A073317 -id:A073317 - cp's OEIS frontend

A073318 a(n) = 2^phi(n) - Sum_{j=0..n} binomial(phi(n), phi(j)).

0, -1, -2, -3, -6, -4, -10, -13, -26, -14, -183, -15, -22, -57, -210, -211, -1730, -58, 25160, -240, -3356, -949, 238031, -241, -256823, -3918, -143243, -3919, 46326924, -242, 281620682, -61817, -639769, -61818, -4718174, -4415, 2023569890, -224436, -7556927, -63639, -43279525745, -4416

Offset: 1

Labos Elemer, Jul 26 2002

a(n) > 0 for {19, 23, 29, 31, 37, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}. Does this hold only for special primes?

No: composites for which a(n) > 0 include 121, 289, 437, 529, 667, 671, 697, 703, 713, 731, .... - Robert Israel, Jan 23 2021

Robert Israel, Table of n, a(n) for n = 1..3322

Cf. A066781, A073318.

g[x_] := EulerPhi[x] Table[Apply[Plus, Table[Binomial[g[n], g[j]], {j, 0, n}]], {n, 1, 50}]

a(n) = A066781(n) - A073317(n).

A073319 Numbers n such that A073318(n) = 2^phi(n) - Sum_{j=0..n} binomial(phi(n), phi(j)) is positive.

Original entry on oeis.org

19, 23, 29, 31, 37, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293

Offset: 1

Labos Elemer, Jul 26 2002

nonn

			Several values are composites: 121, 289, 437, 529, ..., 961, 989. Primes like 2, ..., 17, 41 are not here.

Cf. A066781, A073317, A073318.

g[x_] := EulerPhi[x] Do[s=2^g[n]-Apply[Plus, Table[Binomial[g[n], g[j]], {j, 0, n}]]; If[Sign[s]==1&&!PrimeQ[n], k=k+1; Print[{k, n, PrimeQ[n]}]], {n, 1, 1000}]

Solutions to A066781(x) - A073317(x) > 0.

cp's OEIS Frontend

A073318 a(n) = 2^phi(n) - Sum_{j=0..n} binomial(phi(n), phi(j)).

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