A257926 -id:A257926 - cp's OEIS frontend

A260078 Least positive integer k such that prime(kn)-1+(prime(hn)-1) = prime(in)-1 and prime(kn)-1-(prime(hn)-1) = prime(jn)-1 for some positive integers h,i,j.

3, 3, 15, 5, 25, 29, 32, 20, 41, 87, 17, 61, 18, 100, 58, 10, 82, 82, 45, 74, 166, 20, 28, 338, 18, 35, 159, 290, 64, 29, 353, 311, 75, 41, 42, 492, 107, 155, 77, 364, 100, 330, 145, 474, 502, 332, 227, 553, 238, 92, 121, 597, 338, 339, 452, 164, 239, 832, 221, 243

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, if m and n > 0 are integers with gcd(6,m) = 1, then the set {prime(k*n)+m: k = 1,2,3,...} contains two distinct elements x and y with x+y and x-y also in the set.

			a(2) = 3 since prime(3*2)-1+(prime(2*2)-1) = 12+6 = 18 = prime(4*2)-1, and prime(3*2)-1-(prime(2*2)-1) = 12-6 = 6 = prime(2*2)-1.
a(3) = 15 since prime(15*3)-1+(prime(12*3)-1) = 196+150 = 346 = prime(23*3)-1, and prime(15*3)-1-(prime(12*3)-1) = 196 -150 = 46 = prime(5*3)-1.
a(200) = 3319 since prime(3319*200)-1+(prime(2821*200)-1) = 9987120+8389110 = 18376230 = prime(5869*200)-1, and prime(3319*200)-1-(prime(2821*200)-1) = 9987120-8389110 = 1598010 = prime(605*200)-1.

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A257926, A257938.

f[n_]:=Prime[n]-1
PQ[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[bb];k=k+1;Do[If[PQ[n,f[k*n]+f[j*n]+1]&&PQ[n,f[k*n]-f[j*n]+1],Goto[aa]],{j,1,k-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,60}]

A260080 Least positive integer k such that prime(kn)^2 - 2 = prime(in)prime(jn) for some integers 0 < i < j.

Original entry on oeis.org

5, 18, 18, 9, 115, 208, 69, 373, 68, 430, 8, 214, 57, 1887, 1255, 295, 880, 542, 5612, 767, 1562, 40, 853, 884, 753, 4332, 4750, 6077, 799, 1394, 639, 5442, 4785, 440, 7417, 1290, 15830, 27745, 3927, 5701, 1891, 22008, 8243, 6031, 9172, 5949, 43286, 20778, 9876, 12472

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0.

			a(1) = 5 since prime(5*1)^2-2 = 11^2-2 = 119 = 7*17 = prime(4*1)*prime(7*1).
a(66) = 149073 since prime(149073*66)^2-2 = 176365951^2-2 = 31104948672134399 = 3160879*9840600881 = prime(3448*66)*prime(9840600881*66).

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..105
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A062326, A253257, A257926, A257938, A260078.

Dv[n_]:=Divisors[Prime[n]^2-2]
L[n_]:=Length[Dv[n]]
P[k_,n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n],2]]&&Mod[PrimePi[Part[Dv[k*n],2]],n]==0&&PrimeQ[Part[Dv[k*n],3]]&&Mod[PrimePi[Part[Dv[k*n],3]],n]==0
Do[k=0;Label[bb];k=k+1;If[P[k,n],Goto[aa]];Goto[bb];Label[aa];Print[n," ", k];Continue,{n,1,50}]

A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

Original entry on oeis.org

2, 2, 2, 21, 9, 10, 12, 14, 47, 32, 32, 171, 177, 175, 64, 187, 330, 206, 77, 467, 4, 126, 127, 355, 279, 982, 249, 1930, 105, 109, 659, 801, 269, 777, 703, 125, 819, 1347, 904, 1153, 549, 2344, 757, 1301, 1793, 303, 105, 3168, 2645, 3055, 110, 1619, 1580, 2423, 220, 965, 1397, 84, 988, 322

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n there are distinct positive integers i,j,k such that (prime(i*n)+m)*(prime(j*n)+m) = (prime(k*n)+m)^2.

			a(4) = 21 since (prime(21*4)-1)^2 = 432^2 = 18*10368 = (prime(2*4)-1)*(prime(318*4)-1).
a(61) = 15160 since (prime(15160*61)-1)^2 = 14242116^2 = 47316*4286876916 = (prime(80*61)-1)*(prime(3326491*61)-1).

