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2, 6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478
a(n) = sigma(n) + phi(n).
+10
43
2, 4, 6, 9, 10, 14, 14, 19, 19, 22, 22, 32, 26, 30, 32, 39, 34, 45, 38, 50, 44, 46, 46, 68, 51, 54, 58, 68, 58, 80, 62, 79, 68, 70, 72, 103, 74, 78, 80, 106, 82, 108, 86, 104, 102, 94, 94, 140, 99, 113, 104, 122, 106, 138, 112, 144, 116, 118, 118, 184, 122, 126, 140
COMMENTS
a(n) = 2n for n listed in A008578, the prime numbers at the beginning of the 20th century. When a(n) = a(n + 1), n is probably listed in A066198, numbers n where phi changes as fast as sigma (the only exceptions below 10000 are 2 and 854). - Alonso del Arte, Nov 16 2005 If n is semiprime, a(n) = 2n+1+ceiling(sqrt(n))-floor(sqrt(n)). - Wesley Ivan Hurt, May 05 2015
REFERENCES
K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.
LINKS
A. Makowski, Aufgaben 339, Elemente der Mathematik 15 (1960), pp. 39-40.
FORMULA
G.f.: Sum_{k>=1} (mu(k) + 1)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 29 2017
EXAMPLE
a(10) = 22 because there are 4 coprimes to 10 below 10, the divisors of 10 add up to 18, and 4 + 18 = 22.
PROG
(PARI) for (n=1, 1000, write("b065387.txt", n, " ", sigma(n) + eulerphi(n)) ) \\ Harry J. Smith, Oct 17 2009 (Magma) [DivisorSigma(1, k)+EulerPhi(k):k in [1..65]]; // Marius A. Burtea, Feb 09 2020
Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).
+10
8
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
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
EXAMPLE
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.
MATHEMATICA
Select[Range[2600], DivisorSigma[1, # ]+EulerPhi[ # ]==DivisorSigma[1, #+1]+EulerPhi[ #+1]&]
PROG
(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*/
Numbers n such that phi(n+1)-phi(n)=sigma(n+1)-sigma(n).
+10
2
2, 854, 751358, 1421637, 8775206, 8892195, 16485944, 31845344, 95494035, 277653495, 380438505, 744048855, 1091725394, 1615353002, 2284844925, 2491028745, 6345217034, 8490513014, 12784909335, 14177454885, 15669084375, 17694356295, 17836667354, 24180347115
COMMENTS
This sequence is a subsequence of A066198.
MATHEMATICA
de[n_]:=DivisorSigma[1, n]-EulerPhi[n]; Do[If[de[n]==de[n+1], Print[n]], {n, 50000000}] (* Firoozbakht *)
Select[Range[10^6], (EulerPhi[# + 1] - EulerPhi[#]) == (DivisorSigma[1, # + 1] - DivisorSigma[1, #]) &] (* Alonso del Arte, Feb 08 2012 *)
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