A227927 -id:A227927 - cp's OEIS frontend

A230201 Numbers k such that sigma(phi(k)) < k.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 46, 48, 50, 54, 58, 60, 64, 66, 68, 70, 72, 78, 80, 84, 90, 92, 94, 96, 98, 100, 102, 106, 108, 110, 114, 118, 120, 126, 128, 130, 132, 136, 138, 140, 144, 150, 156, 160, 162, 166, 168

Offset: 1

Vladimir Letsko, Oct 11 2013

nonn

All terms are even. However, sigma(phi(k)) may be equal to k for an odd number k if k+2 is a Fermat prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001229, A018784, A062402, A066694, A227011, A227927, A230203.

Maple
```
for n do if sigma(phi(n))
				
```

Select[Range[200], DivisorSigma[1, EulerPhi[#]] < # &] (* T. D. Noe, Oct 14 2013 *)

isok(n) = sigma(eulerphi(n)) < n; \\ Michel Marcus, Oct 12 2013

A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) > phi(sigma(m))/sigma(phi(m)) for all k

Original entry on oeis.org

1, 3, 5, 11, 13, 17, 29, 41, 181, 209, 377, 779, 3239, 4469, 5249, 15539, 43259, 58589, 119279, 169679, 174719, 461369, 692687, 955499, 1258949, 1859129, 1917299, 3925463, 7991693, 8986469, 13244069, 16732169, 30629363, 44137523, 48466987, 64018433, 68787773

Offset: 1

Vladimir Letsko, Oct 09 2013

nonn

These are the indices where the rational function A062401(n)/A062402(n) drops below the minimum set by all earlier ratios.

a(2) to a(9) are primes. However all known terms beginning from a(10) are composite.

			5 is in the sequence because phi(sigma(5))/sigma(phi(5)) = 2/7 and for all k < 5, phi(sigma(k))/sigma(phi(k)) > 2/7.

Donovan Johnson, Table of n, a(n) for n = 1..48 (terms < 10^10)

Cf. A227927, A062401, A062402, A033632, A229238.

A062401 := proc(n)
    numtheory[phi](numtheory[sigma](n))
end proc:
A062402 := proc(n)
    numtheory[sigma](numtheory[phi](n))
end proc:
s := proc(n)
    A062401(n)/A062402(n) ;
end proc:
r := 100000000000000000000000000000 ;
for n from 1 do
    if s(n) < r then
        printf("%d,\n",n) ;
        r := s(n) ;
    end if;
end do:

f(n)=eulerphi(sigma(n=factor(n)))/sigma(eulerphi(n))
is(n)=my(t=f(n)); for(k=1,n-1,if(f(k)<=t, return(0))); 1 \\ Charles R Greathouse IV, Nov 27 2013

a(33)-a(37) from Donovan Johnson, Oct 11 2013

A230203 Numbers k such that sigma(phi(k)) > k.

Original entry on oeis.org

5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 52, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 93, 95, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112, 113, 115, 116, 117, 119

Offset: 1

Vladimir Letsko, Oct 11 2013

nonn

It seems that the odd number k is not in the sequence if and only if k+2 is a Fermat prime (A019434).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010 (phi), A000203 (sigma).

Cf. A230201, A001229, A018784, A019434, A066694, A227011, A227927, A062402, A230203, A018784.

for n do if sigma(phi(n))>n then print{n} fi od:

Select[Range[150],DivisorSigma[1,EulerPhi[#]]>#&] (* Harvey P. Dale, Apr 14 2019 *)

is(k) = sigma(eulerphi(k)) > k; \\ Amiram Eldar, Apr 04 2024

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A230201 Numbers k such that sigma(phi(k)) < k.

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A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) > phi(sigma(m))/sigma(phi(m)) for all k

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A230203 Numbers k such that sigma(phi(k)) > k.

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