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User: Connor Zapfel

Connor Zapfel has authored 2 sequences.

A286217 Product of the n-th sexy prime triple.

1729, 11339, 49321, 146969, 386389, 1089019, 1221191, 3864241, 5171489, 12640949, 18181979, 21243961, 43974269, 51881689, 178433279, 208506509, 230324329, 278421569, 393806449, 849244031, 932539661, 1341880019, 1416207439, 1672403471, 1829232539, 2111885999

Offset: 1

Connor Zapfel, May 04 2017

A sexy prime triple is such that p, p+6, and p+12 are primes but p+18 is not a prime. - Harvey P. Dale, Oct 13 2024

			The first sexy prime triple is (7, 13, 19) so a(1) = 7*13*19 = 1729.

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

Cf. A023201, A046118, A046119, A046120.

Select[Prime@ Range@ 500, PrimeQ[# + {6, 12, 18}] == {True, True, False} &] // # (#+6) (#+12) & (* Giovanni Resta, May 05 2017 *)
Times@@Take[#,3]&/@(Select[Table[p+{0,6,12,18},{p,Prime[Range[250]]}],Boole[PrimeQ[#]]=={1,1,1,0}&]) (* Harvey P. Dale, Oct 13 2024 *)

a(n) = s(n)*(s(n)+6)*(s(n)+12), where s = A046118.

a(n) = A046118(n) * A046119(n) * A046120(n).

A260905 Totients of the Blum integers.

Original entry on oeis.org

12, 20, 36, 44, 60, 60, 84, 108, 92, 132, 116, 132, 180, 140, 180, 156, 164, 220, 252, 204, 212, 276, 300, 252, 260, 348, 276, 396, 300, 396, 420, 324, 420, 332, 460, 356, 468, 380, 492, 540, 396, 420, 580, 444, 452, 660, 476, 612, 660, 636, 500, 700, 524

Offset: 1

Connor Zapfel, Nov 17 2015

			For the first Blum integer, a(1) = phi(21) = 12.

Robert Israel, Table of n, a(n) for n = 1..10000
Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996.
Wikipedia, Blum Integers
Wikipedia, Euler Phi Function

Cf. A000010, A016105, A195758, A195759.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [seq(4*i+3, i=0.. floor(N/12 - 3/4))]):
Pairs:= select(t -> t[1]*t[2]<=N, [seq(seq([Primes[i],Primes[j]],j=i+1..nops(Primes)),i=1..nops(Primes))]):
map(t -> (t[1]-1)*(t[2]-1), sort(Pairs,(s,t) -> s[1]*s[2] < t[1]*t[2])); # Robert Israel, Nov 18 2015

EulerPhi@ With[{lim = 820}, Select[Union[Times @@@ Subsets[Select[Prime@ Range@ PrimePi@ NextPrime[lim/3], Mod[#, 4] == 3 &], {2}]], # <= lim &]] (* Michael De Vlieger, Nov 18 2015, after Harvey P. Dale at A016105 *)
EulerPhi[Select[4Range[5, 197] + 1, PrimeNu[#] == 2 && MoebiusMu[#] == 1 && Mod[FactorInteger[#][[1, 1]], 4] != 1 &]] (* Alonso del Arte, Nov 18 2015 *)

use ntheory ":all"; forcomposites { say euler_phi($) if ($ % 4) == 1 && is_square_free($) && scalar(factor($)) == 2 && !scalar(grep { ($ % 4) != 3 } factor($)); } 1000; # Dana Jacobsen, Dec 10 2015

from sympy import factorint, totient
def isBlum(n):
    fn = factorint(n)
    return len(fn) == sum(fn.values()) == 2 and all(f%4 == 3 for f in fn)
print([totient(k) for k in range(790) if isBlum(k)]) # Michael S. Branicky, Dec 20 2021

a(n) = phi(Blum(n)) = A000010(A016105(n)), where phi(n) is Euler's totient function and Blum(n) is the n-th Blum integer.

a(n) = (A195758(n)-1)*(A195759(n)-1). - Jianing Song, Sep 16 2019

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User: Connor Zapfel

A286217 Product of the n-th sexy prime triple.

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