A045763 -id:A045763 - cp's OEIS frontend

A272619 Irregular array read by rows: n-th row contains (in ascending order) the numbers 1 <= k < n such that at least one prime divisor p of k also divides n and at least one prime divisor q of k is coprime to n.

0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 0, 10, 0, 6, 10, 12, 6, 10, 12, 6, 10, 12, 14, 0, 10, 14, 15, 0, 6, 12, 14, 15, 18, 6, 12, 14, 15, 18, 6, 10, 12, 14, 18, 20, 0, 10, 14, 15, 20, 21, 22, 10, 15, 20, 6, 10, 12, 14, 18, 20, 22, 24, 6, 12, 15, 18, 21, 24, 6, 10, 12, 18, 20, 21, 22, 24, 26, 0, 14, 21, 22

Offset: 1

Michael De Vlieger, May 03 2016

The k are the "semitotatives" of n as counted by A243823(n).
All nonzero terms k are composite and pertain to composite rows n. This is because prime k must either divide or be coprime to n, and k = 1 is both a divisor of and coprime to n. Further, the terms k must have at least two distinct prime divisors p and q.
Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.
Row n for prime powers p^e contains all the numbers k in the corresponding row of A133995. There is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e, thus none of the terms of the corresponding row of A133995 are in A272618(n).
Rows n = 4 and 6 are special cases of composite n that contains zero. 4 is the smallest composite number; there are no composites k < n. 6 has the prime divisors 2 and 3, thus 5 is the smallest prime coprime to 6; the product of the minimum prime divisor and minimum prime coprime to 6 is 10, which exceeds 6 and falls outside the considered range. The situation is not so for composite n > 6. Thus rows n for composite n > 6 contain at least 1 nonzero value.
The smallest k of row n = A096014(n) < n, i.e., those values of A096014(n) pertaining to composite n > 6, a product of the smallest prime divisor p of n and the smallest prime q coprime to n. The smallest k of n are even squarefree semiprimes since 2 either divides n or is coprime to n and k is by definition a number with at least two distinct primes. The smallest k = 2p for p^2 sets record values for A096014(n) when we ignore values pertaining to prime n, n = 4, and n = 6.
In base n, 1/a(n) has a mixed recurrent expansion.

			For n = 12, the numbers 1 <= k < n such that the prime divisors p of k also divide n are {2, 3, 4, 6, 8, 9}; {2, 3, 4, 6} divide n = 12, thus row n = 12 is {8, 9}.
n:   k
1:   0
2:   0
3:   0
4:   0
5:   0
6:   0
7:   0
8:   6
9:   6
10:  6
11:  0
12: 10
13:  0
14:  6 10 12
15:  6 10 12
16:  6 10 12 14
17:  0
18: 10 14 15
19:  0
20:  6 12 14 15 18

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 144-5, Theorem 136.

Michael De Vlieger, Table of n, a(n) for n = 1..10447 (rows 1 to 256, flattened).
M. De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
M. De Vlieger, Neutral Numbers.
M. De Vlieger, Sequence page.

The union of nonzero terms of a(n) and A272618 = A133995, thus A243822(n) + A243823(n) = A045763(n).

Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n, Function[m, And[! SubsetQ[r, First /@ FactorInteger@ m], 1 < GCD[m, n] < n]]]], {n, 30}] /. {} -> {0} // Flatten (* Michael De Vlieger, May 03 2016 *)

A243823 Quantity of "semitotatives," numbers m < n that are products of at least one prime divisor p of n and one prime q coprime to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 3, 3, 4, 0, 3, 0, 5, 5, 6, 0, 6, 3, 8, 6, 9, 0, 5, 0, 11, 8, 11, 7, 11, 0, 13, 10, 14, 0, 12, 0, 16, 14, 17, 0, 18, 5, 19, 14, 20, 0, 21, 11, 22, 16, 23, 0, 19, 0, 25, 20, 26, 13, 25, 0, 27, 20, 27, 0, 31, 0, 30, 27, 31, 13, 32, 0, 35, 23, 34, 0, 33, 17, 36, 25, 38, 0, 35, 15, 39, 27, 40, 19, 45, 0, 44, 32, 46

Offset: 1

Michael De Vlieger, Jun 11 2014

nonn

Semitotatives m < n have a regular factor that is the product of prime divisors of n, and a coprime factor that is the product of primes q that are coprime to n.

