A137816 -id:A137816 - cp's OEIS frontend

A137815 Year numbers: numbers n such that phi(n) = 2 phi(sigma(n)).

5, 13, 37, 61, 65, 73, 119, 157, 185, 193, 277, 305, 313, 365, 397, 421, 457, 481, 541, 613, 661, 673, 733, 757, 785, 793, 877, 949, 965, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1385, 1453, 1547, 1565, 1615, 1621, 1657, 1753, 1873, 1933, 1985, 1993

Offset: 1

R. K. Guy, R. J. Mathar and M. F. Hasler, Feb 11 2008

Following D. Iannucci, n is called a "year number" if phi(n) / phi(sigma(n)) = 2 (thus 365 is a year number, explaining the terminology).
D. Iannucci asks: Are there any even year numbers? Are there any odd year numbers that are not squarefree?
Remark: If n = q_1 q_2 ... q_k is a product of odd primes such that (q_j + 1)/2 is an odd prime for all j, then n is a year number.
Solution: for nonsquarefree year numbers, see A137816. See A137817-A137819 for year numbers with cubes, 4th powers, 5th powers.
Eric Landquist found year numbers divisible by 7^2, 7^3 and 7^4, as well as 120781449 = 3^8 * 41 * 449.
The existence of even year numbers is still open, but Eric checked all 200-smooth even integers with a single large prime up to 10^8 and found no year numbers among them.
See also references in A082897 (perfect totient numbers).

R. K. Guy, "Euler's Totient Function", "Solutions of phi(m)=sigma(n)", "Iterations of phi and sigma", "Behavior of phi(sigma(n)) and sigma(phi(n))". =A7 B36-B42 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, pp. 138-151, 2004.
Doug Iannucci, in: Gerry Myerson (ed.), 2007 Western Number Theory problems set.

M. F. Hasler, Table of n, a(n) for n=1,...,7499.

Cf. A137816-A137819, A006872 (phi(sigma(n)) = phi(n)), A067704 (phi(sigma(n)) = 2 phi(n)), A082897.

Select[Range[2000],EulerPhi[#]==2EulerPhi[DivisorSigma[1,#]]&]  (* Harvey P. Dale, Mar 18 2011 *)

for( n=1,10^7, eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))

A137819 Year numbers, i.e., phi(n) = 2 phi(sigma(n)), divisible by a 5th prime power.

Original entry on oeis.org

1811079, 4473387, 67009923, 77242167, 88605819, 110475819, 120781449, 132208767, 134082297, 165515319, 183408867, 225548469, 232275681, 272876607, 284339403, 326557251, 349538247, 402371793, 425844621, 501668883, 566867727

Offset: 1

R. K. Guy, R. J. Mathar and M. F. Hasler, Feb 11 2008

nonn

See A137815 for more comments and references about "year numbers". This is the subsequence of elements of A137815 divisible by a 5th prime power. As such, it is of course also a subsequence of A137818 (those divisible by a 4th prime power).

There are only 5 such numbers below 10^8.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A137815-A137818.

for( i=1,#A137816, vecmax(factor(A137816[i])[,2])>4 & print1(A137816[i])", ")

for( n=1,10^9, issquarefree(n) && next; vecmax(factor(n)[,2])>4 || next; eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))

More terms from Russell Pochis (joebobpumpkin(AT)gmail.com), May 30 2010

Typo in data corrected by D. S. McNeil, Aug 17 2010

A067704 Numbers n such that phi(sigma(n)) = 2*phi(n).

Original entry on oeis.org

2, 6, 8, 9, 24, 28, 70, 78, 128, 140, 222, 234, 280, 312, 350, 366, 384, 438, 496, 525, 666, 864, 888, 910, 918, 936, 942, 1036, 1098, 1158, 1232, 1314, 1400, 1464, 1575, 1662, 1708, 1752, 1824, 1836, 1878, 1900, 1938, 2044, 2382, 2480, 2526, 2590, 2664

Offset: 1

Benoit Cloitre, Feb 05 2002

nonn

M. F. Hasler and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 110 terms from Hasler)

Cf. A137815, A137816.

Select[Range[3000],EulerPhi[DivisorSigma[1,#]]==2EulerPhi[#]&] (* Harvey P. Dale, Oct 15 2020 *)

is(n)=2*eulerphi(n=factor(n))==eulerphi(sigma(n)) \\ Charles R Greathouse IV, Nov 27 2013

A137817 Cubeful "year numbers" (such that phi(n) = 2 phi(sigma(n)): A137815).

Original entry on oeis.org

295569, 295947, 1586763, 1811079, 1964375, 2069469, 4473387, 5854375, 6820555, 7285923, 10936053, 12233457, 13260625, 18029709, 18052767, 21153663, 21576537, 21604131, 22182093, 22877451, 23744043, 25536875, 28340307, 34102775

Offset: 1

R. K. Guy, R. J. Mathar and M. F. Hasler, Feb 11 2008

nonn

See A137815 for general comments and references about "year numbers". This is the subsequence of cubeful elements of A137815, i.e. its intersection with A046099. As such, it is of course also a subsequence of A137816.

There are only 56 such numbers below 10^8.

M. F. Hasler and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 56 terms from _M. F. Hasler_)

Cf. A137815-A137819, A006872. A046099. A067704.

for( i=1,#A137816, vecmax( factor( A137816[i] )[,2])>2 && print1(A137816[i]", "))

for( i=1,#A046099, eulerphi(A046099[i])==2*eulerphi(sigma(A046099[i])) && print1( A046099[i] ", "))

for( n=1,10^9, issquarefree(n) && next; vecmax(factor(n)[,2])>2 || next; eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))

A137818 Non-biquadratefree "year numbers": phi(n) = 2 phi(sigma(n)) and p^4 | n for some p>1.

Original entry on oeis.org

295569, 1811079, 1964375, 2069469, 4473387, 5854375, 10936053, 13260625, 18029709, 21576537, 22182093, 25536875, 35595625, 46404333, 49648383, 55094375, 57044817, 58650625, 67009923, 69166467, 72681875, 76106875

Offset: 1

R. K. Guy, R. J. Mathar and M. F. Hasler, Feb 11 2008

nonn

See A137815 for general comments and references about "year numbers". This is the subsequence of elements of A137815 divisible by a biquadrateful number, i.e. its intersection with A046101 (numbers divisible by the 4th power of some prime). As such, it is of course also a subsequence of A137817 and a fortiori of A137816.

There are only 28 such numbers below 10^8.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A137815-A137819, A046101.

for( i=1,#A137816, vecmax( factor( A137816[i])[,2])>3 && print1(A137816[i]", "))

for( i=1,#A046099, eulerphi(A046099[i])==2*eulerphi(sigma(A046099[i])) && print1( A046099[i] ", "))

for( n=1,10^9, issquarefree(n) && next; vecmax(factor(n)[,2])>3 || next; eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))

cp's OEIS Frontend

A137815 Year numbers: numbers n such that phi(n) = 2 phi(sigma(n)).

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A137819 Year numbers, i.e., phi(n) = 2 phi(sigma(n)), divisible by a 5th prime power.

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A067704 Numbers n such that phi(sigma(n)) = 2*phi(n).

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A137817 Cubeful "year numbers" (such that phi(n) = 2 phi(sigma(n)): A137815).

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A137818 Non-biquadratefree "year numbers": phi(n) = 2 phi(sigma(n)) and p^4 | n for some p>1.

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