A065508 -id:A065508 - cp's OEIS frontend

A243544 Primes p such that p^2 - p + 1 is semiprime.

5, 11, 29, 37, 41, 43, 53, 61, 71, 73, 83, 97, 109, 113, 127, 137, 149, 157, 167, 181, 191, 211, 223, 229, 241, 271, 277, 281, 307, 317, 331, 359, 389, 421, 433, 443, 461, 463, 487, 499, 547, 557, 571, 587, 601, 617, 631, 659, 661, 683, 691, 701, 709, 733, 757

Offset: 1

K. D. Bajpai, Jun 06 2014

nonn

Intersection of A000040 and A180748.

			11 is in the sequence because 11 is prime and 11^2 - 11 + 1 = 111 = 3 * 37 is semiprime.
29 is in the sequence because 29 is prime and 29^2 - 29 + 1 = 813 = 3 * 271 is semiprime.
17 is not in the sequence though 17 is prime, because 17^2 - 17 + 1 = 273 = 3 * 7 * 13, has more than two prime factors.

K. D. Bajpai, Table of n, a(n) for n = 1..7090

Cf. A000040, A001358, A053182, A053184, A065508, A091567, A180748.

with(numtheory): A243544 := proc() local a; a:=ithprime(n);  if bigomega(a^2-a+1)=2 then RETURN (a); fi; end: seq(A243544 (), n=1..200);

c = 0; Do[k = Prime[n]; If[PrimeOmega[k^2 - k + 1] == 2, c++; Print[c, " ", k]], {n, 1, 30000}];
Select[Prime[Range[150]],PrimeOmega[#^2-#+1]==2&] (* Harvey P. Dale, Oct 22 2024 *)

s=[]; forprime(p=2, 800, if(bigomega(p^2-p+1)==2, s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

A290767 Primes p such that p^2 +/- p +/- 1 are all nonprimes.

Original entry on oeis.org

23, 37, 43, 73, 107, 109, 113, 137, 157, 179, 211, 223, 227, 229, 239, 251, 257, 271, 277, 283, 311, 313, 317, 347, 353, 367, 389, 439, 443, 467, 503, 509, 521, 523, 547, 557, 563, 577, 587, 593, 601, 631, 653, 661, 719, 733, 757, 797, 811, 821, 823, 829, 853, 859, 877, 883

Offset: 1

Ralf Steiner, Aug 10 2017

nonn

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

Cf. A000040, A053184, A053182, A065508, A091567, A236056.

select(p -> isprime(p) and not ormap(isprime, [p^2+p+1,p^2+p-1,p^2-p+1,p^2-p-1]), [2,seq(i,i=3..1000,2)]); # Robert Israel, Aug 10 2017

Select[Prime[Range[1000]], ! (PrimeQ[#^2 + # + 1] || PrimeQ[#^2 + # - 1] ||PrimeQ[#^2 - # + 1] || PrimeQ[#^2 - # - 1]) &]
Select[Prime[Range[200]],NoneTrue[{#^2+#+1,#^2+#-1,#^2-#+1,#^2-#-1},PrimeQ]&] (* Harvey P. Dale, Oct 13 2024 *)

is(n) = my(v=[n^2+n+1, n^2+n-1, n^2-n+1, n^2-n-1]); for(k=1, #v, if(ispseudoprime(v[k]), return(0))); 1
forprime(p=1, 900, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 10 2017

Intersection of the complements of A053184, A053182, A065508, and A091567 within the primes A000040.

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, ", ");););}

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A243544 Primes p such that p^2 - p + 1 is semiprime.

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A290767 Primes p such that p^2 +/- p +/- 1 are all nonprimes.

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

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