The unit fractions of semitotatives have a mixed recurrent expansion in base n (See Hardy & Wright).

			For n = 10 with prime divisors {2, 5} and prime totatives {3, 7}, the only semitotative is 6. For n = 16, with the prime divisor 2 and the prime totatives {3, 5, 7, 11, 13}, there are four semitotatives {6, 10, 12, 14}.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth Edition, Oxford University Press, 2008, pages 144-145 (last part of Theorem 136).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
M. De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.

Cf. A045763, A243822.

f:= n -> n + 1 - numtheory:-phi(n) - add(numtheory:-mobius(k)*floor(n/k), k=select(t -> igcd(n,t)=1, [$1..n])):
map(f, [$1..100]); # Robert Israel, May 10 2016

Table[n + 1 - EulerPhi@ n - Total[MoebiusMu[#] Floor[n/#] &@ Select[Range@ n, CoprimeQ[#, n] &]], {n, 120}] (* Michael De Vlieger, May 10 2016 *)

a(n) = A045763(n) - A243822(n).

a(n) = n + 1 - phi(n) - Sum_{1 <= k <= n, gcd(n, k) = 1} mu(k)*floor(n/k). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.

A133995 Irregular array read by rows: n-th row contains (in numerical order) the positive integers <= n which are neither divisors of n nor are coprime to n. A 0 is put into row n if there are no such integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 6, 6, 4, 6, 8, 0, 8, 9, 10, 0, 4, 6, 8, 10, 12, 6, 9, 10, 12, 6, 10, 12, 14, 0, 4, 8, 10, 12, 14, 15, 16, 0, 6, 8, 12, 14, 15, 16, 18, 6, 9, 12, 14, 15, 18, 4, 6, 8, 10, 12, 14, 16, 18, 20, 0, 9, 10, 14, 15, 16, 18, 20, 21, 22, 10, 15, 20, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 6, 12, 15, 18, 21, 24

Offset: 1

Leroy Quet, Oct 01 2007

Row n has length A264441(n).
The number of nonzero entries in row n is A045763(n).
Row n has a 0 if every positive integer <= n is coprime to n or divides n.
From Michael De Vlieger, Aug 19 2017: (Start)
When row n is not empty (and here represented by 0), the terms of row n are composite, since primes p < n must either divide or be coprime to n and the empty product 1 both divides and is coprime to all numbers. For the following, let prime p divide n and prime q be coprime to n.
Row n is empty for n < 8 except n = 6.
There are two distinct species of term m of row n. The first are nondivisor regular numbers g in A272618(n) that divide some integer power e > 1 of n. In other words, these numbers are products of primes p that also divide n and no primes q that are coprime to n, yet g itself does not divide n. Prime powers n = p^k cannot have numbers g in A272618(n) since they have only one distinct prime divisor p; all regular numbers g = p^e with 0 <= e <= k divide p^k. The smallest n = 6 for which there is a number in A272618. The number 4 is the smallest composite and is equal to n = 4 thus must divide it; 4 is coprime to 5. The number 4 is neither coprime to nor a divisor of 6.
The second are numbers h in A272619(n) that are products of at least one prime p that divides n and one prime q that is coprime to n.
The smallest n = 8 for which there is a number in A272619 is 8; the number 6 is the product of the smallest two distinct primes. 6 divides 6 and is coprime to 7. The number 6 is neither coprime to nor a divisor of the prime power 8; 4 divides 8 and does not appear in a(8).
There can be no other species since primes p <= n divide n and q < n are coprime to n, and products of primes q exclusive of any p are coprime to n.
As a consequence of these two species, rows 1 <= n <= 5 and n = 7 are empty and thus have 0 in row n.
(End)