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A257926, A257928, A257938, A260078, A260080.

Dv[n_]:=Divisors[(Prime[n]-1)^2]
L[n_]:=Length[Dv[n]]
P[k_,n_,i_]:=PrimeQ[Part[Dv[k*n],i]+1]&&Mod[PrimePi[Part[Dv[k*n],i]+1],n]==0
Do[k=0;Label[bb];k=k+1; Do[If[P[k,n,i]&&P[k,n,L[k*n]-i+1],Goto[aa]],{i,1,L[k*n]/2}];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

A261282 Least positive integer k such that prime(k)prime(kn) = prime(p)+2 for some prime p.

Original entry on oeis.org

14, 60, 135, 41, 199, 2, 2, 2, 61, 2, 183, 25, 15, 12, 47, 143, 110, 294, 117, 88, 22, 402, 26, 269, 116, 145, 164, 6, 10, 488, 2, 44, 120, 4, 127, 144, 119, 704, 1058, 368, 104, 2, 6, 214, 4, 129, 2, 3, 301, 2, 2, 466, 20, 107, 280, 14, 337, 12, 22, 12, 242, 1705, 415, 10, 115, 50, 2, 420, 4, 15

Offset: 1

Zhi-Wei Sun, Aug 14 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(m)*prime(n) = prime(p)+2 for some prime p.
For example, 14/19 = 24528/33288, and prime(24528)*prime(33288) = 281153*392723 = 110415249619 = prime(4528436431)+2 with 4528436431 prime.
The conjecture implies that there are infinitely many primes p such that prime(p)+2 is a product of two primes. Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes.

			a(2) = 60 since prime(60)*prime(60*2) = 281*659 = 185179 = prime(16763)+2 with 16763 prime.

Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A109611, A257926, A259487, A261281.

f[n_]:=Prime[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]

A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

Original entry on oeis.org

3, 9, 4, 127, 98, 133, 55, 78, 65, 85, 375, 109, 251, 283, 105, 462, 681, 149, 156, 213, 525, 209, 205, 381, 757, 313, 252, 615, 61, 737, 478, 1754, 406, 1197, 131, 420, 492, 503, 127, 119, 549, 1748, 95, 442, 2740, 555, 677, 1258, 163, 816, 1649, 710, 203, 126, 628, 582, 1004, 135, 837, 1000

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any positive integers a, m and n, there are integers i,j,k > 0 with i > j such that (prime(i*n)+prime(j*n))/2 (or (prime(i*n)-prime(j*n))/2) is equal to a*prime(k*n)^m.

			a(1) = 3 since (prime(2*1)+prime(3*1))/2 = (3+5)/2 = 2^2 = prime(1*1)^2.
a(158) = 8405 since (prime(778*158)+prime(8405*158))/2 = (1625551+20967091)/2 = 3361^2 = prime(3*158)^2.

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..160
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A257926, A257938, A260078, A260080, A260082.

PQ[n_,m_]:=PrimeQ[Sqrt[m]]&&Mod[PrimePi[Sqrt[m]],n]==0
Do[k=0;Label[bb];k=k+1;Do[If[PQ[n,(Prime[k*n]+Prime[j*n])/2],Goto[aa]];Continue,{j,1,k-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,60}]

A321855 Number of permutations f of {1,...,n} such that prime(k)*prime(f(k)) - 2 is prime for every k = 1,...,n.

