A069230 -id:A069230 - cp's OEIS frontend

A336435 a(n) is the least k such that A069230(k) = n.

1, 3, 2, 4, 14, 6, 16, 63, 20, 18, 12, 64, 102, 66, 24, 40, 100, 208, 112, 36, 650, 444, 200, 198, 228, 160, 84, 72, 96, 60, 1610, 320, 1110, 192, 1218, 324, 400, 728, 462, 144, 280, 264, 270, 168, 120, 882, 828, 468, 980, 588, 288, 252, 300, 1040, 1104, 180, 880

Offset: 0

David A. Corneth, Sep 20 2020

nonn

			a(10) = 12 as A069230(12) = 10 as there are 10 primes between (exclusive) 12 and 12 + tau(12)^2 = 12 + 6^2 = 12 + 36 = 48 namely the 10 primes 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 and k = 12 is the least k such that there are ten primes between (exclusive) k and k + tau(k)^2 where tau is the number of divisors (Cf. A000005).

Amiram Eldar, Table of n, a(n) for n = 0..2500

Cf. A000005, A069230.

A069230(n) = {my(cnt = 0); for(k = n+1, n+numdiv(n)^2-1, cnt += isprime(k)); cnt;};
list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = A069230(k) + 1; if(i <= nmax + 1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 29 2025

A069231 Numbers n such that there are exactly 3 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

Original entry on oeis.org

4, 9, 21, 51, 55, 62, 74, 77, 82, 86, 87, 91, 106, 122, 123, 129, 134, 142, 143, 145, 146, 155, 158, 159, 161, 177, 183, 214, 215, 217, 237, 249, 254, 259, 265, 274, 278, 298, 299, 301, 309, 334, 335, 339, 341, 343, 358, 365, 371, 377, 382, 386, 394, 395, 407

Offset: 1

Benoit Cloitre, Apr 13 2002

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

Cf. A000005, A049591, A069230, A069232, A069233.

filter:= n -> nops(select(isprime, [$(n+1) .. (n+numtheory:-tau(n)^2-1)]))=3:
select(filter, [$1..1000]); # Robert Israel, Jan 05 2018

fQ[n_] := Block[{r = Range[n, n + DivisorSigma[0, n]^2]}, If[ PrimeQ@ n, r = Rest@ r]; If[ PrimeQ[ r[[-1]]], r = Most@ r]; Length@ Select[r, PrimeQ] == 3]; Select[Range@410, fQ] (* Robert G. Wilson v, Jan 05 2018 *)

isok(n) = #select(x->isprime(x), vector(numdiv(n)^2-1, k, k+n)) == 3; \\ Michel Marcus, Jun 18 2017

A069232 Numbers n such that there are exactly 2 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

Original entry on oeis.org

2, 25, 85, 118, 119, 133, 141, 194, 209, 213, 235, 247, 253, 323, 326, 327, 329, 355, 362, 381, 391, 393, 398, 411, 413, 415, 422, 445, 466, 473, 481, 482, 493, 501, 502, 511, 514, 515, 517, 519, 533, 535, 537, 538, 542, 543, 545, 551, 553, 573, 579, 581, 583

Offset: 1

Benoit Cloitre, Apr 13 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A049591, A069230, A069231, A069233.

Select[Range[900],Length[Select[Range[#+1,#+DivisorSigma[0,#]^2-1],PrimeQ]] == 2&] (* Harvey P. Dale, Sep 20 2020 *)

isok(n) = #select(x->isprime(x), vector(numdiv(n)^2-1, k, k+n)) == 2; \\ Michel Marcus, Jun 18 2017

Missing term 2 added by Michel Marcus, Jun 18 2017

A069233 Numbers k such that there is exactly 1 prime p satisfying the inequality k < p < k + tau(k)^2 where tau(k) = A000005(k).

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 49, 59, 71, 101, 107, 111, 115, 121, 137, 149, 169, 179, 191, 197, 201, 202, 203, 205, 206, 227, 239, 269, 281, 287, 289, 291, 295, 311, 314, 319, 321, 347, 361, 403, 419, 431, 461, 469, 471, 505, 521, 526, 527, 569, 599, 617, 622, 623

Offset: 1

Benoit Cloitre, Apr 13 2002

Numbers k such that A069230(k) = 1. - Amiram Eldar, Jan 29 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A049591, A069230, A069231, A069232.

q[k_] := Count[k + Range[1, DivisorSigma[0, k]^2 - 1], ?PrimeQ] == 1; Select[Range[640], q] (* _Amiram Eldar, Jan 29 2025 *)

isok(n) = #select(x->isprime(x), vector(numdiv(n)^2-1, k, k+n)) == 1; \\ Michel Marcus, Jun 18 2017

cp's OEIS Frontend

A336435 a(n) is the least k such that A069230(k) = n.

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A069231 Numbers n such that there are exactly 3 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

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A069232 Numbers n such that there are exactly 2 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

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A069233 Numbers k such that there is exactly 1 prime p satisfying the inequality k < p < k + tau(k)^2 where tau(k) = A000005(k).

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