A083250 -id:A083250 - cp's OEIS frontend

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.

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

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 8, 1, 8, 1, 8, 1, 4, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 8, 1, 2, 1, 8, 1, 8, 1, 32, 1, 2, 1, 4, 1, 4, 1, 32, 1, 2, 1, 4, 1, 32, 1, 32, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 16, 1, 64, 1, 8, 1

Offset: 1

Labos Elemer, Apr 28 2003

nonn

a(n)=1 if and only if n is odd or n = 2. - Robert Israel, May 31 2018

			Different from A069177, analogous sequence with totient, instead of cototient.

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

Cf. A000010, A051953, A009195, A083250, A007283, A050339, A053576, A069177.

f:= n -> padic:-ordp(n - numtheory:-phi(n), 2):
map(f, [$1..100]); # Robert Israel, May 31 2018

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

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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A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

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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.

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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.

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