			The divisors of 12 are: 1,2,3,4,6,12. The positive integers which are <= 12 and are coprime to 12 are: 1,5,7,11. So row 12 contains the positive integers <= 12 which are in neither of these two lists: 8,9,10.
The irregular triangle T(n, k) begins:
n\k 1  2  3  4  5  6  7 ...
1:  0
2:  0
3:  0
4:  0
5:  0
6:  4
7:  0
8:  6
9:  6
10: 4  6  8
11: 0
12: 8  9 10
13: 0
14: 4  6  8 10 12
15: 6  9 10 12
16: 6 10 12 14
17: 0
18: 4  8 10 12 14 15 16
19: 0
20: 6  8 12 14 15 16 18
... formatted by _Wolfdieter Lang_, Jan 16 2016

Robert Israel, Table of n, a(n) for n = 1..10014 (rows 1 to 237, flattened)

Cf. A045763, A133994, A264441, A272618, A272619.

row:= proc(n) local r;
   r:= remove(t -> member(igcd(t, n), [1, t]), [$1..n]):
   if r = [] then 0 else op(r) fi
end proc:
A:= [seq](row(n), n=1..30); # Robert Israel, Jan 19 2016

Table[Select[Range@ n, Nor[Divisible[n, #], CoprimeQ[n, #]] &] /. {} -> {0}, {n, 27}] // Flatten (* Michael De Vlieger, Aug 19 2017 *)

a(n) = union(A272618(n), A272619(n)). - Michael De Vlieger, Aug 19 2017

More terms from Alvin Hoover Belt, Jan 21 2008

Edited by Wolfdieter Lang, Jan 16 2016

A073762 a(n) = 24*n - 12.

Original entry on oeis.org

12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 852, 876, 900, 924, 948, 972, 996, 1020, 1044, 1068, 1092, 1116, 1140, 1164, 1188, 1212

Offset: 1

Labos Elemer, Aug 08 2002

Previous name: "Smallest unrelated number belonging to a term of this sequence equals 8."
This is also the list of numbers k such that A259748(k)/k = 5/12. - José María Grau Ribas, Jul 12 2015.
Also the total number of line segments creating a stellated octahedron, where the length of each stellated edge equals n-1, and where the octahedron has 12 edges, each fixed at unit length. - Peter M. Chema, Apr 28 2016

			URSet[12] = {8,9,10} so 12 is here.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016825, A045763, A073758, A073759, A073760, A259748.

[24*n-12: n in [1..60]]; // Vincenzo Librandi, Jun 15 2011

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Do[s=Min[unr[n]]; If[Equal[s, 8], Print[n]], {n, 1, 1000}]
Range[12, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)

a(n)=24*n-12 \\ Charles R Greathouse IV, Jun 14 2011

x='x+O('x^100); Vec(12*(1+x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015

Min{URS[m]} = 8, where UNR[m] = Complement[RRS[m], Divisors[m]].
a(n) = 24*n - 12. - Max Alekseyev, Mar 03 2007
a(n) = 12*A005408(n-1). - Danny Rorabaugh, Oct 22 2015
G.f.: 12*x*(1 + x)/(1 - x)^2. - Ilya Gutkovskiy, Apr 28 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/48. - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 12*(exp(x)*(2*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A361098 Intersection of A360765 and A360768.

Original entry on oeis.org

36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 338, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 468, 480, 484, 486, 490, 500, 504, 507, 525

Offset: 1

Michael De Vlieger, Mar 15 2023

nonn

Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.
Subset of A126706. All terms are neither prime powers nor squarefree.
From Michael De Vlieger, Aug 03 2023: (Start)
Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.
This sequence contains {A002182 \ A168263}. (End)

