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A262447 Primes p such that pi(p^2) = pi(q^2) + pi(r^2) for some distinct primes q and r. 7
13, 53, 73, 131, 199, 277, 281, 283, 313, 353, 641, 643, 647, 701, 773, 839, 887, 977, 1033, 1103, 1117, 1163, 1187, 1223, 1259, 1409, 1433, 1439, 1487, 1489, 1583, 1721, 1913, 1931, 2239, 2243, 2269, 2309, 2371, 2441, 2473, 2477, 2621, 2683, 2707, 2797, 2843, 2851, 2953, 3049, 3137, 3257, 3307, 3499, 3511, 3613, 3659, 3769, 3779, 3911 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: The sequence has infinitely many terms.

See also A262408 and A262443 for related conjectures.

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

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Zhi-Wei Sun)

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

a(1) = 13 since pi(13^2) = pi(169) = 39 = 9 + 30 = pi(5^2) + pi(11^2) with 13, 5 and 11 distinct primes.

MATHEMATICA

f[n_]:=PrimePi[Prime[n]^2]

T[n_]:=Table[f[k], {k, 1, n}]

n=0; Do[Do[If[2*f[k]>=f[m], Goto[aa]]; If[MemberQ[T[m-1], f[m]-f[k]], n=n+1; Print[n, " ", Prime[m]]; Goto[aa]]; Continue, {k, 1, m-1}]; Label[aa]; Continue, {m, 1, 541}]

CROSSREFS

Cf. A000040, A000290, A000720, A262408, A262443.

Sequence in context: A207977 A056255 A141885 * A165352 A262287 A307749

Adjacent sequences:  A262444 A262445 A262446 * A262448 A262449 A262450

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 23 2015

STATUS

approved

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Last modified October 4 21:22 EDT 2022. Contains 357240 sequences. (Running on oeis4.)