cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A260120 Least integer k > 0 such that (prime(k*n)-1)^2 = prime(j*n)-1 for some j > 0.

Original entry on oeis.org

1, 2, 14, 1, 12, 9, 30, 198, 69, 83, 66, 132, 44, 15, 4, 99, 71, 88, 339, 230, 10, 33, 167, 66, 42, 22, 126, 442, 318, 1185, 29, 289, 37, 174, 157, 44, 146, 301, 171, 403, 2, 5, 26, 699, 573, 144, 338, 33, 2032, 1212, 404, 11, 135, 267, 380, 221, 447, 159, 898, 1397
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 17 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 0. In general, if a,b,c and m are integers with a > 0, gcd(a,b,c-m) = 1 and c == (a+b+1)*(m+1) (mod 2) such that b^2-4a*(c-m) is not a square and gcd(a*m-b,b^2+b-a*c-1) is not divisible by 3, then for any positive integer n there are two elements x and y of the set {prime(k*n)+m: k = 1,2,3,...} with a*x^2+b*x+c = y.
This implies the conjecture in A259731.

Examples

			a(3) = 14 since (prime(14*3)-1)^2 = 180^2 = prime(3477)-1 = prime(1159*3)-1.
a(63) = 5162 since (prime(5162*63)-1)^2 = 4642456^2 = 21552397711936 = prime(726521033763)-1 = prime(11532079901*63)-1.

References

  • 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.

Links

Crossrefs

Cf. A000040, A002496, A259731, A260082, A260121.

Programs

  • Mathematica
    P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[aa];k=k+1; If[P[n,(Prime[k*n]-1)^2+1],Goto[bb]];Goto[aa];Label[bb];Print[n, " ", k];Continue,{n,1,60}]

A261395 Primes p such that (prime(p)-1)^2 = (prime(q)-1)*(prime(r)-1) for some pair of distinct primes q and r.

Original entry on oeis.org

13, 47, 137, 191, 193, 223, 227, 313, 701, 857, 907, 947, 991, 1009, 1069, 1291, 1531, 1889, 2281, 2411, 2447, 2647, 3181, 3389, 3539, 3593, 4093, 4099, 4409, 4481, 4603, 4721, 5557, 5647, 6581, 6793, 6869, 6961, 7211, 7349, 7523, 7723, 7753, 8461, 8537, 8543, 8807, 9137, 9241, 9281
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 17 2015

Keywords

Comments

Conjecture: Let d be any nonzero integer. Then there are infinitely many prime triples (p,q,r) with p,q,r distinct such that (prime(p)+d)^2 = (prime(q)+d)*(prime(r)+d). In other words, the set {prime(p)+d: p is prime} contains infinitely many nontrivial three-term geometric progressions.

Examples

			a(1) = 13 since (prime(13)-1)^2 = (41-1)^2 = 1600 = (5-1)*(401-1) = (prime(3)-1)*(prime(79)-1) with 13, 3, 79 all prime.
a(2) = 47 since (prime(47)-1)^2 = 210^2 = 44100 = 30*1470 = (prime(11)-1)*(prime(233)-1) with 47, 11, 233 all prime.

References

  • 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.

Links

Crossrefs

Cf. A000040, A000290, A260082, A261352, A261385.

Programs

  • Mathematica
    Dv[n_]:=Divisors[n]
L[n_]:=Length[Dv[n]]
f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0;Do[Do[If[PQ[Part[Dv[f[k]^2],i]+1]&&PQ[Part[Dv[f[k]^2],L[f[k]^2]-i+1]+1],n=n+1;Print[n," ",Prime[k]];Goto[aa]];Continue,{i,1,(L[f[k]^2]-1)/2}];
Label[aa];Continue,{k,1,1150}]

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.

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

Views

Author

Zhi-Wei Sun, Jul 15 2015

Keywords

Comments

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.

Examples

			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.

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    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}]
Showing 1-3 of 3 results.

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