A145749 Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).

6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 902, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2013, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578

Offset: 1

Author

Farideh Firoozbakht, Nov 01 2008

Keywords

• easy
• nonn

Comments

If n/2 is an odd prime and n+1 is prime then n is in the sequence, the proof is easy. 8,188,285,902,2013,... are terms of the sequence which they aren't of such form. This sequence is a subsequence of A066198.

If p is an odd Sophie Germain prime then 2*p is in the sequence. There is no term of the sequence which is of the form 2*p where p is prime and p isn't Sophie Germain prime. A244438 gives terms of the sequence which isn't of the form 2*p where p is prime. - Farideh Firoozbakht, Aug 14 2014

Examples

			10 is in the sequence because phi(10) + sigma(10) = 4 + 18 = 22 and phi(11) + sigma(11) = 10 + 12 = 22 also.
12 is not in the sequence because phi(12) + sigma(12) = 4 + 28 = 32 but phi(13) + sigma(13) = 12 + 14 = 26.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A065387, A066198, A145748, A005384.

Programs

Mathematica

Select[Range[2600],DivisorSigma[1,# ]+EulerPhi[ # ]==DivisorSigma[1,#+1]+EulerPhi[ #+1]&]

PARI

for(n=1,10^4, s=eulerphi(n)+sigma(n); if(s==eulerphi(n+1)+sigma(n+1), print1(n,", "))) /* Derek Orr, Aug 14 2014*/

Formula

{n: A065387(n)=A065387(n+1)}.