A053182 -id:A053182 - cp's OEIS frontend

A238399 a(n) is the number of primes occurring between A053182(n) and A053183(n) (excluding the endpoints).

2, 3, 7, 55, 255, 478, 663, 984, 1237, 1955, 3021, 3214, 8312, 13519, 38267, 40805, 45400, 47444, 48835, 55269, 56758, 59032, 66067, 92141, 93063, 103620, 106611, 108602, 112713, 140874, 151335, 163314, 178215, 183330, 211350, 235410, 244165, 265160, 275971

Offset: 1

Torlach Rush, Feb 26 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..2500

Cf. A053182, A053183

(PrimePi[#^2 + #] - PrimePi[#]) & /@  Select[Prime@Range@500, PrimeQ[#^2 + # + 1] &] (* Giovanni Resta, Feb 27 2014 *)

a(33)-a(39) from Giovanni Resta, Feb 27 2014

A147683 Numbers n with property that 6n-1 is in A053182.

Original entry on oeis.org

1, 3, 7, 10, 12, 15, 17, 22, 28, 29, 49, 64, 113, 117, 124, 127, 129, 138, 140, 143, 152, 182, 183, 194, 197, 199, 203, 229, 238, 248, 260, 264, 285, 302, 308, 322, 329, 355, 379, 385, 390, 402, 444, 455, 465, 493, 495, 502, 507, 523, 537, 542, 568, 575, 582

Offset: 1

Zak Seidov, Nov 10 2008

nonn

Cf. A053182.

isok(n) = isprime(p=6*n-1) && isprime(p^2+p+1); \\ Michel Marcus, Oct 15 2013

A053184 Primes p such that p^2+p-1 is prime.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 31, 41, 53, 59, 83, 89, 101, 103, 131, 149, 163, 181, 191, 193, 199, 233, 241, 263, 281, 331, 349, 373, 401, 419, 431, 433, 449, 461, 463, 499, 541, 569, 571, 659, 673, 683, 691, 709, 739, 743, 761, 769, 809, 863, 881, 941, 1013, 1039

Offset: 1

Enoch Haga, Mar 01 2000

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A053182, A065508, A091567.

[p: p in PrimesUpTo(1050) | IsPrime(p^2+p-1)]; // Bruno Berselli, Nov 08 2011

Select[Prime[Range[175]], PrimeQ[#^2+#-1]&] (* Bruno Berselli, Nov 08 2011 *)

isA053184(n)=isprime(n) && isprime(n^2+n-1) \\ Michael B. Porter, May 10 2010

A091567 Primes p such that p^2-p-1 is prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 29, 31, 47, 61, 67, 71, 97, 101, 127, 131, 139, 149, 181, 197, 241, 269, 307, 331, 359, 379, 397, 409, 419, 421, 449, 457, 479, 487, 491, 599, 607, 617, 619, 641, 647, 677, 709, 751, 787, 839, 857, 907, 947, 967, 977, 997, 1051, 1061, 1091

Offset: 1

T. D. Noe, Jan 21 2004

nonn

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A053182 (p^2+p+1 prime), A053184 (p^2+p-1 prime), A065508 (p^2-p+1 prime).

Cf. A091568 (corresponding primes of the form p^2-p-1).

Select[Prime[Range[250]], PrimeQ[ #^2-#-1]&] (* Alexander Adamchuk, Jul 27 2006 *)

isA091567(n)=isprime(n) && isprime(n^2-n-1) \\ Michael B. Porter, May 12 2010

A053183 Primes of the form p^2 + p + 1 when p is prime.

Original entry on oeis.org

7, 13, 31, 307, 1723, 3541, 5113, 8011, 10303, 17293, 28057, 30103, 86143, 147073, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 735307, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1886503, 2037757

Offset: 1

Enoch Haga, Mar 01 2000

Also primes in A001001. - Philippe Deléham, Feb 21 2004
This sequence is a subsequence of A002383. These numbers are repunit primes 111_n, so they are Brazilian primes belonging to A085104. - Bernard Schott, Dec 21 2012
Also, primes in A060800. - Zak Seidov, Mar 21 2014
Also subsequence of A002061, A193574. - Hartmut F. W. Hoft, May 05 2017
As p^2 + p + 1 is the sum of divisors of p^2 for any prime p, this is a subsequence of A023195. - Peter Munn, Feb 16 2018

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A002383, A002384, A023195, A053182, A060800, A065764, A085104, A182253, A185632.

