A065508 -id:A065508 - cp's OEIS frontend

A053182 Primes p such that p^2 + p + 1 is prime.

2, 3, 5, 17, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

Enoch Haga, Mar 01 2000

Roger Horn computed the first 776 terms of this sequence around 1961 to test (with Paul Bateman) their conjecture on the density of simultaneous primes in polynomials. - Charles R Greathouse IV, Apr 05 2011
Starting with a(3)=5 all terms are of the form 6k-1, k in A147683. - Zak Seidov, Nov 10 2008
Primes p such that the sum of divisors of p^2 (sigma(p^2) = A000203(p^2) = p^2+p+1) is prime. - Claudio Meller, Apr 07 2011
The generated prime numbers p^2 + p + 1 are exactly A053183. - Bernard Schott, Dec 20 2012
Positive squarefree k such that the sum of divisors of k^2 is prime. - Peter Munn, Feb 02 2018

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 2650 terms from M. F. Hasler).
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.
Paolo Santonastaso and Ferdinando Zullo, Linearized trinomials with maximum kernel, arXiv:2012.14861 [math.NT], 2020.

Cf. A053184, A065508, A091567, A147683, A188596.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p+1)]; // Vincenzo Librandi, Aug 06 2010

Select[Prime[Range[427]], PrimeQ[#^2+#+1]&] (* Bruno Berselli, Nov 08 2011 *)

isA053182(n)=isprime(n) && isprime(n^2+n+1)  \\ Michael B. Porter, Apr 23 2010

c=0; forprime(p=1,default(primelimit), isprime(p^2+p+1) & write("/tmp/b053182.txt",c++," "p))  \\ M. F. Hasler, Apr 07 2011

List changed to cross-reference by Franklin T. Adams-Watters, May 12 2010

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

A127724 k-imperfect numbers for some k >= 1.

Original entry on oeis.org

1, 2, 6, 12, 40, 120, 126, 252, 880, 2520, 2640, 10880, 30240, 32640, 37800, 37926, 55440, 75852, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120, 715816960, 866829600

Offset: 1

T. D. Noe, Jan 25 2007

For prime powers p^e, define a multiplicative function rho(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e. A number n is called k-imperfect if there is an integer k such that n = k*rho(n). Sequence A061020 gives a signed version of the rho function. As with multiperfect numbers (A007691), 2-imperfect numbers are also called imperfect numbers. As shown by Iannucci, when rho(n) is prime, there is sometimes a technique for generating larger imperfect numbers.
Zhou and Zhu find 5 more terms, which are in the b-file. - T. D. Noe, Mar 31 2009
Does this sequence follow Benford's law? - David A. Corneth, Oct 30 2017
If a term t has a prime factor p from A065508 with exponent 1 and does not have the corresponding prime factor q from A074268, then t*p*q is also a term. - Michel Marcus, Nov 22 2017
For n >= 1, the least n-imperfect numbers are 1, 2, 6, 993803899780063855042560. - Michel Marcus, Feb 13 2018
For any m > 0, if n*p^(2m-1) is k-imperfect, q = rho(p^(2m)) is prime and gcd(pq,n) = 1, then n*p^(2m)*q is also k-imperfect. - M. F. Hasler, Feb 13 2020

			126 = 2*3^2*7, rho(126) = (2-1)*(9-3+1)*(7-1) = 42.  3*42 = 126, so 126 is 3-imperfect. - _Jud McCranie_ Sep 07 2019

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer, 1994, B1.

Giovanni Resta, Table of n, a(n) for n = 1..50 (terms < 10^13, a(1)-a(39) from T. D. Noe (from Iannucci, Zhou, and Zhu), a(40)-a(44) from Donovan Johnson)
David A. Corneth, Conjectured to be the terms up to 10^28
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
Donovan Johnson, 43 terms > 2*10^11
Andrew Lelechenko, 4-imperfect numbers, Apr 19 2014.
Michel Marcus, More 4-imperfect numbers, Nov 07 2017.
Michel Marcus, Least known integers with small denominator-fractional k's, Feb 13 2018.
Allan Wechsler, Some progress in k-imperfect numbers (A127724), Seqfan, Feb 13 2020. Gives a first instance of a 5-imperfect number.
Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.

Cf. A127725 (2-imperfect numbers), A127726 (3-imperfect numbers), A127727 (related primes), A309806 (the k values).
Cf. A061020 (signed version of rho function), A206369 (the rho function).
Cf. A065508, A074268.

