)^3 + 6 = 13^3 + 6 = 2197 + 6 = 2203, which is prime.
24 is in the list because prime (24)^3 + 24 = 89^3 + 24 = 704969 + 24 = 704993, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A030078, A064402, A141526.

[n : n in [1..2000] | IsPrime(NthPrime(n)^3 +n)];

select(n -> isprime(ithprime(n)^3 + n), [seq(n,n=1..2000)]);

Select[Range[2000], PrimeQ[Prime[#]^3 + #] &]

for(n = 1,2000, if(isprime(prime(n)^3 + n), print1(n,", ")));

A381792 Numbers k such that k + prime(k) is prime and k + semiprime(k) is semiprime.

Original entry on oeis.org

4, 6, 18, 24, 34, 72, 96, 98, 116, 130, 150, 172, 200, 206, 270, 290, 350, 356, 362, 386, 410, 420, 450, 504, 508, 554, 576, 618, 666, 682, 720, 738, 754, 782, 784, 808, 820, 832, 858, 892, 960, 962, 984, 1016, 1050, 1102, 1110, 1154, 1162, 1168, 1176, 1184, 1206, 1256, 1284, 1296, 1302, 1360

Offset: 1

Zak Seidov and Robert Israel, Mar 07 2025

nonn

All terms are even.

			a(3) = 18 is a term because the 18-th prime and 18-th semiprime are 61 and 51 respectively, 18 + 61 = 79 is prime and 18 + 51 = 69 = 3 * 23 is semiprime.

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

Intersection of A064402 and A100915.

Cf. A000040, A001222, A001358,

N:= 100: # for a(1) .. a(N)
with(priqueue):
initialize(pq);
insert([-4,2,2],pq);
p:= 1:
R:= NULL: count:= 0:
for n from 1 while count < N do
  p:= nextprime(p);
  t:= extract(pq);
  if n::even and isprime(n + p) and numtheory:-bigomega(n - t[1])=2 then R:= R, n; count:= count+1 fi;
  q:= nextprime(t[3]);
  if t[2] = t[3] then insert([-q^2,q,q],pq) fi;
  insert([-t[2]*q,t[2],q],pq);
od:
R;

lim=1360;i=1;Do[Until[PrimeOmega[i]==2,i++];Sp[n]=i,{n,lim}];Select[Range[lim],PrimeQ[#+Prime[#]]&&PrimeOmega[#+Sp[#]]==2&] (* James C. McMahon, Mar 09 2025 *)

A001222(a(n) + A000040(a(n))) = 1 and A001222(a(n) + A001358(a(n))) = 2.

A187766 Even numbers k such that prime(k) -+ k are both composite.

Original entry on oeis.org

12, 20, 36, 38, 40, 46, 52, 56, 58, 60, 62, 64, 74, 78, 80, 86, 88, 112, 118, 120, 122, 124, 128, 132, 134, 136, 138, 140, 142, 146, 156, 160, 162, 164, 166, 170, 176, 182, 184, 186, 188, 190, 194, 198, 208, 210, 212, 216

Offset: 1

Zak Seidov, Jan 04 2013

nonn

Even numbers not in A064269 and not in A064402.

Interestingly, prime(12) = 37 and both 37 - 12 = 25 and 37 + 12 = 49 are squares. Is there another such n?

Cf. A064269, A064402.

Select[2*Range[200],!PrimeQ[Prime[#]+#]&&!PrimeQ[Prime[#]-#]&] (* Harvey P. Dale, Dec 24 2013 *)

{forstep(i=2,220,2,p=prime(i);if(!isprime(p-i)&&!isprime(p+i),print1(i", ")))}

A381630 a(n) is the least k such that the sum of k and the k-th number with n prime factors (counted with multiplicity) has n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 4, 8, 14, 16, 16, 96, 80, 304, 448, 640, 1984, 544, 2048, 3584, 20480, 9216, 49152, 65536, 524288, 1245184, 3309568, 204800, 1179648, 28311552, 2426880, 29360128, 6291456, 27787264, 125829120, 67108864, 327155712, 1073741824

Offset: 1

Robert Israel, Mar 07 2025

			a(3) = 8 because the 8th number with 3 prime factors (the 8th triprime) is 42 = 2*3*7, 8 + 42 = 50 = 2 * 5^2 also has 3 prime factors, and 8 is the smallest number that works.

Cf. A001222, A064402, A100915.

f:= proc(n) uses priqueue; local pq,k,t,i,q;
    initialize(pq);
    insert([-2^n,2$n],pq);
    for k from 1 do
       t:= extract(pq);
       if numtheory:-bigomega(k-t[1])=n then return k fi;
       q:= nextprime(t[-1]);
       for i from 1 to n while t[-i] = t[-1] do
         insert([t[1]*(q/t[-1])^i,op(t[2..n+1-i]),q$i],pq);
       od
    od
end proc:
map(f, [$1..30]); # Robert Israel, Mar 07 2025

a(32) from Jinyuan Wang, Mar 09 2025

a(33) from Jinyuan Wang, Mar 21 2025

cp's OEIS Frontend

A238881 Number of odd primes p < 2*n with prime(n*(p+1)/2) + n*(p+1)/2 prime.

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A231326 Primes p such that p - 2*k is also prime, where p is k-th prime.

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A231506 Primes p such that p + 3*k and p - 3*k, both are primes, where p is k-th prime.

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A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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A231432 Primes p such that abs(p - 3*k) is also prime, where p is the k-th prime.

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A254867 Numbers n such that prime(n) + n and prime(n) + n^2 are prime.

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A261832 Numbers n such that prime(n)^3 + n is prime.

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A381792 Numbers k such that k + prime(k) is prime and k + semiprime(k) is semiprime.

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A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.

A231506 Primes p such that p + 3k and p - 3k, both are primes, where p is k-th prime.