A260905 Totients of the Blum integers.

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

Author

Connor Zapfel, Nov 17 2015

Keywords

• nonn
• easy

Examples

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

Links

• 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

Crossrefs

Cf. A000010, A016105, A195758, A195759.

Programs

Maple

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

Mathematica

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 *)

Perl

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

Python

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

Formula

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