Jamie Morken - cp's OEIS frontend

A343496 First point of the straight lines in A340649.

5, 31, 194, 1061, 6456, 40080, 251721, 1617206, 10553419, 69709769, 465769825

Offset: 1

Simon Strandgaard and Jamie Morken, Apr 17 2021

prime(a(n)+1) - prime(a(n)) = n*2. E.g., for n=4: prime(a(4)+1) - prime(a(4)) = 4*2, prime(1062) - prime(1061) = 4*2, 8521 - 8513 = 8.

			For n=1, consider k's with prime gap 1*2 = 2, i.e., k's such that A001223(k)=2. k=5 is the first place where A001223(k)=2 and A141042(k)=A340649(k), so a(1)=5.
For n=2, consider k's with prime gap 2*2 = 4, i.e., k's such that A001223(k)=4. k=31 is the first place where A001223(k)=4 and A141042(k)=A340649(k), so a(2)=31.
For n=3, consider k's with prime gap 3*2 = 6, i.e., k's such that A001223(k)=6. k=194 is the first place where A001223(k)=6 and A141042(k)=A340649(k), so a(3)=194.

Simon Strandgaard, Visualization of the first 5 terms.

Cf. A000040, A001223, A141042, A340649, A029707, A029709, A320701, A320702, A320703.

n = 1
last_prime = 2
find_gap = 2
result = []
Prime.each(10_000) do |prime|
    next if prime == 2
    gap = prime - last_prime
    if gap == find_gap
        value = (n * prime) % last_prime
        if value == n * gap
            result << n
            find_gap += 2
        end
    end
    n += 1
    last_prime = prime
end
p result

a(n) = smallest k that satisfies A001223(k) = 2*n and A340649(k) = A141042(k).

A335334 Sum of the integers in the reduced residue system of A002110(n).

Original entry on oeis.org

1, 6, 120, 5040, 554400, 86486400, 23524300800, 8045310873600, 4070927302041600, 3305592969257779200, 3074201461409734656000, 4094836346597766561792000, 6715531608420337161338880000, 12128250084807128913378017280000

Offset: 1

Jamie Morken, Jun 02 2020

nonn

Sum of the integers up to A002110(n) and coprime to A002110(n).

The sequence gives the sum of row n of A286941(n).

			For n = 3: A002110(3) = 30, the reduced residue system of 30 is {1, 7, 11, 13, 17, 19, 23, 29}. The sum is a(3) = 120.

Michael De Vlieger, Table of n, a(n) for n = 1..196
Eric Weisstein's World of Mathematics, Totative.

Cf. A023896, A002110, A005867, A070826, A038110, A058250, A286941.

n = 15;
A002110 = Drop[FoldList[Times, 1, Prime[Range[n]]], 1];
A005867 = Drop[EulerPhi@FoldList[Times, 1, Prime@Range@n], 1];
A002110*A005867/2
(* Second program: *)
Map[# EulerPhi[#]/2 &, FoldList[Times, Prime@ Range@ 14]] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = my(P=factorback(primes(n))); P*eulerphi(P)/2; \\ Michel Marcus, Jun 02 2020

a(n) = A023896(A002110(n)).
a(n) = A002110(n)*A005867(n)/2 = A070826(n)*A005867(n).
a(n) = (A002110(n)*A038110(n+1)/2)*A058250(n).

A335261 Irregular triangle S(n,k) = denominators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 4, 1, 4, 2, 4, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 16, 8, 16, 4, 16, 8, 16, 2, 16, 8, 16, 4, 16, 8, 16, 1, 16, 8, 16, 4, 16, 8, 16

Offset: 1

Michael De Vlieger, Jamie Morken, May 30 2020

Alternatively, denominators of k*A060753(n)/A038110(n) for 1 <= k <= A005867(n).
Let m = A038110(n). For row n, the primitive denominators d | m.
The mean of row n is related to the mean of row n of A309497: A060753(n+1)/(A038110(n+1)*2) = Mean(A309497(n))/A038110(n+1).

