A110176 -id:A110176 - 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

A066435 Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0.

Original entry on oeis.org

2, 1, 0, 5, 4, 2, 14, 2, 0, 5, 22, 43, 26, 7, 0, 496, 34, 2, 38, 37, 0, 11, 46, 6, 50, 13, 0, 4, 26, 10, 62, 929, 282, 17, 28, 252, 20, 19, 0, 101, 8, 14, 12, 19, 17, 23, 38, 307, 98, 25, 54, 65, 106, 51, 14, 14, 0, 29, 118, 66, 56, 30, 0, 8128, 22, 22, 44, 85, 66, 35, 135, 18

Offset: 1

Joseph L. Pe, Dec 27 2001

It would be nice to remove the word "Conjectured" from the description - N. J. A. Sloane.
The values of a(3), a(9), a(15) and a(21) listed above, namely 0, are conjectural. There is no natural number k < 10^6 satisfying the "homomorphic condition" sigma(n+k)=sigma(n)+sigma(k) for n=3,9,15,21.
The terms for which there is no solution k < 10^6 are n = 3, 9, 15, 21, 27, 39, 57, 63, 81, 93, 105, 117, 165, 171, 183, 189, 201, 219, 225, 243,..., which all satisfy n=3 (mod 6). - T. D. Noe, Jan 20 2004
All n<1000 and k<10^10 have been tested. The largest term is a(837)=4631925025. Sequence A110108 gives the n for which there is no solution k<10^10.
All n<1000 and k<10^11 have been tested. The largest term is a(711)=21004780114. - Donovan Johnson, Aug 29 2012

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B15.

Cf. A091554 (primes p such that k=2p is the smallest solution to sigma(p+k)=sigma(p)+sigma(k)).

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

a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], DivisorSigma[ 1, n + # ] == DivisorSigma[ 1, n ] + DivisorSigma[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]

More terms from T. D. Noe, Jan 20 2004

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

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

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A066435 Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0.

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A110177 Number of solutions 0

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