Cf. A002061, A193574. - Hartmut F. W. Hoft, May 05 2017

a053183[n_] := Select[Map[Prime[#]^2 + Prime[#] + 1&, Range[n]], PrimeQ]
a053183[225] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[Table[p^2+p+1,{p,Prime[Range[300]]}],PrimeQ] (* Harvey P. Dale, Aug 15 2017 *)

a(n) = A053182(n)^2 + A053182(n) + 1.

A065508 Primes p such that p^2 - p + 1 is prime.

Original entry on oeis.org

2, 3, 7, 13, 67, 79, 139, 151, 163, 193, 337, 349, 379, 457, 541, 613, 643, 727, 769, 919, 991, 1021, 1093, 1117, 1201, 1231, 1381, 1423, 1549, 1567, 1597, 1621, 1693, 1747, 1789, 1801, 1933, 1987, 2011, 2017, 2113, 2137, 2143, 2239, 2281, 2557, 2647

Offset: 1

Vladeta Jovovic, Nov 26 2001

The primes p^2 - p + 1 are in A074268.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A053182, A053184, A091567, A074268.

[n: n in [0..10000]| IsPrime(n) and IsPrime(n^2 - n + 1)] // Vincenzo Librandi, Aug 07 2010

Select[Prime[Range[500]],PrimeQ[#^2-#+1]&] (* Harvey P. Dale, Oct 06 2015 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (isprime(p^2 - p + 1), write("b065508.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 20 2009

A055638 Numbers k for which sigma(k^2) is prime.

Original entry on oeis.org

2, 3, 4, 5, 8, 17, 27, 41, 49, 59, 64, 71, 89, 101, 125, 131, 167, 169, 173, 256, 289, 293, 383, 512, 529, 677, 701, 729, 743, 761, 773, 827, 839, 841, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1849, 1931

Offset: 1

Robert G. Wilson v, Jun 07 2000

nonn

sigma(n) is the sum of the divisors of n (A000203).
If sigma(x) is prime, then x=2 or x=p^(2m), an even power of a prime, cf. A023194. This sequence lists the values n = p^m such that sigma(n^2) is prime, i.e., sqrt( A023194 \ {2} ). The corresponding primes sigma(n^2)=A062700(n) are 1+p+...+p^(2m) = (p^(2m+1)-1)/(p-1), and any prime of that form (cf. A023195) corresponds to a term p^m is in this sequence. - M. F. Hasler, Oct 14 2014
This is a subsequence of A000961, see A248963 for its complement therein. - M. F. Hasler, Oct 19 2014
a(n) nearly always has digitsum of the form 2 mod 3. Specifically, 99.8% of the first 33733 entries examined conformed. The first exceptions are 3, 4, 27, 49, 64, 169, 256, 289, 529, 729. The exceptions (examined) appear to be integer powers themselves excepting the initial 3. Similarly, except for the initial 3, all entries of A023195 appear to have digitsum = 1 mod 3. - Bill McEachen, Mar 05 2017, Mar 20 2025
Number of terms < 10^k: 5, 13, 36, 137, 735, 4730, 33732, 253393, ..., . Robert G. Wilson v, Mar 09 2017
Primes in the sequence are A053182. - Thomas Ordowski, Nov 18 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 4730 terms from T. D. Noe)

Cf. A023194 (sigma(n) is prime).
Cf. A023195 (primes of the form sigma(n)), A062700 (in order of appearance).
Cf. A000961, A248963.

[n: n in [1..2000] | IsPrime(SumOfDivisors(n^2))]; // Vincenzo Librandi, Oct 18 2014

Select[Range[2000], PrimeQ[DivisorSigma[1, #^2]] &]

for(n=1,9999,isprime(sigma(n^2))&&print1(n",")) \\ M. F. Hasler, Oct 18 2014

a(n) = sqrt(A023194(n+1)).

