A085713 -id:A085713 - cp's OEIS frontend

A007367 Numbers k such that phi(x) = k has exactly 3 solutions.

2, 44, 56, 92, 104, 116, 140, 164, 204, 212, 260, 296, 332, 344, 356, 380, 392, 444, 452, 476, 524, 536, 564, 584, 588, 620, 632, 684, 692, 716, 744, 764, 776, 836, 860, 884, 932, 956, 980, 1004, 1016, 1112, 1124, 1136, 1172, 1196, 1284, 1292, 1304

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

From Torlach Rush, Jul 23 2018: (Start)
For known terms:
- The greatest common divisor of the three solutions is the distance of the middle solution from the least solution and is half the distance of the middle solution to the largest solution.
- If the number of distinct prime factors of k equals the number of solutions of k = phi(x), then the greatest common divisor of the solutions is the least solution divided by the number of solutions.
- Except for a(1), if the largest prime factor is the same for all solutions and is equal to the greatest common divisor of all solutions then the distance from a(n) to the least solution is gcd({k: phi(k) = a(n)}) + 2.  (End)
By Ford's theorem on Euler totient function, this sequence is infinite. - Jianing Song, Jul 18 2018

			phi(69) = phi(92) = phi(138) = 44, so 44 is a term.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 44, p. 17, Ellipses, Paris, 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Ford's theorem.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000010, A002202, A058277, A085713.

Number of solutions: A007617 (0), A007366 (2), this sequence (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a007367 n = a007367_list !! (n-1)
a007367_list = map fst $
               filter ((== 3) . snd) $ zip a002202_list a058277_list
-- Reinhard Zumkeller, Nov 25 2015

a = Table[ 0, {1500} ]; Do[ p = EulerPhi[ n ]; If[ p < 1501, a[ [ p ] ]++ ], {n, 1, 1500} ]; Select[ Range[ 1500 ], a[ [ # ] ] == 3 & ]
Take[Select[Tally[EulerPhi[Range[50000]]],#[[2]]==3&][[All,1]],50]//Sort (* Harvey P. Dale, Apr 02 2018 *)

is(k) = invphiNum(k) == 3 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A303746 Totients t for which {x: phi(x)=t} share the same largest prime factor.

Original entry on oeis.org

10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 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, 294, 296, 306, 310, 316, 328, 330

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.
A303747 a restriction of this sequence gives a relation T = (P * TS) - TS where T is a term, P is the corresponding prime replicator and TS is the starting or seed totient. The relation fails for a(202) = 1210. 1210 does not equal (11 * a(19)) - a(19), i.e., (11 * 110) - 110.
For known terms, the greatest common divisor of the solutions of a(n) is either a power of the largest prime factor of solutions of a(n), or is evenly divisible by same.

			10 is a term because the largest prime factor of 11 and 22, the solutions of phi(10) is 11.
2 is not a term because there is no common largest prime factor of 3, 4 and 6, the solutions of phi(2).

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
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, A085713.

Subsequence of A303745.

filter:= proc(n) local L,q;
  L:= numtheory:-invphi(n);
  if nops(L) = 0 then return false fi;
  q:= max(numtheory:-factorset(L[1]));
  andmap(t -> max(numtheory:-factorset(t))=q, L[2..-1]);
end proc:
select(filter, [seq(i,i=2..1000,2)]); # Robert Israel, Jun 25 2018

isok(n) = if (n > 1, #Set(apply(x->vecmax(factor(x)[,1]), invphi(n))) == 1); \\ Michel Marcus, May 13 2018

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

A305726 Totients t the solutions of which cannot be divided by a prime to yield the solutions of a totient < t.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 88, 96, 100, 108, 112, 120, 128, 132, 144, 156, 160, 162, 168, 176, 180, 192, 200, 216, 220, 224, 232, 240, 252, 256, 264, 272, 276, 280, 288, 294, 300, 312, 320, 324, 336, 342, 348, 352, 360, 384, 396, 400

Offset: 1

Torlach Rush, Jun 21 2018

nonn

This sequence is the complement of A303746.

			1 is a term because there is no prime that divides the solutions of phi(x) = 1 (1 and 2).
2 is a term because there is no prime that divides the solutions of phi(x) = 2 (3, 4, and 6).
10 is not a term because 11 divides solutions of phi(x) = 10 (11 and 22) yielding 1 and 2 the solutions of phi(x) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (for invphi pari script).
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, A085713, A303746.

isok(n) = {if (istotient(n), v = invphi(n); g = gcd(v); if (g == 1, return (1)); f = factor(g); for (k=1, #f~, vv = v/f[k,1]; nt = eulerphi(vv[1]); if (vv == invphi(nt), return(0));); return (1););} \\ Michel Marcus, Jul 25 2018

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A007367 Numbers k such that phi(x) = k has exactly 3 solutions.

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A303746 Totients t for which {x: phi(x)=t} share the same largest prime factor.

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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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A305726 Totients t the solutions of which cannot be divided by a prime to yield the solutions of a totient < t.

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