A058340 -id:A058340 - cp's OEIS frontend

A175178 a(n)=Values of cardinality of rooted trees CRT for successive primes.

1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 2, 4, 6, 1, 1, 2, 9, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 6, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 16, 1, 9, 2, 1, 1, 1, 1, 1, 7, 1, 19, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 6, 3, 1, 1, 2, 1, 11, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1

Offset: 1

Artur Jasinski, Mar 01 2010

nonn

Karpenko A.S. 2006. Lukasiewicz's Logics and Prime Numbers (English translation).
Karpenko A.S. 2000. Lukasiewicz's Logics and Prime Numbers (Russian).

A058340, A058340, A096827, A175177.

A303747 Totients t for which gcd({x: phi(x)=t}) equals the largest prime factor of each member of {x: phi(x)=t}.

Original entry on oeis.org

10, 22, 28, 30, 44, 46, 52, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 116, 126, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 184, 190, 196, 198, 204, 208, 210, 212, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 296, 306, 310, 316, 328, 330, 332, 344, 346

Offset: 1

Torlach Rush, Apr 29 2018

nonn

Terms of this sequence are totients selected by prime replicators of totients not terms of this sequence.
Following are some examples of terms and their corresponding prime replicators for increasing cardinality of solutions:
#({x: phi(x)=t}) = 2: {(10,11),(22,23),(28,29),(30,31),(46,47),(52,53),...}
#({x: phi(x)=t}) = 3: {(44,23),(56,29),(92,47),(104,53),(116,59),(140,71),...}
#({x: phi(x)=t}) = 4: {(184,47),(208,53),(328,83),(424,107),(664,167),...}
#({x: phi(x)=t}) = 5: {(368,47),(416,53),(656,83),(848,107),(1328,167),...}
#({x: phi(x)=t}) = 6: {(984,83),(1272,107),(6024,503),(7824,653),...}
...
Denote the starting or seed totient for each of the above TS and we have {1,2,4,8,12,...}. We then have a relation between all of the terms (T) and their corresponding primes (P), which is T = (P * TS) - TS.
The values of the GCD of the solutions of terms of this sequence are the terms of A058340.

			10 is a term because the largest prime factor of 11 and 22, the solutions of phi(x)=10 is 11 which is also the greatest common divisor of the solutions of phi(x)=10.
54 is not a term because while 3 is the largest prime factor of solutions phi(x)=54, 3 <> gcd({x: phi(x)=54}) = 81.

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP scripts for miscellaneous problems
K. B. Stolarski and S. Greenbaum, A Ratio Associated with phi(x) = n, The Fibonacci Quarterly, Volume 23, Number 3, August 1985, pp. 265-269.

Cf. A000010, A002202, A058340, A085713.

Intersection of A303745 and A303746.

filter:= proc(n) local L,q;
  L:= numtheory:-invphi(n);
  if nops(L) = 0 then return false fi;
  q:= igcd(op(L));
  if not isprime(q) then return false fi;
  andmap(t -> max(numtheory:-factorset(t))=q, L);
end proc:
select(filter, [seq(i,i=2..1000,2)]); # Robert Israel, Jun 25 2018

isok(n) = my(v=invphi(n)); ((g=gcd(v)) > 1) && (s = Set(apply(x->vecmax(factor(x)[,1]), invphi(n)))) && (#s == 1) && (s[1] == g); \\ Michel Marcus, May 13 2018

Definition clarified by Robert Israel, Jun 25 2018

A068014 Nonprimes n such that 1+phi(n) and -1 + sigma(n) are prime numbers.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 40, 46, 55, 57, 58, 60, 63, 74, 76, 86, 88, 93, 111, 114, 117, 118, 124, 126, 135, 145, 153, 158, 166, 178, 184, 186, 190, 194, 198, 206, 208, 209, 216, 221, 224, 230, 232, 238, 250, 252, 254, 260, 266, 270, 278, 280, 295, 297, 298

Offset: 1

Labos Elemer, Feb 08 2002

nonn

1+A000010(n) and -1+A000203(n) are primes but n is nonprime.

