A109606 -id:A109606 - cp's OEIS frontend

A078892 Numbers n such that phi(n) - 1 is prime, where phi is Euler's totient function (A000010).

5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 33, 35, 36, 38, 39, 42, 43, 44, 45, 49, 50, 51, 52, 54, 56, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 77, 78, 80, 81, 84, 86, 90, 91, 92, 93, 95, 96, 98, 99, 102, 103, 104, 105, 109, 111, 112, 117

Offset: 1

Reinhard Zumkeller, Dec 12 2002

For all primes p: p is in the sequence iff p is the greater member of a twin prime pair (A006512), see A078893.

Union of A006512 and A078893. - Ray Chandler, May 26 2008

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

Cf. A000010, A000040, A008864, A109606, A006512, A039698, A072281, A078893.

[n: n in [1..200] | IsPrime(EulerPhi(n)-1)]; // Vincenzo Librandi, Aug 13 2013

Select[Range[200], PrimeQ[EulerPhi[#] - 1]&] (* Vincenzo Librandi, Aug 13 2013 *)

is(n)=isprime(eulerphi(n)-1) \\ Charles R Greathouse IV, Feb 21 2013

A078893 Composite numbers k such that phi(k) - 1 is prime, where phi is Euler's totient function (A000010).

Original entry on oeis.org

8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 33, 35, 36, 38, 39, 42, 44, 45, 49, 50, 51, 52, 54, 56, 62, 64, 65, 66, 68, 69, 70, 72, 77, 78, 80, 81, 84, 86, 90, 91, 92, 93, 95, 96, 98, 99, 102, 104, 105, 111, 112, 117, 120, 121, 122, 123, 124, 129, 130, 133

Offset: 1

Reinhard Zumkeller, Dec 12 2002

nonn

A078892 with the primes removed. - Ray Chandler, May 26 2008

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

Cf. A000010, A000040, A008864, A109606, A078892, A066071, A068019.

Select[Range[150],CompositeQ[#]&&PrimeQ[EulerPhi[#]-1]&] (* Harvey P. Dale, Dec 28 2021 *)

is(n)=!isprime(n) && isprime(eulerphi(n)-1) \\ Charles R Greathouse IV, Feb 21 2013

A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, -1, -2, -5, 1, -8, 1, -9, -13, -9, 1, -16, 1, -22, -21, -17, 1, -28, -14, -21, -20, -36, 1, -43, 1, -31, -37, -29, -45, -49, 1, -33, -45, -62, 1, -67, 1, -64, -64, -41, 1, -69, -34, -64, -61, -78, 1, -68, -77, -96, -69, -53, 1, -88, 1, -57, -96, -79, -93, -115, 1, -106, -85, -123, 1, -95, 1, -69

Offset: 1

Antti Karttunen, Jan 05 2021

Conversely, the Dirichlet inverse of this sequence yields a sequence which is one less than A319340, i.e., pointwise sum s(n) = A109606(n) + A023900(n).

a(9796) = 0 is the only zero among the first 2^22 terms.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A023900, A109606, A319340, A340090, A340094, A340197, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));
A340198(n) = if(1==n,1,-sumdiv(n,d,if(dA319340(n/d)-1)*A340198(d),0)));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA319340(n/d)-1) * a(d).

A214287 Primes of the form phi(n)-1 sorted by increasing n, where phi is the Euler totient function.

Original entry on oeis.org

3, 5, 3, 5, 3, 3, 11, 5, 7, 7, 5, 17, 7, 11, 7, 19, 11, 17, 11, 7, 29, 19, 23, 11, 17, 23, 11, 41, 19, 23, 41, 19, 31, 23, 17, 23, 59, 29, 31, 47, 19, 31, 43, 23, 23, 71, 59, 23, 31, 53, 23, 41, 23, 71, 43, 59, 71, 31, 41, 59, 31, 101, 47, 47, 107, 71, 47, 71, 31, 109, 59, 79, 59, 83

Offset: 1

Vincenzo Librandi, Jul 13 2012

Primes in A109606.

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

Cf. A000010, A078892 (associated n), A214288.

Select[Table[EulerPhi[n]-1,{n,1,1000}],PrimeQ]

A336191 Numbers k of the form k = ab (the decimal concatenation of a and b) such that phi(ab) = a*b + 1.