			For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
For prime power n = p^e > 4, e > 0, A360543(n) = p^(e-1) - e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e | n with e > 0.
For n = 18, A360480(n) = | {10, 14, 15} | = 3,
            A360543(n) = | {} | = 0,
            A361235(n) = | {4, 8, 16} | = 3,
            A355432(n) = | {12} | = 1.
Therefore 18 is not in the sequence.
For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
            A360543(n) = | {30} | = 1,
            A361235(n) = | {8, 16, 27, 32} | = 4,
            A355432(n) = | {24} | = 1.
Therefore 36 is the smallest term in the sequence.
Table pertaining to the first 12 terms:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A000005, phi = A000010.
    n |  a + b =  c | d + e = f | g + tau + phi - 1 =  n
  ------------------------------------------------------
   36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 +  9 + 12 - 1 =  36
   48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 =  48
   50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 +  6 + 20 - 1 =  50
   54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 +  8 + 18 - 1 =  54
   72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 =  72
   75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 +  6 + 40 - 1 =  75
   80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 =  80
   96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 =  96
   98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 +  6 + 42 - 1 =  98
  100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 +  9 + 40 - 1 = 100
  108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
  112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the k-th pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).

Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A355432, A360480, A360543, A361235.

Subsets: A095990\{18}, A286708, A303606, A359280.

nn = 2^16;
a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Reap[ Do[n = s[[j]];
    If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
      FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]]

A066760 a(n) = Sum_{1<=k<=n, k is not a divisor of n and k is not coprime to n} k.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 6, 6, 18, 0, 27, 0, 40, 37, 42, 0, 79, 0, 89, 74, 108, 0, 145, 45, 154, 96, 183, 0, 274, 0, 210, 184, 270, 163, 360, 0, 340, 257, 411, 0, 556, 0, 467, 418, 504, 0, 669, 140, 683, 439, 657, 0, 880, 369, 805, 548, 810, 0, 1183, 0, 928, 779, 930, 502

Offset: 1

Jon Perry, Jan 17 2002

This function can be used to prove no p^k is perfect or multi-perfect.

			There are three integers that satisfy this definition for n = 12, namely 8, 9 and 10. These sum to 27, hence a(12) = 27.

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

Cf. A000010, A000203, A000217, A023896, A024816, A045763, A067392.

[0] cat [1 + n*(n+1)/2 - SumOfDivisors(n) - n*EulerPhi(n)/2: n in [2..70]]; // Vincenzo Librandi, Nov 03 2014

f:= n -> 1 + n*(n+1)/2 - numtheory:-sigma(n) - n*numtheory:-phi(n)/2;
0, seq(f(n),n=2..100); # Robert Israel, Nov 02 2014

Table[n(n + 1)/2 + 1 - EulerPhi[n] * n/2 - DivisorSigma[1, n], {n, 2, 65}] (* Robert G. Wilson v, Jul 31 2004 *)
Table[Sum[k * Boole[Not[Divisible[n, k]]] * Boole[GCD[n, k] > 1], {k, n - 1}], {n, 65}] (* Alonso del Arte, Nov 02 2014 *)

a(n)=n*(n + 1 - eulerphi(n))\2 + 1 - sigma(n) \\ Charles R Greathouse IV, Nov 02 2014

a(n) = 1 + n*(n+1)/2 - sigma(n) - n*phi(n)/2.
a(n) = 0 if and only if n = 1, 4 or a prime. - Robert G. Wilson v, Jul 31 2004
a(n) = 1 + A067392(n) - A000203(n). - Amiram Eldar, Dec 03 2023
a(n) = Sum_{k=1..n} k * (1 - floor(1/gcd(n,k))) * (ceiling(n/k) - floor(n/k)). - Wesley Ivan Hurt, Jan 06 2024

Offset corrected to 1 by Michael De Vlieger, Jul 05 2014

A073759 Largest number < n that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 6, 6, 8, 0, 10, 0, 12, 12, 14, 0, 16, 0, 18, 18, 20, 0, 22, 20, 24, 24, 26, 0, 28, 0, 30, 30, 32, 30, 34, 0, 36, 36, 38, 0, 40, 0, 42, 42, 44, 0, 46, 42, 48, 48, 50, 0, 52, 50, 54, 54, 56, 0, 58, 0, 60, 60, 62, 60, 64, 0, 66, 66, 68, 0, 70, 0, 72, 72, 74, 70, 76

Offset: 1

Labos Elemer, Aug 15 2002

nonn

Largest "unrelated" number to n.
From Michael De Vlieger, Mar 28 2016: (Start)
Primes n have no unrelated numbers m < n since all such numbers are coprime to n.
Unrelated numbers m must be composite since primes must either divide or be coprime to n.
m = 1 is not counted as unrelated as it divides and is coprime to n.
a(4) = 0 since 4 is the smallest composite and unrelated numbers m with respect to n must be composite and smaller than n. All other composite n have at least one unrelated number m.
The test for unrelated numbers m that belong to n is 1 < gcd(m, n) < m. (End)

			n = 20: unrelated set to 20 = {6,8,12,14,15,16,18}, largest is a(20) = 18.

