A020490 -id:A020490 - cp's OEIS frontend

A083249 Numbers n with A045763(n) = n + 1 - d(n) - phi(n) < d(n) < phi(n).

5, 7, 9, 11, 13, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Labos Elemer, May 07 2003

For primes this means 0 < 2 < p-1 so primes p greater than 3 are members.
Only two composite solutions below 10000000: n = 9 and n = 16.
From Charles R Greathouse IV, Apr 12 2010: (Start)
d(n) < phi(n) is true for all n > 30 (see A020490), so the main condition is n + 1 - d(n) - phi(n) < d(n). Rewrite this as n - phi(n) < 2d(n) - 1.
If n is composite, then the cototient n - phi(n) >= sqrt(n).
For n > 32760, d(n) < sqrt(n)/2.
So all composite solutions are in 1..32760. Checking these (and applying the other inequality), the only composite members are 9 and 16.
Thus the sequence is the primes greater than 3, together with 9 and 16.
(End)

			n = 9 is a member: 3 divisors, 6 coprimes, 1 (it is 6) unrelated: 6 > 3 > 1;
n = 16 is a member: 5 divisors, 8 coprimes 4 unrelateds ({6, 10, 12, 14}): 8 > 5 > 4.

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

Cf. A000005, A000010, A045763, A073757, A051953.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[r, d]&&Greater[d, u]&&!PrimeQ[n], Print[n, {d, r, u}]], {n, 1, 1000}] (* for composite solutions *) (* corrected by Charles R Greathouse IV, Apr 12 2010 *)
(* Second program: *)
Select[Range@ 272, Function[n, n - (#1 + #2 - 1) < #1 < #2 & @@ {DivisorSigma[0, n], EulerPhi[n]}]] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = if(n>6,prime(n),[5,7,9,11,13,16][n]) \\ Charles R Greathouse IV, Apr 12 2010

Extension, new definition, and edits from Charles R Greathouse IV, Apr 12 2010

A279289 Numbers k such that phi(k) > tau(k).

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Jaroslav Krizek, Dec 09 2016

nonn

Numbers k such that A000010(k) > A000005(k).
There are 11 numbers k such that phi(k) <= tau(k) and 7 numbers k such that phi(k) = tau(k); see A020490 and A020488.
For k >= 31; phi(k) - tau(k) >= 1, see A063070.

			14 is a term because phi(14) = 6 > tau(14) = 4.

Complement of A020490.

Cf. A000005, A000010, A020488, A020491, A063070, A279287, A279288.

[n: n in[1..1000] | EulerPhi(n) gt NumberOfDivisors(n)];

Select[Range@ 77, EulerPhi@ # > DivisorSigma[0, #] &] (* Michael De Vlieger, Dec 11 2016 *)

is(n) = eulerphi(n) > numdiv(n) \\ Felix Fröhlich, Dec 09 2016

a(n)=if(n<20, select(k -> eulerphi(k)>numdiv(k), [5..29])[n], n+11) \\ Charles R Greathouse IV, Dec 16 2016

a(n) = n + 11 for n >= 20.

A279287 a(n) = numerator of (phi(n)/tau(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 5, 2, 6, 3, 2, 8, 8, 1, 9, 4, 3, 5, 11, 1, 20, 3, 9, 2, 14, 1, 15, 8, 5, 4, 6, 4, 18, 9, 6, 2, 20, 3, 21, 10, 4, 11, 23, 8, 14, 10, 8, 4, 26, 9, 10, 3, 9, 7, 29, 4, 30, 15, 6, 32, 12, 5, 33, 16, 11, 3, 35, 2, 36, 9, 20, 6, 15, 3

Offset: 1

Jaroslav Krizek, Dec 09 2016

a(n) = numerator of (A000010(n)/A000005(n)).
See A279288 (denominator of (phi(n)/tau(n))) and A063070 (phi(n)-tau(n)).
a(n) = 1 and A279288(n) = 1 for numbers n in A020488; a(n) > A279288(n) for numbers n in A279289.

			For n = 6: phi(6)/tau(6) = 2/4 = 1/2; a(6) = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A020488, A020490, A020491, A063070, A279288, A279289.

[Numerator(EulerPhi(n)/NumberOfDivisors(n)): n in[1..1000]];

with(numtheory): A279287:=n->numer(phi(n)/sigma(n)): seq(A279287(n), n=1..150); # Wesley Ivan Hurt, Dec 11 2016

Table[Numerator[EulerPhi[n]/DivisorSigma[0, n]], {n, 78}] (* Michael De Vlieger, Dec 09 2016 *)

a(n) = numerator(eulerphi(n)/numdiv(n)) \\ Felix Fröhlich, Dec 09 2016

A279288 a(n) = denominator of (phi(n)/tau(n)).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 2, 1

Offset: 1

Jaroslav Krizek, Dec 09 2016

a(n) = denominator of (A000010(n)/A000005(n)).
See A279287 (numerator of (phi(n)/tau(n))) and A063070 (phi(n)-tau(n)).
a(n) = 1 and A279287(n) = 1 for numbers n in A020488; A279287(n) > a(n) for numbers n in A279289.

			For n = 6: phi(6)/tau(6) = 2/4 = 1/2; a(6) = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A020488, A020490, A020491, A063070, A279287, A279289.

[Denominator(EulerPhi(n)/NumberOfDivisors(n)): n in[1..1000]];

Table[Denominator[EulerPhi[n]/DivisorSigma[0, n]], {n, 120}] (* Michael De Vlieger, Dec 10 2016 *)

a(n) = denominator(eulerphi(n)/numdiv(n)) \\ Felix Fröhlich, Dec 09 2016

a(n) = 1 for numbers in A020491.

A141341 Totally Goldbach numbers: Positive integers k such that for all primes p < k-1 with p not dividing k, k-p is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 18, 24, 30

Offset: 1

Rick L. Shepherd, Jun 25 2008

As Browers et al. point out, A141340 = A141341 union {7,14,16,36,42,48,60,90,210}, A020490 = A141341\{5} and A048597 = A141341\{5,10}. The authors show that the first strategy of Deshouillers et al. to establish a bound (of 10^520) for A141340 is sufficient for then determining the totally Goldbach numbers and "leads us naturally to interesting questions concerning primes in a fixed residue class".

J-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.
David van Golstein Brouwers, John Bamberg and Grant Cairns, Totally Goldbach numbers and related conjectures, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.
Index entries for sequences related to Goldbach conjecture

Cf. A020490, A048597, A141340.

q[k_]:=AllTrue[k-Select[Prime[Range[PrimePi[k-2]]],!Divisible[k,#]&],PrimeQ];Select[Range[30],q[#]&] (* James C. McMahon, Jul 21 2025 *)

A279507 a(n) = floor(phi(n)/tau(n)).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 2, 1, 5, 0, 6, 1, 2, 1, 8, 1, 9, 1, 3, 2, 11, 1, 6, 3, 4, 2, 14, 1, 15, 2, 5, 4, 6, 1, 18, 4, 6, 2, 20, 1, 21, 3, 4, 5, 23, 1, 14, 3, 8, 4, 26, 2, 10, 3, 9, 7, 29, 1, 30, 7, 6, 4, 12, 2, 33, 5, 11, 3, 35, 2, 36, 9, 6, 6, 15, 3, 39, 3

Offset: 1

Jaroslav Krizek, Dec 13 2016

nonn

a(n) = floor(A000010(n)/A000005(n)).
There are 11 numbers n such that phi(n) <= tau(n) and 7 numbers n such that phi(n) = tau(n); see A020490 and A020488.
Sequences b(k) of numbers n such that a(n) = k are finite for all k >=0; see A279508 (the smallest numbers n such that a(n) = k for k>=0) and A279509 (the largest numbers n such that a(n) = k for k>=0).
See A140475 (numbers n such that floor(phi(n)/tau(n)) > floor(phi(m)/tau(m)) for all m < n).

			For n=5; a(5) = floor(phi(5)/tau(5)) = floor(4/2) = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..2000

Cf. A000005, A000010, A020488, A020490, A140475, A279289, A279508, A279509.

[Floor(EulerPhi(n)/NumberOfDivisors(n)): n in[1..100]]

Table[Floor[EulerPhi[n]/DivisorSigma[0, n]], {n,1,25}] (* G. C. Greubel, Dec 13 2016 *)

for(n=1, 25, print1(floor(eulerphi(n)/numdiv(n)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n) > 1 for numbers in A279289.

A279508 a(n) = smallest number k such that floor(phi(k)/tau(k)) = n.

Original entry on oeis.org

2, 1, 5, 7, 27, 11, 13, 58, 17, 19, 55, 23, 65, 106, 29, 31, 85, 142, 37, 158, 41, 43, 115, 47, 119, 125, 53, 133, 145, 59, 61, 254, 262, 67, 274, 71, 73, 298, 1180, 79, 187, 83, 203, 346, 89, 209, 235, 382, 97, 394, 101, 103, 169, 107, 109, 253, 113, 458, 295

Offset: 0

Jaroslav Krizek, Dec 13 2016

nonn

a(n) = the smallest number k such that floor(A000010(k)/A000005(k)) = A279507(k) = n.
Sequences b_n of numbers k such that floor(phi(k)/tau(k)) = n for n = 0..2:
b_0: 2, 4, 6, 12;
b_1: 1, 3, 8, 10, 14, 16, 18, 20, 24, 30, 36, 42, 48, 60;
b_2: 5, 9, 15, 22, 28, 32, 40, 54, 66, 72, 84, 90, 96, 120, 180.
Sequences b_n are finite for all n >=0. See A279509 (largest number k such that floor(phi(k)/tau(k)) = n).
Supersequence of A045344 (primes excluding 3).

			For n = 2; a(2) = 5 because 5 is the smallest number with floor(phi(5) / tau(5)) = floor(4/2) = 2.

Cf. A000005, A000010, A020488, A020490, A045344, A279289, A279507, A279509.

[Min([n: n in[1..100000] | Floor(EulerPhi(n)/NumberOfDivisors(n)) eq k]): k in [0..60]]

Table[k = 1; While[Floor[EulerPhi[k]/DivisorSigma[0, k]] != n, k++]; k, {n, 0, 58}] (* Michael De Vlieger, Dec 14 2016 *)

a(n) = my(k=1); while(floor((eulerphi(k)/numdiv(k)))!=n, k++); k \\ Felix Fröhlich, Dec 14 2016

a((p-1)/2) = p for p = prime > 3.

A279509 a(n) = largest number k such that floor(phi(k)/tau(k)) = n.

Original entry on oeis.org

12, 60, 180, 240, 420, 480, 840, 462, 1260, 1680, 1440, 690, 2520, 2100, 2160, 2310, 3360, 2400, 3780, 5040, 4620, 3600, 3300, 1410, 5460, 4080, 6300, 7560, 5880, 4140, 9240, 2646, 10080, 6600, 6480, 7200, 10920, 8820, 9360, 2370, 13860, 8640, 8160, 15120

Offset: 0

Jaroslav Krizek, Dec 19 2016

nonn

a(n) = largest number k such that floor(A000010(k)/A000005(k)) = A279507(k) = n.
Sequences b_n of numbers k such that floor(phi(k)/tau(k)) = n for n = 0..2:
b_0: 2, 4, 6, 12;
b_1: 1, 3, 8, 10, 14, 16, 18, 20, 24, 30, 36, 42, 48, 60;
b_2: 5, 9, 15, 22, 28, 32, 40, 54, 66, 72, 84, 90, 96, 120, 180.
Sequences b_n are finite for all n >= 0. See A279508 (smallest number k such that floor(phi(k)/tau(k)) = n).

			For n = 1; a(1) = 60 because 60 is the largest number with floor(phi(60)/tau(60)) = floor(16/12) = 1.

Cf. A000005, A000010, A020488, A020490, A279289, A279507, A279508.

[Max([n: n in[1..100000] | Floor(EulerPhi(n) / NumberOfDivisors(n)) eq k]): k in [0..50]]

A275581 Numbers n such that A010846(n) >= n/2.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 18, 30

Offset: 1

Michael De Vlieger, Dec 25 2016

Let 1 <= r <= n be a number "regular to" n, that is, a product of prime divisors p that also divide n.
This sequence includes numbers n such that there are at least as many regulars r among the range 1 <= m <= n as m that are nonregular to n.
Divisors d are a special case of regular r such that d also divides n in addition to all prime divisors p of d also dividing n.
With the exception of r = 1, all regular numbers r are part of the cototient of n.
For prime n, A010846(n) = A000005(n). Composite n > 4 can have nondivisor r. The only prime numbers in a(n) are {2,3}.
Regular r divides some power of n, but reliably divides n^r. When r is a divisor d of n, r | n.
Because the divisors of n are a subset of the regulars of n, it is interesting to compare A010846(n) with A000005(n) = tau(n):
A010846(n) = n for n = (1,2). Tau(n) = n for n = (1,2).
n/2 < A010846(n) < n for n = (3,4,6,10,12,18,30). n/2 < tau(n) < n for n = (3,4,6).
A010846(n) = n/2 for n = 8. Tau(n) = n/2 for n = (8,12).
Sequence A020490 includes numbers n such that tau(n) >= phi(n). The number 24 is in A020490 but not in a(n). This is because A010846(24) = 11 (cf. A162306(24) = {1,2,3,4,6,8,9,12,16,18,24}).
Numbers n such that tau(n) >= n/2: {1,2,3,4,6,8,12}.

Cf. A010846, A020490, A051293, A162306.

Select[Range[10^3], 2 Count[Range@ #, k_ /; PowerMod[#, k, k] == 0] >= # &] (* or *)
Select[Range[10^3], Function[n, 2 Total[MoebiusMu[#] Floor[n/#] &@ Select[Range@ n, CoprimeQ[#, n] &]] >= n]] (* or *)
Select[Range[10^3], Function[n, 2 (1 + Count[Range@ n, m_ /; SubsetQ[FactorInteger[n][[All, 1]], FactorInteger[m][[All, 1]]]]) >= n]]

cp's OEIS Frontend

A083249 Numbers n with A045763(n) = n + 1 - d(n) - phi(n) < d(n) < phi(n).

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A279289 Numbers k such that phi(k) > tau(k).

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A279287 a(n) = numerator of (phi(n)/tau(n)).

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A279288 a(n) = denominator of (phi(n)/tau(n)).

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A141341 Totally Goldbach numbers: Positive integers k such that for all primes p < k-1 with p not dividing k, k-p is prime.

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A279507 a(n) = floor(phi(n)/tau(n)).

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A279508 a(n) = smallest number k such that floor(phi(k)/tau(k)) = n.

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