f[p_,e_]:=Sum[(-1)^(e-k) p^k, {k,0,e}]; rho[n_]:=Times@@(f@@@FactorInteger[n]); Select[Range[10^6], Mod[ #,rho[ # ]]==0&]

isok(n) = denominator(n/sumdiv(n, d, d*(-1)^bigomega(n/d))) == 1; \\ Michel Marcus, Oct 28 2017

upto(ulim) = {res = List([1]); rhomap = Map(); forprime(p = 2, 3, for(i = 1, logint(ulim, p), mapput(rhomap, p^i, rho(p^i)); iterate(p^i, mapget(rhomap, p^i), ulim))); listsort(res, 1); res}
iterate(m, rhoo, ulim) = {my(c); if(m / rhoo == m \ rhoo, listput(res, m); my(frho = factor(rhoo)); for(i = 1, #frho~, if(m%frho[i, 1] != 0, for(e = 1, logint(ulim \ m, frho[i, 1]), if(mapisdefined(rhomap, frho[i, 1]^e) == 0, mapput(rhomap, frho[i, 1]^e, rho(frho[i, 1]^e))); iterate(m * frho[i, 1]^e, rhoo * mapget(rhomap, frho[i, 1]^e), ulim)); next(2))))}
rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res * =q); res} \\ David A. Corneth, Nov 02 2017

A127724_vec=concat(1, select( {is_A127724(n)=!(n%A206369(n))}, [1..10^5]*2))
  /* It is known that the least odd term > 1 is > 10^49. This code defines an efficient function is_A127724, but A127724_vec is better computed with upto(.) */
  A127724(n)=A127724_vec[n] \\ Used in other sequences. - M. F. Hasler, Feb 13 2020

Small correction in name from Michel Marcus, Feb 13 2018

A066071 Nonprime numbers k such that phi(k) + 1 is prime.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 18, 21, 22, 26, 27, 28, 32, 34, 36, 38, 40, 42, 46, 48, 49, 54, 55, 57, 58, 60, 62, 63, 74, 75, 76, 77, 82, 86, 88, 91, 93, 94, 95, 98, 99, 100, 106, 108, 110, 111, 114, 115, 117, 118, 119, 122, 124, 125, 126, 132, 133, 134, 135, 142, 145, 146

Offset: 1

Labos Elemer, Dec 03 2001

nonn

A039698 with the primes removed. For every prime p, 2p is in the sequence. - Ray Chandler, May 26 2008

Includes 3*p for p in A005382 and p^2 for p in A065508. - Robert Israel, Dec 29 2017

			Solutions to 1+phi(x)=13 are {13, 21, 26, 28, 36, 42} of which the 5 composites are in the sequence.

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A000010, A000040, A005382, A006093, A039649, A039698, A058339, A058340, A065508, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

[n: n in [1..200] |not IsPrime(n) and IsPrime(EulerPhi(n)+1)]; // Vincenzo Librandi, Jul 02 2016

select(n -> not isprime(n) and isprime(1+numtheory:-phi(n)), [$1..1000]); # Robert Israel, Dec 29 2017

Select[Complement[Range@ #, Prime@ Range@ PrimePi@ #] &@ 150, PrimeQ[EulerPhi@ # + 1] &] (* Michael De Vlieger, Jul 01 2016 *)

isok(k) = { !isprime(k) && isprime(eulerphi(k) + 1) } \\ Harry J. Smith, Nov 10 2009

A074268 Primes of the form p^2 - p + 1 where p is prime.

Original entry on oeis.org

3, 7, 43, 157, 4423, 6163, 19183, 22651, 26407, 37057, 113233, 121453, 143263, 208393, 292141, 375157, 412807, 527803, 590593, 843643, 981091, 1041421, 1193557, 1246573, 1441201, 1514131, 1905781, 2023507, 2397853, 2453923, 2548813, 2626021, 2864557

Offset: 1

Enoch Haga, Jan 16 2004

Note that p^2 - p + 1 = p * phi(p) + 1 when p is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A065508 (values of p).

Select[Table[p = Prime[n]; p*EulerPhi[p] + 1, {n, 270}], PrimeQ] (* Arkadiusz Wesolowski, Dec 13 2011 *)
Select[Table[p^2-p+1,{p,Prime[Range[300]]}],PrimeQ] (* Harvey P. Dale, Sep 22 2015 *)

A190275 Semiprimes of the form p*(p^2 - p + 1).

Original entry on oeis.org

6, 21, 301, 2041, 296341, 486877, 2666437, 3420301, 4304341, 7152001, 38159521, 42387097, 54296677, 95235601, 158048281, 229971241, 265434901, 383712781, 454166017, 775307917, 972261181, 1063290841, 1304557801, 1392422041, 1730882401, 1863895261, 2631883561, 2879450461, 3714274297, 3845297341, 4070454361, 4256780041, 4849695001, 5328809461, 5722533337, 5838483601, 7218898681, 7841065621

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

This sequence is infinite, assuming Schinzel's Hypothesis H.
Related to Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q + r + 1") setting p = q. Generalization can be achieved by removing semiprimality condition and accepting p^e, e >= 2.
These are semiprimes m = p*q such that 1/p + 1/q - 1/m = p/q. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 22 2019

			a(1) = 6 = 2*3 = 2*(2^2-2+1).
a(2) = 21 = 3*7 = 3*(3^2-3+1).
a(3) = 301 = 7*43 = 7*(7^2-7+1).

