A145748 -id:A145748 - cp's OEIS frontend

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

Farideh Firoozbakht, Nov 01 2008

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

			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.

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

Cf. A065387, A066198, A145748, A005384.

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

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*/

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

A066198 Numbers n where phi changes as fast as sigma, i.e., abs(phi(n+1) - phi(n)) = abs(sigma(n+1) - sigma(n)).

Original entry on oeis.org

2, 6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 854, 862, 886, 902, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2013, 2026, 2038, 2062, 2098, 2206, 2446, 2458

Offset: 1

Joseph L. Pe, Dec 16 2001

nonn

This sequence is the union of two sequences A145748 and A145749. See comment lines of A145749. [Farideh Firoozbakht, Nov 01 2008]

			|phi(7) - phi(6)| = |6 - 2| = |8 - 12| = |sigma(7) - sigma(6)|.
|phi(9) - phi(8)| = |6 - 4| = 2 = |13 - 15| = |sigma(9) - sigma(8)|.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A065387, A145748, A145749.

Select[ Range[ 1, 10^4 ], Abs[ DivisorSigma[ 1, # + 1 ] - DivisorSigma[ 1, # ] ] == Abs[ EulerPhi[ # + 1 ] - EulerPhi[ # ] ] & ]

{ n=0; for (m=1, 10^9, if (abs(eulerphi(m + 1) - eulerphi(m)) == abs(sigma(m + 1) - sigma(m)), write("b066198.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010

More terms from Jason Earls, Jun 05 2002

cp's OEIS Frontend

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

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