			Table begins:
    1;
    1;
    1, 1;
    4, 2, 4, 1, 4, 2, 4, 1;
    8, 4, 8, 2, 8, 4, 8, 1, ..., 8, 1;
    ...
Row n = 4 contains the denominators of (35/8)*k for 1 <= k <= A005867(4): 35/8, 35/4, 105/8, 35/2, 175/8, 105/4, 245/8, 35, 315/8, 175/4, 385/8, 105/2, 455/8, 245/4, 525/8, 70, 595/8, 315/4, 665/8, 175/2, 735/8, 385/4, 805/8, 105, 875/8, 455/4, 945/8, 245/2, 1015/8, 525/4, 1085/8, 140, 1155/8, 595/4, 1225/8, 315/2, 1295/8, 665/4, 1365/8, 175, 1435/8, 735/4, 1505/8, 385/2, 1575/8, 805/4, 1645/8, 210.
The mean of row 4: (A060753(4+1)/(A038110(4+1)*2))*(A005867(4)+1) = (35/(8*2))*(48+1) = (35/16)*49 = 1715/16.

Cf. A000010, A002110, A005867, A038110, A060753, A309497, A335260.

Table[Denominator[P Range[EulerPhi[P]]/EulerPhi[P]], {P, FoldList[Times, Prime@ Range@ 5]}] (* or, more efficiently for larger datasets: *)
Flatten@ Block[{nn = 7, s, t}, s = Array[Numerator@ Product[1 - 1/Prime[k], {k, # - 1}] &, nn]; t = Nest[Append[#, #[[-1]] (Prime[Length@ #] - 1)]&, {1}, nn]; MapIndexed[Function[{m, D, i}, PadRight[{}, t[[i]], ReplacePart[ConstantArray[0, m], Flatten@ Map[Function[d, Map[# -> d &, m/d Select[Range[d], GCD[#, d] == 1 &]]], D]]]] @@ {#1, Divisors@ #1, First[#2]} &, s]]
(* or, to generate a single denominator of T(n,k) *)
f[n_, k_] := #/GCD[#, Mod[k, #]] &@ Numerator@ Product[1 - 1/Prime[i], {i, n - 1}]

T(n,k) = m/GCD(k (mod m), m) with m = A038110(n).

Row lengths: A005867(n).

A335260 Irregular triangle S(n,k) = numerators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

Original entry on oeis.org

1, 2, 3, 6, 15, 15, 45, 15, 75, 45, 105, 30, 35, 35, 105, 35, 175, 105, 245, 35, 315, 175, 385, 105, 455, 245, 525, 70, 595, 315, 665, 175, 735, 385, 805, 105, 875, 455, 945, 245, 1015, 525, 1085, 140, 1155, 595, 1225, 315, 1295, 665, 1365, 175, 1435, 735, 1505

Offset: 1

Michael De Vlieger, Jamie Morken, May 30 2020

Alternatively, numerators of k*A060753(n)/A038110(n) for 1 <= k <= A005867(n).

			Table begins:
     1;
     2;
     3, 6;
    15, 15, 45, 15, 75, 45, 105, 30;
    ...
Row n = 4 contains the numerators of (35/8)*k for 1 <= k <= A005867(4): 35/8, 35/4, 105/8, 35/2, 175/8, 105/4, 245/8, 35, 315/8, 175/4, 385/8, 105/2, 455/8, 245/4, 525/8, 70, 595/8, 315/4, 665/8, 175/2, 735/8, 385/4, 805/8, 105, 875/8, 455/4, 945/8, 245/2, 1015/8, 525/4, 1085/8, 140, 1155/8, 595/4, 1225/8, 315/2, 1295/8, 665/4, 1365/8, 175, 1435/8, 735/4, 1505/8, 385/2, 1575/8, 805/4, 1645/8, 210.