			For n = 38, phi(38) + 1 = 19 and sigma(38) - 1 = 1 + 2 + 19 + 38 - 1 = 59. [corrected by _Peter Munn_, Dec 30 2017]

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A058339, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

Do[s=-1+DivisorSigma[1, n]; s1=1+EulerPhi[n]; If[PrimeQ[s]&&PrimeQ[s1]&&!PrimeQ[n], Print[{n, s1, s}]], {n, 1, 1000}] (* generates sequence and related primes too *)
Select[Range@ 300, And[CompositeQ@ #, AllTrue[{1 + EulerPhi@ #, -1 + DivisorSigma[1, #]}, PrimeQ]] &] (* Michael De Vlieger, Dec 29 2017 *)

isok(n) = !isprime(n) && isprime(1+eulerphi(n)) && isprime(sigma(n)-1); \\ Michel Marcus, Dec 29 2017

A271983 The smaller of a pair n, m such that phi(n) = phi(m) and there is no other k such that phi(n) = phi(k).

Original entry on oeis.org

1, 11, 23, 29, 31, 47, 53, 81, 59, 67, 71, 79, 83, 103, 107, 121, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 343, 307, 311, 317, 331, 361, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479, 491, 499

Offset: 1

Geoffrey Critzer, Apr 17 2016

nonn

If phi(x) = N has exactly two solutions, x = n and x = m, say (see A007366), it is conjectured that one of n and m is odd and the other even.

This sequence differs from A058340 in that it contains nonprime integers. The first few are 81, 121, 343, 361, 529, 649, 841, 961, 1219, 1331, 1537, 1633, ...

			81 is a term because phi(81) = phi(162) = 54 (= A007366(8)).

Cf. A007366, A058340.

(* takes about 2 minutes, can return the sequence up to terms less than 5760=Euler phi(13 primorial) *)
Prepend[Select[
   Table[Flatten[Position[Table[EulerPhi[n], {n, 1, 30030}], m]], {m,
     2, 500, 2}], Length[#] == 2 &][[All, 1]], 1]

Edited by N. J. A. Sloane, Apr 22 2016 at the suggestion of Franklin T. Adams-Watters.

A280587 Composite numbers k such that phi(x) = phi(k) has only 2 solutions: x = k and x = 2*k.

Original entry on oeis.org

81, 121, 343, 361, 529, 649, 841, 961, 1219, 1331, 1537, 1633, 1837, 1849, 1909, 1969, 2047, 2209, 2401, 2497, 2773, 2809, 2959, 3113, 3127, 3151, 3223, 3421, 3481, 3487, 3841, 3901, 3953, 4189, 4321, 4399, 4531, 4661, 4741, 4829, 4897, 4913, 5041, 5129, 5137, 5191, 5269, 5401, 5539, 5753

Offset: 1

Thomas Ordowski and Altug Alkan, Jan 06 2017

nonn

All terms are odd.
If q > 7 is in A005385, then q^2 is in the sequence.
The sequence has a positive natural density. - Information from Carl Pomerance, Jan 07 2017

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

Cf. A000010, A005385, A058340 (such primes).

A173883 a(n) = number of iterations in the sequence of classes of prime numbers for prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 6, 5, 4, 4, 4, 4, 4, 6, 4, 5, 4, 5, 4, 5, 5, 4, 4, 4, 4, 5, 6, 5, 4, 6, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8

Offset: 2

Artur Jasinski, Mar 01 2010

nonn

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006, pp. 98-102.

Cf. A058340, A096827, A175177, A175178, A175179.

Edited, corrected and extended by Arkadiusz Wesolowski, Jan 19 2013

cp's OEIS Frontend

A175178 a(n)=Values of cardinality of rooted trees CRT for successive primes.

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A303747 Totients t for which gcd({x: phi(x)=t}) equals the largest prime factor of each member of {x: phi(x)=t}.

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A068014 Nonprimes n such that 1+phi(n) and -1 + sigma(n) are prime numbers.

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A271983 The smaller of a pair n, m such that phi(n) = phi(m) and there is no other k such that phi(n) = phi(k).

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A280587 Composite numbers k such that phi(x) = phi(k) has only 2 solutions: x = k and x = 2*k.

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A173883 a(n) = number of iterations in the sequence of classes of prime numbers for prime(n).

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