Original entry on oeis.org

1, 1, 2, 3, 5, 12, 2, 3, 65, 248, 448, 1792, 4288, 6468, 27068, 29752, 106066, 447982, 1250762, 6304196, 46613084, 126391780, 504582496, 2270372946, 3028652541, 8941959118, 36442298864, 175008626450, 318369805106, 1974700703920, 6654020288821, 48819526290634, 150577775767875, 574885284627624, 3058310882340228, 15949743649457780

Offset: 1

Zhi-Wei Sun, Nov 19 2018

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for each n > 0, there is an even permutation f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n. Also, for any integer n > 2, there is an odd permutation f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n.
If we let b(n) denote the number of even permutations f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n, then (b(1),...,b(11)) = (1,1,1,1,3,6,1,1,33,125,226).
In 1973 J.-R. Chen proved that there are infinitely many primes p with p + 2 a product of at most two primes, such primes p are now called Chen primes.

			a(7) = 2. The only even permutation of {1,...,7} meeting the requirement is (1,5,7,4,2,6,3) with prime(1)*prime(1) - 2 = 2, prime(2)*prime(5) - 2 = 31, prime(3)*prime(7) - 2 = 83, prime(4)*prime(4) - 2 = 47, prime(5)*prime(2) - 2 = 31, prime(6)*prime(6) - 2 = 167 and prime(7)*prime(3) - 2 = 83 all prime. Also, the only odd permutation of {1,...,7} meeting the requirement is (1,5,7,6,2,4,3) with prime(1)*prime(1) - 2 = 2, prime(2)*prime(5) - 2 = 31, prime(3)*prime(7) - 2 = 83, prime(4)*prime(6) - 2 = 89, prime(5)*prime(2) - 2 = 31, prime(6)*prime(4) - 2 = 89 and prime(7)*prime(3) - 2 = 83 all prime.

Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), pp. 157-176.
Zhi-Wei Sun, Chen primes and permutations, Question 315679 on Mathoverflow, Nov. 19, 2018.
Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018.

Cf. A000040, A109611, A257926, A321597, A321610, A321611, A321727, A321805.

Permanent[m_List]:=With[{v = Array[x, Length[m]]},Coefficient[Times @@ (m.v), Times @@ v]];
a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[Prime[i]*Prime[j]-2]],{i,1,n},{j,1,n}]];
Do[Print[n," ",a[n]],{n,1,27}]

a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(prime(i)*prime(j) - 2))); \\ Jinyuan Wang, Jun 13 2020

a(28)-a(29) from Jinyuan Wang, Jun 13 2020

a(30)-a(36) from Vaclav Kotesovec, Aug 20 2021

cp's OEIS Frontend

A260078 Least positive integer k such that prime(k*n)-1+(prime(h*n)-1) = prime(i*n)-1 and prime(k*n)-1-(prime(h*n)-1) = prime(j*n)-1 for some positive integers h,i,j.

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A260080 Least positive integer k such that prime(k*n)^2 - 2 = prime(i*n)*prime(j*n) for some integers 0 < i < j.

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A260082 Least positive integer k such that (prime(k*n)-1)^2 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j.

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A261282 Least positive integer k such that prime(k)*prime(k*n) = prime(p)+2 for some prime p.

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A258580 Least positive integer k such that (prime(j*n)+prime(k*n))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

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A321855 Number of permutations f of {1,...,n} such that prime(k)*prime(f(k)) - 2 is prime for every k = 1,...,n.

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Extensions

A260078 Least positive integer k such that prime(kn)-1+(prime(hn)-1) = prime(in)-1 and prime(kn)-1-(prime(hn)-1) = prime(jn)-1 for some positive integers h,i,j.

A260080 Least positive integer k such that prime(kn)^2 - 2 = prime(in)prime(jn) for some integers 0 < i < j.

A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

A261282 Least positive integer k such that prime(k)prime(kn) = prime(p)+2 for some prime p.

A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.