Index entries for sequences related to primorial numbers

Cf. A000010, A002110, A005867, A038110, A060753, A309497, A335261.

Table[Numerator[P Range[EulerPhi[P]]/EulerPhi[P]], {P, FoldList[Times, Prime@ Range@ 5]}] (* or, more efficiently for larger datasets: *)
Flatten@ Block[{nn = 7, s, t}, s = Array[Numerator@ Product[1 - 1/Prime[k], {k, # - 1}] &, nn]; t = Nest[Append[#, #[[-1]] (Prime[Length@ #] - 1)]&, {1}, nn]; u = Denominator@ Nest[Append[#, #[[-1]] + (1 - #[[-1]])/Prime[Length@ #]] &, {0}, nn]; MapIndexed[Function[{m, D, i},  u[[i]]*Range[t[[i]]]/ PadRight[{}, t[[i]], ReplacePart[ConstantArray[0, m], Flatten@ Map[Function[d, Map[# -> m/d &, m/d Select[Range[d], GCD[#, d] == 1 &]]], D]]]] @@ {#1, Divisors@ #1, First[#2]} &, s]]
(* or, generate a single numerator of S(n,k): *)
f[n_, k_] := #2 k/GCD[#1, Mod[k, #1]] & @@ {Numerator@ Product[1 - 1/Prime[i], {i, n - 1}], Denominator@ Last@ Nest[Append[#, #[[-1]] + (1 - #[[-1]])/Prime[Length@ #]] &, {0}, n - 1]}

S(n,k) = k*A060753(n)/GCD(k (mod m), m) for m = A038110(n).
Row lengths: A005867(n).
Least numerator in row n: A060753(n), all numerators are multiples j*A060753(n).

A329815 Number of distinct terms in the first difference sequence of the reduced residue system of the n-th primorial.

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 13, 16, 20, 23, 29, 33, 37, 43, 49, 53, 59, 66, 75, 84, 92, 99, 108, 116, 127, 132, 140, 148, 156, 164, 174, 185, 193, 206, 215, 224, 235, 245, 255, 267, 275, 286, 297, 308

Offset: 1

Jamie Morken, Nov 21 2019

This sequence is the number of distinct terms in the first difference sequence for rows n in A286941 and A309497.

Number of distinct terms listed in row n of A331118. - Michael De Vlieger, Jul 11 2020

			For n = 3, A002110(3) = 30, RRS = {1, 7, 11, 13, 17, 19, 23, 29}, dRRS = {6, 4, 2, 4, 2, 4, 6}, so a(3) = 3.

Mario Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT], 2020.

Cf. A061498, A048670, A309497, A286941, A331118.

Primorial[n_] := Times @@ Prime[Range[n]]; Table[Length@ Union@ Differences@ Select[Range@ Primorial[n], CoprimeQ[#, Primorial[n]] &], {n, 7}] (* after Michael De Vlieger Jul 15 2017 from A061498 *)

f(n) = {my(va = select(x->(gcd(n, x)==1), [1..n])); vd = vector(#va-1, k, va[k+1] - va[k]); #Set(vd); } \\ A061498
a(n) = f(prod(i=1, n, prime(i))); \\ Michel Marcus, Dec 19 2019

a(n) = A061498(A002110(n)).

a(n) <= A048670(n)/2.

a(12)-a(44) from Jamie Morken, Jul 11 2020 (after Mario Ziller)

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).

