A088710 -id:A088710 - cp's OEIS frontend

A088711 Numbers m that are a product of two primes j and k such that both m+j+k and m-j-k are primes.

10, 14, 15, 21, 26, 33, 35, 38, 51, 65, 86, 93, 111, 123, 161, 201, 203, 206, 209, 215, 221, 278, 321, 371, 395, 398, 413, 471, 485, 533, 543, 545, 551, 626, 671, 698, 723, 755, 779, 803, 815, 866, 905, 993, 1046, 1286, 1349, 1371, 1383, 1385, 1403, 1461

Offset: 1

Chuck Seggelin, Oct 11 2003

nonn

			a(1)=10 because 10 has only one pair of prime factors (2 and 5) and both 10+2+5 and 10-2-5 (17 and 3) are primes.

Zak Seidov, Table of n, a(n) for n=1..1616, a(n)<200000

Cf. A001358, A088709, A088710.

a[n_]:={1,1}==Last/@FactorInteger[n];b[n_]:=First[First/@FactorInteger[n]]+Last[First/@FactorInteger[n]];Select[Range[6,3000],a[#]&&PrimeQ[#-b[#]]&&PrimeQ[#+b[#]]&] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)
Reap[Do[fi=FactorInteger[n]; la=Last/@fi; If[{1,1}==la, pq=fi[[1,1]]+fi[[2,1]];If[PrimeQ[n-pq] && PrimeQ[n+pq], Sow[n]]], {n,6,200000}]][[2,1]] (* used to create b-file, Zak Seidov, Mar 04 2011 *)

A271568 Squarefree semiprimes n such that phi(n) - 1 is prime.

Original entry on oeis.org

10, 14, 15, 21, 26, 33, 35, 38, 39, 51, 62, 65, 69, 77, 86, 91, 93, 95, 111, 122, 123, 129, 133, 146, 159, 161, 201, 203, 206, 209, 213, 215, 217, 218, 221, 249, 278, 287, 291, 299, 301, 302, 303, 305, 321, 335, 339, 362, 371, 381, 386, 395, 398, 403

Offset: 1

Debapriyay Mukhopadhyay, Jul 13 2016

nonn

Equals (A001358 intersection A078892) - A001248.

Appears to be equal to A088710 without the 9. - R. J. Mathar, Jun 21 2025

			15 is in the sequence, because 15 = 3*5 is a semiprime with omega(15) = 2 and phi(15) - 1 = 2*4 - 1 = 7 is a prime.
21 is in the sequence, because 21 = 3*7 is a semiprime with omega(21) = 2 and phi(21) - 1 = 2*6 - 1 = 11 is a prime.

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

Cf. A001358, A078892, A001248.

[n: n in [1..500] |(EulerPhi(n)+DivisorSigma(1,n)) eq 2*(n+1) and IsPrime(EulerPhi(n)-1)]; // Vincenzo Librandi, Jul 29 2016

with(numtheory):
is_A271568 := n -> issqrfree(n) and bigomega(n) = 2 and isprime(phi(n)-1):
select(is_A271568, [$1..403]); # Peter Luschny, Jul 21 2016

A271568Q = SquareFreeQ[#] && PrimeNu[#] == 2 && PrimeQ[EulerPhi[#] - 1] &; Select[Range[500], A271568Q] (* JungHwan Min, Jul 29 2016 *)

is_a001358(n) = bigomega(n)==2
is_a005117(n) = issquarefree(n)
is_a078892(n) = ispseudoprime(eulerphi(n)-1)
is(n) = is_a001358(n) && is_a005117(n) && is_a078892(n) \\ Felix Fröhlich, Jul 21 2016

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && isprime((f[1,1]-1)*(f[2,1]-1)-1) \\ Charles R Greathouse IV, Jul 21 2016

list(lim)=my(v=List()); forprime(p=2, sqrt(lim), forprime(q=p+1, lim\p, if(isprime((p-1)*(q-1)-1), listput(v, p*q)))); Set(v) \\ Charles R Greathouse IV, Aug 29 2016

New name from Charles R Greathouse IV, Jul 29 2016

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A088711 Numbers m that are a product of two primes j and k such that both m+j+k and m-j-k are primes.

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A271568 Squarefree semiprimes n such that phi(n) - 1 is prime.

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