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A151999
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Numbers k such that every prime that divides phi(k) also divides k.
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6
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
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MAPLE
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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;
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MATHEMATICA
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Rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[1 + Range[300], Mod[Rad[#], Rad[EulerPhi[#]]]==0 &] (* José María Grau Ribas, Jan 09 2012 *)
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PROG
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(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
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CROSSREFS
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Cf. A007947 (radical of n), A007694 (phi(n) divides n, a subsequence).
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KEYWORD
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easy,nonn
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AUTHOR
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J. Luis A. Yebra and J. Jimenez Urroz (yebra(AT)mat.upc.es), Nov 19 2008
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EXTENSIONS
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Deleted erroneous comment and added correct b-file by Paolo P. Lava, Nov 06 2017
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STATUS
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approved
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