A074268 -id:A074268 - cp's OEIS frontend

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

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

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

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

Original entry on oeis.org

3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333, 1641, 1807, 2163, 2757, 3423, 3661, 4423, 4971, 5257, 6163, 6807, 7833, 9313, 10101, 10507, 11343, 11773, 12657, 16003, 17031, 18633, 19183, 22053, 22651, 24493, 26407, 27723, 29757, 31863

Offset: 1

Alexander Adamchuk, Aug 02 2006

nonn

Prime terms belong to A074268, which is a subset of A002383, A087126, A034915, A085104.

In every interval of prime(n)^2 integers, a(n) is the number that are not divisible by prime(n) plus the number that are divisible by prime(n)^2. - Peter Munn, Dec 12 2020

Cf. A002061, A074268, A002383, A087126, A034915, A068468, A085104, A083558.

Mathematica
```
Table[Prime[n]^2-Prime[n]+1,{n,1,100}]
```

a(n) = {my(p = prime(n)); p^2 - p + 1; } \\ Amiram Eldar, Nov 07 2022

a(n) = prime(n)^2 - prime(n) + 1.
a(n) = A036689(n)+1. - R. J. Mathar, Aug 13 2019
Product_{n>=1} (1 - 1/a(n)) = zeta(6)/(zeta(2)*zeta(3)) (A068468). - Amiram Eldar, Nov 07 2022

A258435 Primes of form x^2 - phi(x) in increasing order.

Original entry on oeis.org

3, 7, 43, 157, 1069, 1201, 4177, 4423, 5869, 6163, 8209, 17581, 19183, 22651, 26407, 37057, 48649, 60793, 61837, 82129, 89137, 102829, 113233, 115981, 121453, 141793, 143263, 190573, 208393, 230929, 283609, 292141, 303097, 314401, 337069

Offset: 1

Carlos Eduardo Olivieri, May 30 2015

			a(1) = 3, because  2^2 - 1 = 3, and 1^2 - 1 = 0 is not a prime.
a(2) = 7, since 3^2 = 9, phi(3) = 2, so 9-2 = 7 (prime).
a(3) = 43, since 7^2 = 49, phi(7) = 6, so 49-6 = 43 (prime).
a(6) = 1201, since 35^2 = 1225, phi(35) = 24, so 1225-24 = 1201 (prime).

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

Subset of A258434.
For phi see A000010.
A074268 is a subsequence. - Michel Marcus, Jun 19 2015
Cf. A259145.

[a: n in [1..1000] | IsPrime(a) where a is n^2-EulerPhi(n) ]; // Vincenzo Librandi, Jun 03 2015

lst = Table[n^2 - EulerPhi[n], {n, 1000}]; Select[lst, PrimeQ]
Select[Table[n^2 - EulerPhi[n], {n, 1000}], PrimeQ] (* Vincenzo Librandi, Jun 03 2015 *)

lista(nn) = {for (n=1, nn, if (isprime(p=n^2 -eulerphi(n)), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

More terms from Vincenzo Librandi, Jun 03 2015

Edited by Wolfdieter Lang, Jun 16 2015

A034915 Primes of the form p^k - p + 1 for prime p.

Original entry on oeis.org

3, 7, 31, 43, 79, 127, 157, 241, 337, 727, 1321, 3121, 4423, 6163, 6841, 8191, 19183, 19681, 22651, 26407, 28549, 29761, 37057, 68881, 78121, 113233, 117643, 121453, 130303, 131071, 143263, 208393, 292141, 371281, 375157, 412807, 524287, 527803

Offset: 1

Jud McCranie

nonn

Related to hyperperfect numbers of a certain form.

Since x^k-x+1 is divisible by x^2-x+1 for k==2 (mod 6), none of k=8,14,20,... occur. - Robert Israel, Mar 20 2018

			11^3 - 11 + 1 = 1321 is prime, so 1321 is a term.

Robert Israel, Table of n, a(n) for n = 1..4960
J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.

Contains A074268.

N:= 10^6: # to get all terms <= N
Res:= NULL;
p:= 1:
do
  p:= nextprime(p);
  if p^2-p+1>N then break fi;
  for i from 2 to floor(log[p](N+p-1)) do
    if isprime(p^i-p+1) then Res:= Res, p^i-p+1 fi
  od
od:
sort(convert({Res},list)); # Robert Israel, Mar 20 2018

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.

