A137815 -id:A137815 - cp's OEIS frontend

A137816 Nonsquarefree "year numbers" (numbers n such that phi(n) = 2*phi(sigma(n)): A137815).

5491, 8075, 25317, 27455, 71383, 72283, 76131, 104975, 138575, 193041, 203167, 295569, 295947, 298775, 334951, 356915, 361415, 400843, 451535, 492275, 509575, 572975, 589475, 595975, 654493, 683757, 815975, 862087, 876627, 919075, 936729

Offset: 1

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

nonn

This is the subsequence of nonsquarefree elements of A137815. See there for more comments and references.

There are only 32 such numbers below 10^6, 145 below 10^7 and 785 below 10^8.

M. F. Hasler, Table of n, a(n) for n=1,...,785.

Cf. A137815-A137819, A006872, A067704.

Reap[For[n=1, n<10^6, n++, If[!SquareFreeQ[n] && EulerPhi[n] == 2*EulerPhi[ DivisorSigma[1, n]], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2016 *)

for( n=1,10^6, !issquarefree(n) && eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))

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", "))

A006872 Numbers k such that phi(k) = phi(sigma(k)).

Original entry on oeis.org

1, 3, 15, 26, 39, 45, 74, 104, 111, 117, 122, 146, 175, 183, 195, 219, 296, 314, 333, 357, 386, 471, 488, 549, 554, 555, 579, 584, 585, 608, 626, 646, 657, 794, 831, 842, 914, 915, 939, 962, 1071, 1082, 1095, 1191, 1226, 1256, 1263, 1292, 1322, 1346

Offset: 1

Jeffrey Shallit, N. J. A. Sloane

nonn

S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
R. K. Guy, Unsolved Problems in Number Theory, B42.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..12394 from Marius A. Burtea)
S. W. Golomb, Letter to N. J. A. Sloane, Oct. 1992
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Cf. A000010, A000203, A062401, A353637 (characteristic function).
Positions of zeros in A353636.
Setwise difference of A353684 and A353683, and also of A353685 and A353686.
Intersection of A353684 and A353685.
Subsequences: A260021, A353634, A353635, A353679 (odd terms).
Cf. also A033631, A033632, A067880, A137815, A260020.

a006872 n = a006872_list !! (n-1)
a006872_list = filter (\x -> a000010' x == a000010' (a000203' x)) [1..]
-- Reinhard Zumkeller, Jul 14 2015

[n:n in [1..2000]| EulerPhi(SumOfDivisors(n)) eq EulerPhi(n)]; // Marius A. Burtea, Jan 01 2019

Select[Range@ 1350, EulerPhi@ # == EulerPhi@ DivisorSigma[1, #] &] (* Michael De Vlieger, Jan 01 2019 *)

lista(nn) = {for (i=1, nn, if (eulerphi(i)==eulerphi(sigma(i)), print1(i, ", ")););} \\ Michel Marcus, May 25 2013

More terms from Jud McCranie

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

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", "))

A226953 Leap year numbers: numbers n such that tau(phi(n)) = phi(tau(n))^2, where tau(n) is the number of divisors of n and phi(n) the Euler totient function.

Original entry on oeis.org

1, 2, 9, 14, 15, 18, 20, 22, 46, 94, 118, 166, 214, 231, 248, 286, 308, 310, 334, 344, 350, 351, 358, 366, 372, 392, 399, 405, 406, 430, 454, 483, 490, 494, 516, 518, 522, 526, 532, 536, 568, 595, 598, 632, 638, 644, 654, 663, 666

Offset: 1

Jean-François Alcover, Jun 24 2013

Paraphrasing Doug Iannucci, n is called a "leap year number" if tau(phi(n)) = phi(tau(n))^2 (366 is a leap year number, hence the sequence name). The beast number is a leap year number. The only prime leap year number is 2.

			phi(666)=216, tau(216)=16, tau(666)=12, phi(12)=4, 4^2=16, therefore 666 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A137815 (Doug Iannucci's "year numbers").

Select[Range[1000], DivisorSigma[0, EulerPhi[#]] == EulerPhi[DivisorSigma[0, #]]^2 &]

cp's OEIS Frontend

A137816 Nonsquarefree "year numbers" (numbers n such that phi(n) = 2*phi(sigma(n)): A137815).

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

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A006872 Numbers k such that phi(k) = phi(sigma(k)).

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

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A226953 Leap year numbers: numbers n such that tau(phi(n)) = phi(tau(n))^2, where tau(n) is the number of divisors of n and phi(n) the Euler totient function.

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