Original entry on oeis.org

57, 195, 319, 5595, 11709, 77097, 114765, 1313667, 1348559, 4752465, 10219099, 11031119, 185573199, 2918945715, 3165616929, 12233666703, 16996664613, 18052811909, 20650199699, 38081370319, 58943659521, 195823876095, 236323024041, 242687655369, 342764528277, 677924155713

Offset: 1

M. Farrokhi D. G., Jul 11 2020

Is the sequence infinite?

			phi(57) = 5 * 7 + 1
phi(195) = 1 * 95 + 1 = 19 * 5 + 1
phi(319) = 31 * 9 + 1
phi(5595) = 5 * 595 + 1
phi(11709) = 11 * 709 + 1
phi(77097) = 7 * 7097 + 1
phi(114765) = 11 * 4765 + 1
phi(1313667) = 1313 * 667 + 1
phi(1348559) = 134855 * 9 + 1
phi(4752465) = 47 * 52465 + 1
phi(10219099) = 1021 * 9099 + 1
phi(11031119) = 1103111 * 9 + 1
phi(185573199) = 185 * 573199 + 1

Cf. A000010, A109606, A336192.

seqQ[n_] := Module[{d = IntegerDigits[n]}, MemberQ[Times @@@ Table[FromDigits /@ {Take[d, k], Take[d, -Length[d] + k]}, {k, 1, Length[d] - 1}], EulerPhi[n] - 1]]; Select[Range[10, 10^5], seqQ] (* Amiram Eldar, Jul 11 2020 *)

isok(m) = {my(tm=eulerphi(m)-1, d=digits(m)); for (i=1, #d-1, if (fromdigits(vector(i, k, d[k]))*fromdigits(vector(#d-i, k, d[i+k])) == tm, return(1)););} \\ Michel Marcus, Jul 11 2020

Missing terms 1348559 & 4752465, and a(12) from Amiram Eldar, Jul 11 2020

More terms from Giovanni Resta, Jul 13 2020

A291880 Numbers n such that phi(n) - 1 | sigma(n).

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 20, 22, 40, 76, 80, 108, 160, 204, 320, 640, 1072, 1280, 2560, 4192, 5120, 10240, 20480, 40960, 49344, 81920, 163840, 327680, 655360, 1310720, 2621440, 4197376, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 268460032, 335544320, 671088640, 1073790976, 1342177280, 2684354560, 5368709120

Offset: 1

Paolo P. Lava, Sep 05 2017

nonn

Numbers n such that A109606(n) | A000203(n).
All numbers of the form 5*2^x, with x >= 0, are part of the sequence (A020714).
Values of the ratio sigma(n)/(phi(n)-1) are 4, 7, 2, 12, 5, 6, 6, 4, 6, 4, 6, 8, 6, 8, 6, 6, 4, 6, 6, 4, 6, 6, 6, 6, 8, 6, 6, 6, 6, 6, 6, 4, 6, ...
Sequence contains also terms of the form 2^(n-2)*(2^n+3) where 2^n+3 is a prime and n > 3, like 22, 76, 1072, 4192, 4197376, 268460032. See A057733 for primes of the form 2^n+3. - Michel Marcus, Sep 17 2017

			sigma(1072) = 2108, phi(1072) = 528 and 2108/(528 - 1) = 4.

Cf. A000010, A000203, A020714, A056097, A057733, A109606.

with(numtheory): P:=proc(q) local n; for n from 3 to q do
if type(sigma(n)/(phi(n)-1),integer) then  print(n); fi; od;  end: P(10^7);

Select[Range[3, 10^6], Divisible[DivisorSigma[1, #], EulerPhi[#] - 1] &] (* Michael De Vlieger, Sep 06 2017 *)

isok(n) = denominator(sigma(n)/(eulerphi(n)-1)) == 1; \\ Michel Marcus, Sep 06 2017

a(34)-a(41) from Michel Marcus, Sep 15 2017

a(42)-a(45) from Michel Marcus, Sep 21 2017

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A078892 Numbers n such that phi(n) - 1 is prime, where phi is Euler's totient function (A000010).

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A078893 Composite numbers k such that phi(k) - 1 is prime, where phi is Euler's totient function (A000010).

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A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

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A214287 Primes of the form phi(n)-1 sorted by increasing n, where phi is the Euler totient function.

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A336191 Numbers k of the form k = ab (the decimal concatenation of a and b) such that phi(ab) = a*b + 1.

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A291880 Numbers n such that phi(n) - 1 | sigma(n).

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