Original entry on oeis.org

0, 1, 2, 1, 11, 2, 1, 8, 7, 14, 13, 4, 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8

Offset: 0

Jamie Morken, Aug 05 2019

The sequence is Primorial rows of A308121.
Row n has length A005867(n).
Row n > 1 average value = A060753(n)/2.
Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
First value on row(n) = A161527(n-1).
Last value on row(n) = A038110(n) for n > 2.
For n > 1, A060753(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A060753(n), where a = A005867(n-1) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
Related identity:
  A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

			The triangle starts:
row1: 0;
row2: 1;
row3: 2, 1;
row4: 11, 2, 1, 8, 7, 14, 13, 4;
row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;

Michael De Vlieger, Table of n, a(n) for n = 0..6299 (rows n = 0..6, flattened)
Eric Weisstein's World of Mathematics, Totative

Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.

row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* Amiram Eldar, Aug 10 2019 *)

A307964 Irregular triangle read by rows: T(n,k) = A308121(A024556(n),k).

Original entry on oeis.org

7, 14, 13, 4, 11, 2, 1, 8, 3, 6, 5, 8, 3, 2, 5, 4, -1, 2, 1, 4, 13, 26, 19, 32, 25, 38, 31, 4, 17, 10, 23, 16, 29, 2, -5, 8, 1, 14, 7, 20, 11, 22, 33, 44, 31, 18, 29, 16, 27, 38, 1, 12, 23, 34, -3, 8, 19, 6, 17, 4, -9, 2, 13, 24, 5, 10, 7, 12, 9, 14, 11, 16, 5

Offset: 1

Jamie Morken, Jul 29 2019

The sequence gives odd squarefree composite rows n in A308121, i.e., rows 15, 21, 33, 35, 39, 51, 55, 57, 65, ... given by A024556(n).  These rows are the primitive rows of A308121.
Row n has length A000010(A024556(n)).
For row n:
  T(n, 1) = T(n, 2) / 2.
  T(n, phi(n)) - T(n, phi(n)-1) = T(n, 1).
  T(n, phi(n)/2+1) - T(n, phi(n)/2) = T(n, 1).
From Charlie Neder, Jul 30 2019: (Start)
For row n, T(n, k) + T(n, phi(n)-k) is constant for all k.
For 2 <= k < lpf(A024556(n)), T(n, k) = k*T(n, 1). (End)

			The sequence as an irregular triangle:
1:  7, 14, 13, 4, 11, 2, 1, 8;
2:  3, 6, 5, 8, 3, 2, 5, 4, -1, 2, 1, 4;
3:  13, 26, 19, 32, 25, 38, 31, 4, 17, 10, 23, 16, 29, 2, -5, 8, 1, 14, 7, 20;
4:  11, 22, 33, 44, 31, 18, 29, 16, 27, 38, 1, 12, 23, 34, -3, 8, 19, 6, 17, 4, -9, 2, 13, 24;
5:  5, 10, 7, 12, 9, 14, 11, 16, 5, 2, 7, 4, 9, 6, 11, 8, -3, 2, -1, 4, 1, 6, 3, 8
6:  19, 38, 25, 44, 31, 50, 37, 56, 43, 62, 49, 4, 23, 10, 29, 16, 35, 22, 41, 28, 47, 2, -11, 8, -5, 14, 1, 20, 7, 26, 13, 32;
7:  3, 6, 9, 12, 7, 10, 13, 16, 3, 6, 9, 4, 7, 10, 13, 8, 3, 6, 1, 4, 7, 10, 5, 8, 3, -2, 1, 4, 7, 2, 5, 8, -5, -2, 1, 4, -1, 2, 5, 8;
8:  7, 14, 9, 16, 11, 18, 13, 20, 15, 22, 17, 24, 7, 2, 9, 4, 11, 6, 13, 8, 15, 10, 17, 12, -5, 2, -3, 4, -1, 6, 1, 8, 3, 10, 5, 12;
9:  17, 34, 51, 68, 37, 54, 71, 88, 57, 74, 43, 12, 29, 46, 63, 32, 49, 66, 83, 4, 21, 38, 7, 24, 41, 58, 27, 44, 61, -18, -1, 16, 33, 2, 19, 36, 53, 22, -9, 8, -23, -6, 11, 28, -3, 14, 31, 48;
  ...

