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A058213
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Triangular arrangement of solutions of phi(x) = 2^n (n >= 0), where phi=A000010 is Euler's totient function. Each row corresponds to a particular n and its length is n+2 for 0 <= n <= 31, 32 for n >= 32. (This assumes that there are only 5 Fermat primes.)
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5
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1, 2, 3, 4, 6, 5, 8, 10, 12, 15, 16, 20, 24, 30, 17, 32, 34, 40, 48, 60, 51, 64, 68, 80, 96, 102, 120, 85, 128, 136, 160, 170, 192, 204, 240, 255, 256, 272, 320, 340, 384, 408, 480, 510, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 771, 1024, 1028, 1088
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OFFSET
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0,2
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COMMENTS
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phi(x) is a power of 2 if and only if x is a power of 2 multiplied by a product of distinct Fermat primes. So if, as is conjectured, there are only 5 Fermat primes, then there are only 32 possibilities for the odd part of x, namely the divisors of 2^32-1, given in A004729.
The same numbers, in increasing order, are given in A003401.
The first entry in row n is the n-th divisor of 2^32-1 for 0 <= n <= 31 (A004729) and is 2^(n+1) for n >= 32. The last entry in row n is given in A058215.
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LINKS
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EXAMPLE
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Triangle begins:
{ 1, 2},
{ 3, 4, 6},
{ 5, 8, 10, 12},
{15, 16, 20, 24, 30},
{17, 32, 34, 40, 48, 60},
{51, 64, 68, 80, 96, 102, 120},
{85, 128, 136, 160, 170, 192, 204, 240},
...
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MATHEMATICA
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phiinv[ n_, pl_ ] := Module[ {i, p, e, pe, val}, If[ pl=={}, Return[ If[ n==1, {1}, {} ] ] ]; val={}; p=Last[ pl ]; For[ e=0; pe=1, e==0||Mod[ n, (p-1)pe/p ]==0, e++; pe*=p, val=Join[ val, pe*phiinv[ If[ e==0, n, n*p/pe/(p-1) ], Drop[ pl, -1 ] ] ] ]; Sort[ val ] ]; phiinv[ n_ ] := phiinv[ n, Select[ 1+Divisors[ n ], PrimeQ ] ]; Join@@(phiinv[ 2^# ]&/@Range[ 0, 10 ]) (* phiinv[ n, pl ] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[ n ] = list of x with phi(x)=n *)
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CROSSREFS
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Cf. A000010, A001317, A003401, A004729, A019434, A045544, A047999, A053576, A054432, A058214, A058215.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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