Giovanni Resta, Table of n, a(n) for n = 1..10000
For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A065508 (primes p such that p^2-p+1 is prime).

Cf. A001358 (semiprime), A003415 (arithmetic derivative), A164643, A190272 (n'=a-1), A190273 (n'=a+1), A190274 (n'=p^2-1).

seq(`if`(isprime((ithprime(i)^2-ithprime(i)+1))=true,(ithprime(i)^2-ithprime(i)+1)*ithprime(i),NULL),i=1..300);

p = Select[Prime@ Range@ 500, PrimeQ[#^2 - # + 1] &]; p (p^2 - p + 1) (* Giovanni Resta, Jul 22 2019 *)

forprime(p=2,1e4,if(isprime(k=p^2-p+1),print1(p*k", "))) \\ Charles R Greathouse IV, May 08 2011

A259145 Numbers k such that k^2 - phi(k) is prime, where phi() is A000010.

Original entry on oeis.org

2, 3, 7, 13, 33, 35, 65, 67, 77, 79, 91, 133, 139, 151, 163, 193, 221, 247, 249, 287, 299, 321, 337, 341, 349, 377, 379, 437, 457, 481, 533, 541, 551, 561, 581, 591, 595, 611, 613, 643, 721, 727, 763, 769, 779, 789, 803, 817, 843, 851, 869, 917, 919, 991

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Conjecture: a(n) is a cyclic number (see A003277) for all n.

A065508 is the subsequence of prime terms. - Michel Marcus, Jun 19 2015

			a(1) = 2, since phi(2) = 1, thus 2^2 - 1 = 3 (prime).
a(3) = 7, since phi(7) = 6, thus 7^2 - 6 = 43 (prime).
a(5) = 33, since phi(33) = 20, thus 33^2 - 20 = 1069 (prime).

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

Cf. A000010, A003277, A065508, A258435.

[n: n in [1..1000] | IsPrime(n^2 - EulerPhi(n))]; // Vincenzo Librandi, Jun 21 2015

Select[Range[2000], PrimeQ[#^2 - EulerPhi[#]] &]

main(size)={ v=vector(size); i=0; m=1; while(iAnders Hellström, Jul 08 2015 */

A136240 Numbers n among A006093 such that n^2 + n + 1 is prime.

Original entry on oeis.org

1, 2, 6, 12, 66, 78, 138, 150, 162, 192, 336, 348, 378, 456, 540, 612, 642, 726, 768, 918, 990, 1020, 1092, 1116, 1200, 1230, 1380, 1422, 1548, 1566, 1596, 1620, 1692, 1746, 1788, 1800, 1932, 1986, 2010, 2016, 2112, 2136, 2142, 2238, 2280, 2556, 2646

Offset: 1

Lekraj Beedassy, Dec 23 2007

nonn

See A074268 for the primes associated with a(n).

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

Cf. A136241.

Select[Prime[Range[400]]-1,PrimeQ[#^2+#+1]&] (* Harvey P. Dale, Feb 11 2022 *)

isA136240(n) = {isprime(n+1) && isprime(n^2+n+1)} \\ Michel Marcus, Jul 28 2013

a(n) = A065508(n) - 1.

A259504 Numbers n such that n and n+1 are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

27, 44, 75, 98, 116, 124, 147, 153, 164, 170, 171, 174, 230, 244, 245, 284, 285, 332, 356, 369, 387, 425, 428, 429, 434, 435, 474, 506, 507, 530, 548, 555, 574, 595, 602, 603, 604, 605, 609, 627, 637, 638, 645, 651, 657, 710

Offset: 1

Zak Seidov, Nov 08 2015

nonn

Conjecture: this sequence is infinite.
Number of terms < 10^k: 0, 4, 63, 727, 7014, 64556, 585725, 5284711, ... . - Robert G. Wilson v, Nov 09 2015
a(n) = p^3 where p is prime iff p is in intersection of A065508 and A005383. - Altug Alkan, Nov 24 2015
There are 47753279 terms less than 10^9 and 432841730 terms less than 10^10. - Charles R Greathouse IV, Jun 27 2019

			27=3*3*3, 28=2*2*7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A014612 and A045920.

Cf. A067813.

Select[Range[1000], 3 == PrimeOmega[#] == PrimeOmega[# + 1] &]

forcomposite(n=1, 1e3, if(bigomega(n)==3 && bigomega(n+1)==3, print1(n, ", "))); \\ Altug Alkan, Nov 08 2015

list(lim)=my(v=List(),was=1,is); forfactored(n=28,lim\1+1, is=vecsum(n[2][,2])==3; if(is && was, listput(v,n[1]-1)); was=is); Vec(v) \\ Charles R Greathouse IV, Jun 26 2019

cp's OEIS Frontend

A053182 Primes p such that p^2 + p + 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

A127724 k-imperfect numbers for some k >= 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Extensions

A066071 Nonprime numbers k such that phi(k) + 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A074268 Primes of the form p^2 - p + 1 where p is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A190275 Semiprimes of the form p*(p^2 - p + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs