A083244 -id:A083244 - cp's OEIS frontend

A083243 Numbers k for which there are more divisors and coprimes than other numbers less than k: A045763(k) < A073757(k) or A045763(k) < k/2 or A073757(k) > k/2.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Labos Elemer, May 07 2003

nonn

			30 is not in the sequence because d(30) + phi(30) - 1 = 8 + 8 - 1 = 15. There are as many divisors and coprimes as there are numbers j <= 30 that neither divide nor are coprime to 30.
50 is not here because d(50) + phi(50) - 1 = 6 + 20 - 1 = 25. There are as many divisors and coprimes as there are numbers j < 50 that neither divide nor are coprime to 50.
146 is here because d(146) + phi(146) - 1 = 4 + 72 - 1 = 75; 146/2 = 73, and 75 > 73.
61455 is here because d(61455) + phi(61455) - 1 = 16 + 30720 - 1 = 30735; 61455/2 = 30727 + 1/2, and 30735 > 61455/2.

Cf. A000005, A000010, A045763, A073757, A020488, A083244, A083245.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[n-u, n/2], Print[n, {d, r, u}]], {n, 1, 100}]
(* Second program: *)
Select[Range[120], DivisorSigma[0, #] + EulerPhi[#] - 1 > #/2 &] (* Michael De Vlieger, Aug 22 2023 *)

{ k : d(k) + phi(k) - 1 > k/2 }.

Data corrected and entry edited by Michael De Vlieger, Aug 22 2023

A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2A045763(n) = 2A073757(n)-n.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 6, 7, 4, 11, 6, 13, 4, 7, 8, 17, 4, 19, 6, 9, 4, 23, 6, 19, 4, 15, 6, 29, 0, 31, 10, 13, 4, 19, 4, 37, 4, 15, 6, 41, -4, 43, 6, 13, 4, 47, 2, 39, 0, 19, 6, 53, -4, 31, 6, 21, 4, 59, -6, 61, 4, 19, 12, 37, -12, 67, 6, 25, -8, 71, -2, 73, 4, 15, 6, 49, -16, 79, 2, 35, 4, 83, -14, 49, 4, 31, 6, 89, -20, 59, 6, 33, 4, 55, -10, 97, -4

Offset: 1

Labos Elemer, May 07 2003

sign

There are only 2 cases [n=30, n=50] below 10^7 such that a(n) = 0.

No other zeros found up to 10^9. - Michel Marcus, Jul 30 2017

			n=37, d=2,r=36,u=0, a(37)=2+36-1-0=37>0; primes are fixed points.
n=42, d=8,r=12,u=23,a(42)=8+12-1-23=-4<0, terms of A083244;
n=30, d=8,r=8,u=15, a(30)=0;
n=50, d=6,r=20,u=25,a(50)=0.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083246, A020488.

Table[2*(DivisorSigma[0, w]+EulerPhi[w]-1)-w, {w, 1, 1000}]

a(n) = 2*(numdiv(n)+eulerphi(n)-1) - n; \\ Michel Marcus, Jul 30 2017

a(n) = 2(A000005(n)+A000010(n)-1)-n.

A083246 Numbers n such that at least one of the following four conditions is satisfied: 1# d(n)=phi(n); 2# d(n)=u(n); 3# phi(n)=u(n), or 4# n=2u(n). Here d(n)=A000005(n) is the number of divisors of n, phi(n)=A000010(n) is Euler's totient and u(n)=A045763(n) is the size of the 'unrelated set'.

Original entry on oeis.org

1, 3, 8, 10, 15, 18, 24, 25, 30, 50, 61455

Offset: 1

Labos Elemer, May 07 2003

Is this sequence complete?

			1# d(n)=phi(n) holds for {1,3,8,10,18,24,30}, see A020488;
2# d(n)=u(n) holds for {15,25};
3# phi(n)=u(n) holds for {61455};
4# n=2u(n) holds for {30,50}. No more cases below 10^7.
{n,d,r,u} values for 11 initial terms are as follows:
{1, 1, 1, 0}, {3, 2, 2, 0}, {8, 4, 4, 1}, {10, 4, 4, 3}, {15, 4, 8, 4}, {18, 6, 6, 7}{24, 8, 8, 9}, {25, 3, 20, 3}, {30, 8, 8, 15}, {50, 6, 20, 25}, {61455, 16, 30720, 30720}.

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083246, A020488.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Equal[d, r]||Equal[d, u]||Equal[r, u]||Equal[u, n-u], Print[n(*, {d, r, u}*)]], {n, 1, 10000000}]

is(n)=my(r=eulerphi(n),d=numdiv(n),u=n-r-d+1);d==r||d==u||r==u||2*u==n \\ Charles R Greathouse IV, Feb 21 2013

A083247 Numbers k such that A000010(k) > A045763(k) > A000005(k).

Original entry on oeis.org

14, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 49, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 81, 82, 85, 86, 87, 91, 92, 93, 94, 95, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 128, 129, 133, 134, 135, 141, 142

Offset: 1

Labos Elemer, May 07 2003

Primes are not terms since A045763(p) = 0 < A000005(p) = 2 for a prime p.

			k = 99 is a term since d(k) = 6, phi(k) = 60, unrelateds(k) = 99 - 6 - 60 + 1 = 34, and 60 > 34 > 6 holds.

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

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083245, A083246, A083248, A020488.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[r, u]&&Greater[u, d], Print[n, {d, r, u}]], {n, 1, 1000}]

is(n)=my(r=eulerphi(n),d=numdiv(n),u=n-r-d+1); r>u && u>d \\ Charles R Greathouse IV, Feb 21 2013

A083248 Numbers k such that A045763(k) > A000010(k) > A000005(k).

Original entry on oeis.org

36, 40, 42, 48, 50, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186, 190, 192, 196, 198, 200, 204, 208, 210

Offset: 1

Labos Elemer, May 07 2003

nonn

Primes are not terms since A045763(p) = 0 < A000005(p) = 2 for a prime p.

			k = 100 is a term since d(k) = 9, phi(k) = 40, unrelateds(k) = 100 - 9 - 40 + 1 = 52, and 52 > 40 > 9 holds.

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

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083245, A083246, A083247, A020488.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[u, r]&&Greater[r, d], Print[n, {d, r, u}]], {n, 1, 1000}]

isok(k) = {my(f = factor(k), d = numdiv(f), r = eulerphi(f), u = k - r - d + 1); u > r && r > d;} \\ Amiram Eldar, Feb 08 2025

cp's OEIS Frontend

A083243 Numbers k for which there are more divisors and coprimes than other numbers less than k: A045763(k) < A073757(k) or A045763(k) < k/2 or A073757(k) > k/2.

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A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2*A045763(n) = 2*A073757(n)-n.

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A083246 Numbers n such that at least one of the following four conditions is satisfied: 1# d(n)=phi(n); 2# d(n)=u(n); 3# phi(n)=u(n), or 4# n=2u(n). Here d(n)=A000005(n) is the number of divisors of n, phi(n)=A000010(n) is Euler's totient and u(n)=A045763(n) is the size of the 'unrelated set'.

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A083247 Numbers k such that A000010(k) > A045763(k) > A000005(k).

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A083248 Numbers k such that A045763(k) > A000010(k) > A000005(k).

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A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2A045763(n) = 2A073757(n)-n.