A073757 -id:A073757 - 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

A083244 k is in the sequence iff the number of numbers unrelated to k is larger than that of related ones[=divisors and coprimes] to k: A045763(k) > A073757(k) or A045763(k) > k/2 or A073757(k) < k/2.

Original entry on oeis.org

42, 54, 60, 66, 70, 72, 78, 84, 90, 96, 98, 100, 102, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 180, 182, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 228, 230, 234, 238, 240, 242, 246, 250, 252

Offset: 1

Labos Elemer, May 07 2003

nonn

			k = 42 is a term because d = 8 divisors, r = 12 coprimes and u = 23 unrelated belong to it: u = 23 > 19 = 8 + 12 - 1 = d + r - 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A045763, A073757, A020488, A083243, A083245.

filter:= n -> n > 2*(numtheory:-tau(n) + numtheory:-phi(n)-1):
select(filter, [$1..1000]); # Robert Israel, May 15 2017

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-d-r+1; If[Greater[u, n/2], Print[n, {d, r, u}]], {n, 1, 100}]
(* Second program: *)
Select[Range@ 256, # - (DivisorSigma[0, #] + EulerPhi[#] - 1) > #/2 &] (* Michael De Vlieger, Jul 22 2017 *)

Numbers k such that k - d(k) - phi(k) + 1 > k/2.

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.

A083266 Sum of related numbers (counted in A073757) belonging to n: a(n) = A000203(n) + A023896(n) - 1; related = {divisor-set, RRS}.

Original entry on oeis.org

1, 3, 6, 10, 15, 17, 28, 30, 39, 37, 66, 51, 91, 65, 83, 94, 153, 92, 190, 121, 157, 145, 276, 155, 280, 197, 282, 223, 435, 191, 496, 318, 377, 325, 467, 306, 703, 401, 523, 409, 861, 347, 946, 523, 617, 577, 1128, 507, 1085, 592, 887, 721, 1431, 605, 1171

Offset: 1

Labos Elemer, May 13 2003

Sum of 1 <= m <= n such that gcd(m, n) is either 1 or m. - Michael De Vlieger, Apr 07 2021.

			n=10: related terms = {1,2,5,10,3,7,9}, sum = 1+2+5+10+1+3+7+9-1 = 37 = a(10).

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

Cf. A073757 (count), A083267 (product), A083268 (lcm).

Table[DivisorSigma[1, n] + Total@ Select[Range[2, n - 1], GCD[n, #] == 1 &], {n, 55}] (* or *)
{1}~Join~Array[DivisorSigma[1, #] + # EulerPhi[#]/2 - 1 &, 54, 2] (* Michael De Vlieger, Apr 07 2021 *)

a(n)=if(n>1,sigma(n)+n*eulerphi(n)/2-1,1) \\ Charles R Greathouse IV, Feb 19 2013

A083250 Numbers n for which abs(A045763(n) - A073757(n)) = 3, i.e., signed difference of size of related and unrelated sets to n equals either 3 or -3.

Original entry on oeis.org

3, 825, 1755, 4641, 5313, 56865, 58395, 140049, 159152763, 361701435

Offset: 1

Labos Elemer, May 07 2003

No more terms below 10^9. - Amiram Eldar, Jan 27 2019

			For n=140049: d=40 divisors, r=70026 coprimes and u=70023 unrelated numbers to n; abs(r-u) = 3.

Cf. A000005, A000010, A045763, A073757, A083243-A083249.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 3], Print[n(*, {d, r, u}*)]], {n, 1, 100}]

isok(n) = abs(n-2*eulerphi(n)-2*numdiv(n)+2) == 3; \\ Michel Marcus, Jul 30 2017

a(9)-a(10) from Amiram Eldar, Jan 27 2019

A083251 Numbers n such that abs(A045763(n) - A073757(n)) = 2, i.e., signed difference of size of related and unrelated sets to n equals either 2 or -2.

Original entry on oeis.org

2, 48, 72, 80, 112, 176, 208, 272, 304, 368, 464, 496, 592, 656, 688, 752, 848, 944, 976, 1072, 1136, 1168, 1264, 1328, 1424, 1552, 1616, 1648, 1712, 1744, 1808, 2032, 2096, 2192, 2224, 2384, 2416, 2512, 2608, 2672, 2768, 2864, 2896

Offset: 1

Labos Elemer, May 07 2003

nonn

			For n=2896: d=10 divisors, r=1440 coprimes, u=1447 unrelated or n - u = r + d - 1 = 1449 related numbers to n; thus abs(1449 - 1447) = 2.

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

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 2], Print[n(*, {d, r, u}*)]], {n, 1, 3000}]

a(n) = 8 * (A076274(n-1) + 1) for n > 3, as proved by Lawrence Sze. - Ralf Stephan, Nov 16 2004

A083267 Product of related numbers (counted in A073757) belonging to n; related = {divisor-set, RRS}: a(n) = A007955(n)*A001783(n).

Original entry on oeis.org

1, 2, 6, 24, 120, 180, 5040, 6720, 60480, 18900, 39916800, 665280, 6227020800, 3783780, 201801600, 2075673600, 355687428096000, 496215720, 121645100408832000, 69837768000, 20858213376000, 604969665300, 25852016738884976640000, 12336143339520, 5170403347776995328000

Offset: 1

Labos Elemer, May 13 2003

nonn

			For n = 10: related terms = {1,2,5,10,3,7,9}, product = 1*2*5*10*1*3*7*9 = 18900 = a(10).