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

Cf. A045763, A073758.

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Table[Max[Join[{0}, unr[w]]], {w, 1, 78}]
Table[t = Select[r = Range[n - 1], Divisible[n, #] || GCD[n, #] == 1 &]; Max[Join[{0}, Complement[r, t]]], {n, 78}] (* Jayanta Basu, Jul 09 2013 *)
Table[SelectFirst[Range[n - 2, 2, -1], 1 < GCD[#, n] < # &] /. n_ /; MissingQ@ n -> 0, {n, 100}] (* Michael De Vlieger, Mar 28 2016, Version 10.2 *)

a(n) = {forstep(k=n-2, 1, -1, if ((gcd(n,k) != 1) && (n % k), return (k));); 0;} \\ Michel Marcus, Mar 29 2016

Name clarified by Sean A. Irvine, Dec 18 2024

A053571 Sum of totient functions over arguments running through numbers unrelated to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 2, 8, 0, 14, 0, 16, 16, 16, 0, 36, 0, 38, 32, 44, 0, 68, 20, 62, 40, 78, 0, 136, 0, 80, 82, 110, 78, 170, 0, 138, 116, 190, 0, 264, 0, 192, 198, 204, 0, 310, 66, 294, 196, 272, 0, 398, 182, 358, 248, 328, 0, 584, 0, 372, 372, 372, 248, 658, 0, 468

Offset: 1

Labos Elemer, Jan 17 2000

nonn

Phi-summation over numbers not exceeding n are given in A002088, over divisor-set of n would give n, over RRS or unrelated numbers to n give newer values: at n=36 these values are {396,36,191,170}. This is a further way of Phi-summation.

			n=36 and its "unrelated-set" is {8,10,14,15,16,20,21,22,24,26,27,28,30,32,33,34} and their totients are {4,4,6,8,8,8,12,10,8,12,18,12,8,16,20,16}. The sum of these values is 170, so a(36)=170. For primes the unrelated set is empty and Phi-sum over empty set is 0.

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

Cf. A000010, A002088, A045763.

A045763_row :=proc(n)
    a := {} ;
    dvs := numtheory[divisors](n) ;
    for m from 2 to n-1 do
        if igcd(m,n) >1 and not m in dvs then
            a := a union {m} ;
        end if;
    end do:
    a;
end proc:
A053571 := proc(n)
    add(numtheory[phi](a), a=A045763_row(n)) ;
end proc: # R. J. Mathar, Jan 09 2017

Table[Total@ EulerPhi@ Select[Range@ n, 1 < GCD[#, n] < # &], {n, 68}] (* Michael De Vlieger, Mar 05 2017 *)

A073758 Smallest number that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 6, 6, 4, 0, 8, 0, 4, 6, 6, 0, 4, 0, 6, 6, 4, 0, 9, 10, 4, 6, 6, 0, 4, 0, 6, 6, 4, 10, 8, 0, 4, 6, 6, 0, 4, 0, 6, 6, 4, 0, 9, 14, 4, 6, 6, 0, 4, 10, 6, 6, 4, 0, 8, 0, 4, 6, 6, 10, 4, 0, 6, 6, 4, 0, 10, 0, 4, 6, 6, 14, 4, 0, 6, 6, 4, 0, 8, 10, 4, 6, 6, 0, 4, 14, 6, 6, 4, 10, 9, 0, 4, 6

