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

1,1

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000010, A016105, A195758, A195759.

Sequence in context: A299028 A082296 A187766 * A211415 A209973 A282754

Adjacent sequences: A260902 A260903 A260904 * A260906 A260907 A260908

Keyword

nonn,easy

Author

Connor Zapfel, Nov 17 2015

Status

approved