Equal to A000961 \ A248963. - M. F. Hasler, Oct 19 2014

Minor edits by M. F. Hasler, Oct 18 2014

A348749 Odd numbers k for which A064989(sigma(k)) > A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

9, 25, 45, 49, 75, 81, 117, 121, 225, 243, 289, 325, 333, 405, 441, 529, 549, 605, 625, 657, 675, 729, 841, 925, 1053, 1089, 1125, 1215, 1225, 1413, 1445, 1521, 1525, 1575, 1665, 1681, 1737, 1825, 1875, 2025, 2205, 2401, 2475, 2493, 2601, 2817, 2825, 2925, 2997, 3025, 3033, 3125, 3249, 3481, 3573, 3645, 3675, 3789

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Sequence obtained when A003961 is applied to A348739 and the terms are sorted into ascending order.
From Robert Israel, Nov 12 2024: (Start)
If a and b are terms and are coprime, then a * b is a term.
If p > 2 is in A053182, Legendre's conjecture implies p^2 is in this sequence. (End)

Cf. A000203, A003961, A053182, A064989, A326042, A348739, A348748, A348939 (terms of A228058 that occur here).

Cf. also A348742, A348754.

g:= prevprime: g(2):= 1:
f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  mul(g(t[1])^t[2],t=F)
end proc:
select(t -> f(numtheory:-sigma(t)) > f(t), [seq(i,i=1..4000,2)]); # Robert Israel, Nov 12 2024

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 4000, 2], s[DivisorSigma[1, #]] > s[#] &] (* Amiram Eldar, Nov 04 2021 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA348749(n) = ((n%2)&&(A064989(sigma(n)) > A064989(n)));

A065405 Composite numbers k such that the sum of the divisors of k^2 is a prime.

Original entry on oeis.org

4, 8, 27, 49, 64, 125, 169, 256, 289, 512, 529, 729, 841, 1849, 2197, 3125, 4913, 5329, 6241, 6889, 15625, 16129, 29791, 32768, 37249, 51529, 57121, 69169, 76729, 113569, 117649, 128881, 139129, 157609, 192721, 208849, 226981, 229441, 253009

Offset: 1

Labos Elemer, Nov 06 2001

nonn

All these composite numbers k should be prime powers because if k=a*b with gcd(a,b)=1, then sigma(aabb) = sigma(aa)*sigma(bb) cannot be a prime; 46 of the 236 prime powers below 1000000 are here.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..70 from Harry J. Smith)

Cf. A062700, A000203, A065403, A065404, A053182, A053183, A028982.

Select[ Range[3 10^5], ! PrimeQ[ # ] && PrimeQ[ DivisorSigma[1, #^2]] & ]

isok(k) = { !isprime(k) && isprime(sigma(k^2)) } \\ Harry J. Smith, Oct 18 2009

sigma(a(n)^2) = sigma(A065404(n)) = A065403(n) is prime.

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

Original entry on oeis.org

3, 43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Dec 18 2012

nonn

These are the primes associated with A182253.
All these numbers are in A002383 but not in A053183.
All the numbers n^2 + n + 1 = 111_n with n >= 2 are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A182253.

Cf. A085104, A125134.

Select[Table[If[PrimeQ[n],Nothing,n^2+n+1],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 02 2023 *)

lista(nn) = {for (n = 1, nn, if (! isprime(n) && isprime(p = n^2 + n + 1), print1(p, ", ");););} \\ Michel Marcus, Sep 04 2013

cp's OEIS Frontend

A238399 a(n) is the number of primes occurring between A053182(n) and A053183(n) (excluding the endpoints).

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A147683 Numbers n with property that 6n-1 is in A053182.

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PARI

A053184 Primes p such that p^2+p-1 is prime.

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Magma

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PARI

A091567 Primes p such that p^2-p-1 is prime.

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Mathematica

PARI

A053183 Primes of the form p^2 + p + 1 when p is prime.

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Formula

A065508 Primes p such that p^2 - p + 1 is prime.

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Magma

Mathematica

PARI

A055638 Numbers k for which sigma(k^2) is prime.

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Magma

Mathematica

PARI

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Extensions

A348749 Odd numbers k for which A064989(sigma(k)) > A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

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