A217141 -id:A217141 - cp's OEIS frontend

A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

48, 68, 72, 78, 86, 88, 114, 143, 144, 156, 157, 164, 168, 186, 192, 203, 216, 222, 247, 273, 292, 356, 402, 432, 444, 450, 452, 456, 612, 654, 728, 732, 762, 798, 834, 864, 876, 884, 932, 942, 964, 1032, 1054, 1080, 1086, 1124, 1147, 1152, 1194, 1209, 1220

Offset: 1

Michel Marcus, Sep 27 2012

Most of numbers n in this sequence are divisible by 2, and it appears that n/2 belongs to A179188. The other ones are listed in sequence A217141.

Proof of the comment: If n is even and not a multiple of 4 then phi(n)=phi(n/2). If n is a multiple of 4 then phi(n)=2 * phi(n/2). So when k is a multiple of 4 and phi(n)=phi(n+k), then phi(n/2)=phi(n/2+k/2). QED. This also applies to A179186, A179202. - Jud McCranie, Dec 30 2012

Jud McCranie, Table of n, a(n) for n = 1..10000
Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Cf. A000010, A179188, A217141, A007015.

[n: n in [1..3000] | EulerPhi(n) eq EulerPhi(n+12)]; // Vincenzo Librandi, Sep 08 2016

Select[Range[1, 5000], EulerPhi[#] == EulerPhi[# + 12] &] (* Vincenzo Librandi, Jun 24 2014 *)

{op=vector(N=12); for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), print1(n-N, ", ")))}

A217140 a(n) = m/n where m is the least number divisible by n such that phi(m) = phi(m+6n).

Original entry on oeis.org

24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 60, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36

Offset: 1

Michel Marcus and Jonathan Sondow, Oct 01 2012

nonn

			A179188(1)=24 is divisible by 1 and the quotient is 24, so a(1)=24.
A217139(1)=48 is divisible by 2 and the quotient is 24, so a(2)=24.
The first solution to phi(n)=phi(n+18) to be divisible by 3 is 72 and the quotient is 24, so a(3)=24.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A007015, A179188, A217139, A217141, A217068.

A217142 Least number m such that phi(m) = phi(m+6n) and m is not divisible by n.

Original entry on oeis.org

143, 52, 101, 124, 104, 123, 183, 156, 248, 144, 208, 267, 241, 365, 219, 248, 312, 306, 496, 369, 288, 432, 241, 543, 369, 468, 482, 386, 730, 444, 438, 432, 496, 1220, 624, 779, 612, 801, 915, 744, 723, 582, 576, 1095, 864, 488, 482, 641, 1086, 674, 738

Offset: 2

Michel Marcus, Sep 27 2012

nonn

Donovan Johnson, Table of n, a(n) for n = 2..1000

Cf. A000010, A179188, A217068, A217139, A217141.

fpr(Nmax, lim) = {for (i=2, Nmax,N = i*6;op = vector(N);f = 0;for (n=1, lim, if (op[n%N+1]+0==op[n%N+1]=eulerphi(n), if ((n-N) % i != 0, f = n-N;break;);););print1(f, ", "););}

A217619 a(n) = m/(12n) where m is the least multiple of n that satisfies phi(m) = phi(m+6n).

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2

Offset: 1

Jonathan Sondow and Michel Marcus, Oct 09 2012

nonn

It appears that A217140(n) is divisible by 12 for all n.

			A179188(1)=24 is divisible by 1 and the quotient 24 when divided by 12 gives 2, so a(1)=2.
A217139(1)=48 is divisible by 2 and the quotient 24 when divided by 12 gives 2, so a(2)=2.
A217140(5)=36 and 36/12=3, so a(5)=3.

S. W. Graham, J. J. Holt and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.

Cf. A179188, A217139, A217141, A217068, A217140.

a(n) = A217140(n)/12.

cp's OEIS Frontend

A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

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A217140 a(n) = m/n where m is the least number divisible by n such that phi(m) = phi(m+6n).

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A217142 Least number m such that phi(m) = phi(m+6n) and m is not divisible by n.

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A217619 a(n) = m/(12n) where m is the least multiple of n that satisfies phi(m) = phi(m+6n).

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cp's OEIS Frontend

A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

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Author

Keywords

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Programs

Magma

Mathematica

PARI

A217140 a(n) = m/n where m is the least number divisible by n such that phi(m) = phi(m+6n).

Views

Author

Keywords

Examples

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Crossrefs

A217142 Least number m such that phi(m) = phi(m+6n) and m is not divisible by n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A217619 a(n) = m/(12*n) where m is the least multiple of n that satisfies phi(m) = phi(m+6*n).

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A217619 a(n) = m/(12n) where m is the least multiple of n that satisfies phi(m) = phi(m+6n).