A179188 -id:A179188 - cp's OEIS frontend

A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

3, 5, 8720288051472, 9134280520365, 41544070492925, 42466684755492, 51363581614342, 68616494581632, 113312918293575, 210911076210835, 215517565688425, 294988451482725, 383617980270525, 432759876053505, 442863123838135, 532068058516992, 892813363927485, 923102743748185, 929531173876305

Offset: 1

Michel Marcus and Giovanni Resta, Feb 29 2020

nonn

10^15 < a(20) <= 1089641067389872.
Also terms: 1248817919303952, 1332436545865422, 1394926716616125, 1868522795664525, 1950445682260072.
a(4) and a(9) appear in Kevin Ford's paper.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
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.
Mathematics StackExchange, Conjecture on the gap between integers having the same number of co-primes, Sep 25 2019.

Cf. A000010, A007015.
Cf. A001274, A001494, A179186, A179187, A179188, A179189, A179202, A330429.
Cf. A276503, A276504, A217139.

Select[Range[100000], EulerPhi[#] == EulerPhi[# + 3] &] (* Alonso del Arte, Mar 01 2020 *)

isok(k) = eulerphi(k) == eulerphi(k+3); \\ Michel Marcus, Feb 29 2020

A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

9, 15, 1005079920836, 13695542245376, 26160864154416, 27402841561095, 27599063056565, 110263115897935, 124632211478775, 127400054266476, 154090744843026, 205849483744896, 231019991767556, 339938754880725, 459718637643265, 632733228632505, 646552697065275, 683008674773416, 884965354448175

Offset: 1

Giovanni Resta, Mar 01 2020

nonn

a(20) > 10^15.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
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. A000010, A007015.
Cf. A001274, A001494, A330251, A179186, A179187, A179188, A179189, A179202.
Cf. A276503, A276504, A217139.

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

A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

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A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

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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

A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

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Mathematica

PARI

A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

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Author

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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).