A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 29, 31, 41, 43, 109, 131, 157, 181, 271, 307, 379, 811, 929, 991, 1721, 1723, 2161, 2861, 3539, 3541, 3659, 4421, 4423, 4969, 5113, 6163, 6971, 8009, 8011, 9311, 10099, 10301, 10303, 10711, 16001, 17029, 17291, 17293, 19181, 19183, 22051, 22349, 22651

Offset: 1

Ralf Steiner, Aug 11 2017

nonn

This sequence contains prime chains and prime trees using an appropriate mapping form p^2 +- p +- 1 in each step, such as the chain: 3 -> 5 -> 19 -> 379 -> 143263 -> 20524143907 and the tree: 41 -> {1721, 1723}.

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

Cf. A000040.

Union of A053183, A053185, A074268, A091568.

{p^2+(-1)^k*p+(-1)^s:p in PrimesUpTo(150), s,k in [1..2]|IsPrime(p^2+(-1)^k*p+(-1)^s)}; //  Marius A. Burtea, Nov 28 2019

select(isprime, [3,seq(op([p^2-p-1,p^2-p+1,p^2+p-1,p^2+p+1]),p=select(isprime,[seq(i,i=3..1000,2)]))]); # Robert Israel, Nov 27 2019

Select[Union[Flatten[{(#^2 + # + 1 ), (#^2 + # - 1 ), (#^2 - # + 1 ), (#^2 - # - 1 )}] &[Prime[Range[100]]]], (PrimeQ[#]) &]

A297868 Prime powers p^e with odd exponent e such that rho(p^(e+1)) is prime, where rho is A206369.

Original entry on oeis.org

8, 27, 32, 125, 243, 512, 1331, 2048, 32768, 50653, 79507, 103823, 131072, 161051, 177147, 357911, 1419857, 2097152, 2248091, 3869893, 11089567, 15813251, 16974593, 20511149, 28934443, 69343957, 115501303, 147008443, 263374721, 536870912, 844596301, 1284365503, 1305751357

Offset: 1

Michel Marcus, Jan 07 2018

nonn

Along with A065508, these are the integers mentioned at the bottom of page 4 of the Iannucci link. Let x = p^e, and q = rho(p^(e+1)), then x/rho(x) = (x*p*q)/rho(x*p*q). An example with A065508 is 3, for which rho(3) is 7, so 3 and 3*3*7 have the same x/rho(x) ratio, 3/2.
Note that there are other "rho-friendly pairs" that have a different, yet simple, form like for instance 7^5 and 7^8*117307.
Number of terms < 10^k: 1, 3, 6, 8, 11, 16, 20, 26, 31, 46, 73, 110, 198, 327, 611, 1157, 2135, 4107, 7724, 14771, 28610, etc. - Robert G. Wilson v, Jan 07 2018

			8=2^3 is a term because rho(2*8)=11 is prime, so 8 and 8*2*11 have the same x/rho(x) ratio, 8/5.

Robert G. Wilson v, Table of n, a(n) for n = 1..28610
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.

Cf. A065508, A074268, A206369.

rho[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &]; fQ[n_] := Block[{p = FactorInteger[n][[1, 1]]}, PrimeQ[ rho[p n]]]; mx = 10^9; lst = Sort@ Flatten@ Table[ Prime[n]^e, {n, PrimePi[mx^(1/3)]}, {e, 3, Floor@ Log[ Prime@ n, mx], 2}]; Select[lst, fQ] (* Robert G. Wilson v, Jan 07 2018 *)

rhope(p, e) = my(s=1); for(i=1, e, s=s*p + (-1)^i); s;
lista(nn) = {for (n=1, nn, if ((e = isprimepower(n,&p)) && (e > 1) && (e % 2) && isprime(rhope(p,e+1)), print1(n, ", ");););}

cp's OEIS Frontend

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

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

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A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

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A258435 Primes of form x^2 - phi(x) in increasing order.

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Extensions

A034915 Primes of the form p^k - p + 1 for prime p.

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

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A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

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Maple