Offset: 1

Labos Elemer, Aug 08 2002

nonn

Original name: Smallest number of "unrelated set" belonging to n [=URS(n)]. Least number, neither divisor nor relatively prime to n. Or a(n)=0 if unrelated set is empty.
From Michael De Vlieger, Mar 28 2016 (Start):
Primes n have no unrelated numbers m < n since all such numbers are coprime to n.
Unrelated numbers m must be composite since primes must either divide or be coprime to n.
m = 1 is not counted as unrelated as it divides and is coprime to n.
a(4) = 0 since 4 is the smallest composite and unrelated numbers m with respect to n must be composite and smaller than n. All other composite n have at least one unrelated number m.
The test for unrelated numbers m that belong to n is 1 < gcd(m, n) < m.
a(6) = A073759(6), a(8) = A073759(8), a(9) = A073759(9). (End)

			a(20) = 6 since it is the smallest term of the set of numbers m that neither divide nor are coprime to 20, i.e., {6, 8, 12, 14, 15, 16, 18}.

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

Cf. A045763.

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Table[Min[unr[w]], {w, 1, 128}] (* + or -Infinity is replaced by 0 *)
Table[SelectFirst[Range[4, n - 2], 1 < GCD[#, n] < # &] /. n_ /; MissingQ@ n -> 0, {n, 99}] (* Michael De Vlieger, Mar 28 2016, Version 10.2 *)

a(n) = {for(k=1, n-1, if ((gcd(n,k) != 1) && (n % k), return (k));); 0;} \\ Michel Marcus, Mar 29 2016

A299992 Composite n with A001221(n) > 1 for which A243822(n) < A000005(n).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 56, 57, 63, 65, 68, 69, 72, 75, 76, 77, 80, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 108, 111, 112, 115, 116, 117, 119, 123, 124, 129, 133, 135, 136, 141, 143, 144

Offset: 1

Michael De Vlieger, Feb 26 2018

nonn

Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A045763(n).
Primes p have 2 divisors {1, p}; these two numbers constitute the cototient of p: there are no nondivisors in the cototient.
Prime powers p^i have (i + 1) divisors; all smaller powers of the same prime p, i.e., p^j with 0 <= j <= i, also divide p^i. These numbers constitute the cototient of p^i; there are no nondivisors in the cototient.
Therefore, we can ignore cases where n has no nondivisors in the cototient, since they obviously have more divisors than nondivisors therein.
This sequence lists (composite) numbers n with omega(n) > 1 that have fewer nondivisors k in the cototient of n than divisors d.
The smallest odd term is 15.
The number m = 1001 is the smallest term with A001221(m) = 3. No term less than 36,000,000 has A001221(m) > 3.
The following terms m are the smallest to have A001222(m) = {2, 3, 4, ...}: {6, 12, 24, 48, 96, 192, 384, 1152, 2304, 4608, 13824, 27648, 55296, 110592, 331776, 663552, 1327104, 3981312, 7962624, 15925248, ...}
Number of terms less than 10^k for 0 <= k <= 7: {0, 2, 44, 319, 2171, 15545, 119469, 969749}.

			6 is the first term since it is the smallest number with more than one distinct prime divisor that has more divisors (4) than numbers in A243822(6) = 1.

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

Cf. A000005, A001221, A001222, A010846, A045763, A162306, A243822, A272618, A299990, A299991.

Select[Range@ 144, Function[n, And[PrimeNu[n] > 1, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] < 2 DivisorSigma[0, n]]]]

cp's OEIS Frontend

A272619 Irregular array read by rows: n-th row contains (in ascending order) the numbers 1 <= k < n such that at least one prime divisor p of k also divides n and at least one prime divisor q of k is coprime to n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A243823 Quantity of "semitotatives," numbers m < n that are products of at least one prime divisor p of n and one prime q coprime to n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A133995 Irregular array read by rows: n-th row contains (in numerical order) the positive integers <= n which are neither divisors of n nor are coprime to n. A 0 is put into row n if there are no such integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A073762 a(n) = 24*n - 12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A361098 Intersection of A360765 and A360768.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A066760 a(n) = Sum_{1<=k<=n, k is not a divisor of n and k is not coprime to n} k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A073759 Largest number < n that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.

Views

Author