Jamie Morken, Table of n, a(n) for n = 1..9768

Cf. A024556, A308121, A309497, A000010, A076511, A076512.

rowsToCheck = 340;
A024556 =
Complement[Select[Range[3, rowsToCheck, 2], SquareFreeQ],
  Prime[Range[
    PrimePi[rowsToCheck]]]]; (* after Harvey P. Dale , Jan 26 2011 *)
A308121 =
Table[With[{a = n/GCD[n, #], b = Numerator[#/n]},
     MapIndexed[a First@#2 - b #1 &,
      Flatten@Position[GCD[Table[Mod[k, n], {k, n - 1}], n],
         1] /. {} -> {1}]] &@EulerPhi@n, {n,
   rowsToCheck}]; (* after Michael De Vlieger, Jun 06 2019 *)
A307964 = {};
For[i = 1, i <= Length[A024556], i++,
AppendTo[A307964, A308121[[A024556[[i]]]]]]
A307964flattened = Flatten[A307964]
(* Jamie Morken, Apr 20 2021 *)

A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3, 7, 14, 13, 4, 11, 2, 1, 8

Offset: 1

Jamie Morken, May 13 2019

Row n has length A000010(n).
Row n > 1 has sum = n*A076512(n)/2.
First value on row(n) = A076511(n).
Last value on row(n) = A076512(n) for n > 1.
For n > 1, A109395(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
  x + y = A109395(n), where a = A000010(n) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
From Charlie Neder, Jun 05 2019: (Start)
If p is a prime dividing n, then row p*n consists of p copies of row n.
Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)

			The sequence as an irregular triangle:
  n/k 1, 2, 3, 4, ...
   1: 0
   2: 1
   3: 1, 2
   4: 1, 1
   5: 1, 2, 3, 4
   6: 2, 1
   7: 1, 2, 3, 4, 5, 6
   8: 1, 1, 1, 1
   9: 1, 2, 1, 2, 1, 2
  10: 3, 4, 1, 2
  11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  12: 2, 1, 2, 1
  13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  14: 4, 5, 6, 1, 2, 3
  15: 7, 14, 13, 4, 11, 2, 1, 8
  ...
  Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.

Jamie Morken, Table of n, a(n) for n = 1..13413 (Rows n = 1..210 of triangle, flattened)

Cf. A000010, A109395, A038566, A076511, A076512.

Flatten@ Table[With[{a = n/GCD[n, #], b = Numerator[#/n]}, MapIndexed[a First@ #2 - b #1 &, Flatten@ Position[GCD[Table[Mod[k, n], {k, n - 1}], n], 1] /. {} -> {1}]] &@ EulerPhi@ n, {n, 15}] (* Michael De Vlieger, Jun 06 2019 *)

vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ Michel Marcus, May 14 2019

A307388 Length of the period of decimal representation of Product_{k=1..n} A038111(k)/A038110(k).

Original entry on oeis.org

1, 27, 729, 59049, 43046721, 31381059609, 68630377364883, 150094635296999121, 328256967394537077627, 717897987691852588770249, 4710128697246244834921603689, 92709463147897837085761925410587, 3649600726280146254718103955713167842

Offset: 9

Jamie Morken, Apr 06 2019

The offset is 9 since for 0 < n < 5, the product is an integer, and for 4 < n < 9 the decimal expansion ends with zeros.

			For example for n=9 with (2/1) * (6/1) * (15/1) * (105/4) * (385/8) * (1001/16) * (17017/192) * (323323/3072) * (7436429/55296) = 2759414170256180364552625 / 154618822656 = 17846560482454.30745852273604315188195970323350694444444444444... so a(9) = 1.

Jamie Morken, Mathematica Stack Exchange question

Cf. A038111, A038110, A051626, A058250.

