A110174 -id:A110174 - cp's OEIS frontend

A110173 Least k such that phi(n) = phi(k) + phi(n-k) for 0

0, 0, 1, 2, 0, 0, 0, 4, 4, 4, 0, 6, 0, 4, 5, 8, 0, 6, 0, 6, 5, 6, 0, 6, 6, 4, 11, 6, 0, 0, 0, 16, 6, 8, 10, 12, 0, 4, 13, 12, 0, 12, 0, 6, 7, 8, 0, 12, 0, 10, 16, 6, 0, 6, 26, 12, 19, 26, 0, 30, 0, 4, 12, 32, 24, 24, 0, 6, 23, 28, 0, 18, 0, 10, 12, 8, 24, 12, 0, 24, 0, 8, 0, 24, 8, 4, 6, 12, 0, 30

Offset: 1

T. D. Noe, Jul 15 2005

nonn

Sequence A110174 gives the number of solutions 0A110175.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A066426 (least k such that phi(n)+phi(k)=phi(n+k)), A110174.

Cf. also A110176.

a[n_] := Select[Range[n-1], EulerPhi[n]==EulerPhi[n-# ]+EulerPhi[ # ]&]; Table[s=a[n]; If[Length[s]==0, 0, First[s]], {n, 150}]

A110173(n) = { my(ph=eulerphi(n)); for(k=1,n-1,if(ph == (eulerphi(k)+eulerphi(n-k)), return(k))); (0); }; \\ Antti Karttunen, Feb 20 2023

A110177 Number of solutions 0

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 2, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 4

Offset: 1

T. D. Noe, Jul 15 2005

nonn

The number of solutions is always even because k=n/2 cannot be a solution for even n.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A110176 (least k such that sigma(n)=sigma(k)+sigma(n-k)).

Cf. also A110174.

a[n_] := Select[Range[n-1], DivisorSigma[1, n]==DivisorSigma[1, n-# ]+DivisorSigma[1, # ]&]; Table[Length[a[n]], {n, 150}]

A110177(n) = { my(x=sigma(n)); sum(k=1,n-1,(x == (sigma(k)+sigma(n-k)))); }; \\ Antti Karttunen, Feb 20 2023

cp's OEIS Frontend

A110173 Least k such that phi(n) = phi(k) + phi(n-k) for 0

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