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A151999
Numbers k such that every prime that divides phi(k) also divides k.
6
1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 32, 34, 36, 40, 42, 48, 50, 54, 60, 64, 68, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 114, 120, 126, 128, 136, 144, 150, 156, 160, 162, 168, 170, 180, 192, 200, 204, 210, 216, 220, 222, 228, 234, 240, 250, 252, 256, 270
OFFSET
1,2
COMMENTS
Alternative descriptions:
(a) For every prime p|n and every prime q|p-1 we have q|n;
(b) Numbers n such that radical of phi(n) divides radical of n, where phi is Euler's totient function and radical is the squarefree kernel function.
These numbers are "valid bases".
Numbers n such that radical of phi(n) divides n. - Michel Marcus, Nov 06 2017
Pollack and Pomerance call these numbers "phi-deficient numbers". - Amiram Eldar, Jun 02 2020
LINKS
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
J. Jimenez Urroz and J. Luis A. Yebra, On the equation a^x=x mod b^n, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.8.
MAPLE
A151999 := proc(n)
if n = 1 then
2;
else
for a from procname(n-1)+1 do
pdvs := numtheory[factorset](a) ;
aworks := true;
for p in numtheory[factorset](a) do
for q in numtheory[factorset](p-1) do
if a mod q = 0 then
;
else
aworks := false;
end if;
end do:
end do:
if aworks then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, Jun 01 2013
MATHEMATICA
Rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[1 + Range[300], Mod[Rad[#], Rad[EulerPhi[#]]]==0 &] (* José María Grau Ribas, Jan 09 2012 *)
PROG
(PARI) isok(n) = {fp = factor(n); for (i=1, #fp[, 1], fq = factor(fp[i, 1] - 1); for (j=1, #fq[, 1], if (n % fq[j, 1], return (0)); ); ); return (1); } \\ Michel Marcus, Jun 01 2013
(PARI) isok(n) = (n % factorback(factor(eulerphi(n))[, 1])) == 0; \\ Michel Marcus, Nov 04 2017
(Magma) [n: n in [1..300] | forall{d: d in PrimeDivisors(EulerPhi(n)) | IsIntegral(n/d)}]; // Bruno Berselli, Nov 04 2017
(Sage)
for n in range(1, 271):
if euler_phi(n)**2 == euler_phi(euler_phi(n) * n): print(n, end=', ') # Torlach Rush, Oct 01 2024
CROSSREFS
Cf. A007947 (radical of n), A007694 (phi(n) divides n, a subsequence).
Cf. A080400 (radical of phi(n)).
Cf. A152000.
Sequence in context: A213708 A371176 A239063 * A267817 A177917 A071594
KEYWORD
easy,nonn
AUTHOR
J. Luis A. Yebra and J. Jimenez Urroz (yebra(AT)mat.upc.es), Nov 19 2008
EXTENSIONS
Corrected by Michel Marcus, Jun 01 2013
Edited by N. J. A. Sloane, Jun 02 2013 at the suggestion of Michel Marcus, merging this with A204010
Deleted erroneous comment and added correct b-file by Paolo P. Lava, Nov 06 2017
STATUS
approved