Amiram Eldar, Table of n, a(n) for n = 1..450

Cf. A073757 (count), A083266 (sum), A083268 (LCM), A083267 (product), A001783, A007955.

a[n_] := n^(DivisorSigma[0, n]/2) * Times@@ Select[Range[n], CoprimeQ[n, #] &]; Array[a, 30] (* Amiram Eldar, Jun 20 2024 *)

More terms from Amiram Eldar, Jun 20 2024

A083268 a(n) is the lcm of related numbers to n (counted in A073757): related = {divisor-set, RRS}.

Original entry on oeis.org

1, 2, 6, 12, 60, 30, 420, 840, 2520, 630, 27720, 4620, 360360, 90090, 120120, 720720, 12252240, 1531530, 232792560, 58198140, 77597520, 29099070, 5354228880, 892371480, 26771144400, 3346393050, 80313433200, 20078358300, 2329089562800

Offset: 1

Labos Elemer, May 13 2003

nonn

			For n = 10: related terms = {1,2,5,10,3,7,9}; lcm(10,1,3,7,9) = 630 = a(10).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A073757 (count), A083266 (sum), A083268 (LCM), A083267 (product), A038610.

a[n_] := LCM @@ Join[{n}, Select[Range[n], CoprimeQ[n, #] &]]; Array[a, 30] (* Amiram Eldar, Jun 20 2024 *)

a(n)=my(t=lcm([1..n])/n,g); while((g=gcd(t,n))>1,t/=g); t*n \\ Charles R Greathouse IV, Nov 14 2014

a(n) = lcm(n, A038610(n)).

A083252 Numbers k for which abs(A045763(k) - A073757(k)) = 5, i.e., signed difference of size of related and unrelated sets to k equals either 5 or -5.

Original entry on oeis.org

5, 105, 315, 182835, 960075, 7838265, 4291166265

Offset: 1

Labos Elemer, May 07 2003

a(7), if it exists, is > 10^9. - Vaclav Kotesovec, Sep 06 2019

			For k = 960075: d = 36 divisors, r = 480000 coprimes, u = 480040 unrelated; k - u = r + d - 1 = 480035 related numbers to k; thus abs(480040 - 480035) = 5.

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250, A083251.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 5], Print[n(*, {d, r, u}*)]], {n, 1, 3000}]

isok(n) = abs(n-2*eulerphi(n)-2*numdiv(n)+2) == 5; \\ Michel Marcus, Jul 29 2017

a(6) from Michel Marcus, Jul 29 2017

a(7) from Amiram Eldar, Feb 02 2025

A083253 Smallest number k for which abs(A045763(k) - A073757(k)) = n, i.e., signed difference of size of related and unrelated sets to k equals either n or -n.

Original entry on oeis.org

30, 1, 2, 3, 4, 5, 8, 7, 16, 21, 32, 11, 64, 13, 84, 27, 78, 17, 200, 19, 90, 57, 140, 23, 102, 69, 120, 435, 114, 29, 132, 31, 126, 93, 392, 81, 138, 37, 156, 49, 230, 41, 168, 43, 322, 129, 260, 47, 150, 77, 180, 795, 186, 53, 204, 95, 198, 885, 280, 59, 434, 61, 228, 183

Offset: 0

Labos Elemer, May 07 2003

nonn

a(258) > 10^5. - Michael De Vlieger, Jul 31 2017

			A045763(x) - A073757(x) = 0 is first satisfied at x = 30 = a(0).

Michael De Vlieger, Table of n, a(n) for n = 0..257

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250, A083251, A083252.

With[{s = Table[Abs[n - 2 (DivisorSigma[0, n] + EulerPhi[n] - 1)], {n, 10^3}]}, TakeWhile[#, # > 0 &] &@ Flatten@ Map[FirstPosition[s, #] /. k_ /; MissingQ@ k -> 0 &, Range[0, Max@ s]]] (* Michael De Vlieger, Jul 31 2017 *)

a(n) = {my(k = 1); while (abs(k - 2*(numdiv(k) + eulerphi(k) - 1)) != n, k++); k;} \\ Michel Marcus, Aug 01 2017

a(n) = min{x; abs(A045763(x) - A073757(x)) = n}.

a(p) = p, for p prime.

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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A083244 k is in the sequence iff the number of numbers unrelated to k is larger than that of related ones[=divisors and coprimes] to 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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A083266 Sum of related numbers (counted in A073757) belonging to n: a(n) = A000203(n) + A023896(n) - 1; related = {divisor-set, RRS}.

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A083250 Numbers n for which abs(A045763(n) - A073757(n)) = 3, i.e., signed difference of size of related and unrelated sets to n equals either 3 or -3.

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A083251 Numbers n such that abs(A045763(n) - A073757(n)) = 2, i.e., signed difference of size of related and unrelated sets to n equals either 2 or -2.

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A083267 Product of related numbers (counted in A073757) belonging to n; related = {divisor-set, RRS}: a(n) = A007955(n)*A001783(n).

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A083268 a(n) is the lcm of related numbers to n (counted in A073757): related = {divisor-set, RRS}.

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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.