Primorial[n_] := Times @@ Prime[Range[n]]
ClearAll[iter]
ClearAll[fracPer, vp];
(*p-adic order*)
vp[p_?PrimeQ, n_Integer] :=
  Length@NestWhileList[#/p &, n/p, IntegerQ] - 1;
(*fraction decimal expansion period*)
fracPer[q_Integer] := 0;
fracPer[q_Rational] := Module[{den, p2, p5}, den = Denominator[q];
   p2 = vp[2, den];
   p5 = vp[5, den];
   den = den/2^p2/5^p5;
   If[den == 1, 0, MultiplicativeOrder[10, den]]];
iter[{periods_, frac_, n_}] := {{periods, fracPer[#]}, #, n + 1} &[
   frac*Primorial[n]/EulerPhi[Primorial[Max[1, n - 1]]]];
Flatten@First@
  Nest[iter, {0, Primorial[0]/EulerPhi[Primorial[0]], 0}, 50]

A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 6, 2, 7, 8, 9, 3, 10, 11, 1, 12, 4, 13, 14, 15, 5, 16, 2, 17, 18, 6, 19, 20, 21, 7, 22, 23, 1, 24, 8, 25, 26, 3, 27, 9, 28, 29, 30, 10, 31, 4, 32, 33, 11, 34, 35, 36, 12, 37, 2, 38, 39, 13, 40, 41, 5, 42, 14, 43, 44, 3, 45, 15, 46, 6, 47, 48, 16, 49, 50, 51, 17, 52, 53, 54, 18, 55, 56, 7

Offset: 1

Jamie Morken and Vincenzo Librandi, Feb 09 2019

Composites with least prime factor p are on that row of A083140 which begins with p
Sequence with similar values: A122005.
Sequence written as a jagged array A with new row when a(n) > a(n+1):
  1,  2,  3,
  1,  4,  5,  6,
  2,  7,  8,  9,
  3, 10, 11,
  1, 12,
  4, 13, 14, 15,
  5, 16,
  2, 17, 18,
  6, 19, 20, 21,
  7, 22, 23,
  1, 24,
  8, 25, 26,
  3, 27,
  9, 28, 29, 30.
A153196 is the list B of the first values in successive rows with length 4.
  B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)
For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).
A243811 is the list of the second values in successive rows with length 4.
A047845 is the list of values in the second column and A104279 is the list of values in the third column of the jagged array starting on the second row.
Sequence written as an irregular triangle C with new row when a(n)=1:
  1,2,3,
  1,4,5,6,2,7,8,9,3,10,11,
  1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,
  1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7,57,19,58,4,59.
A243887 is the last value in each row of C.
The second value D on the row n > 1 of the irregular triangle C is a(A053683(n)) or equivalently A084921(n). For example for row 3 of the irregular triangle:
  D = a(A053683(3)) = a(16) = 12 or D = A084921(3) = 12. This is the number of composites < A066872(3) with the same least prime factor p as the A053683(3) = 16th composite, A066872(3) = 26.
The number of values in each row of the irregular triangle C begins: 3,11,18,57,39,98,61,141,265,104,351,268,...
The second row of the irregular triangle C is A117385(b) for 3 < b < 15.
The third row of the irregular triangle C has similar values as A117385 in different order.

			First composite 4, least prime factor is 2, first case for 2 so a(1)=1.
Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.
Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.
Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.
Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.

Jamie Morken, Table of n, a(n) for n = 1..10000

Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.

counts = {}
values = {}
For[i = 2, i < 130, i = i + 1,
If[PrimeQ[i], ,
x = PrimePi[FactorInteger[i][[1, 1]]];
  If[Length[counts] >= x,
   counts[[x]] = counts[[x]] + 1;
   AppendTo[values, counts[[x]]], AppendTo[counts, 1];
   AppendTo[values, 1]]]]
   (* Print[counts] *)
   Print[values]

c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = my(c=c(n), lpf = vecmin(factor(c)[,1]), nb=0); for(k=2, c, if (!isprime(k) && vecmin(factor(k)[,1])==lpf, nb++)); nb; \\ Michel Marcus, Feb 10 2019

a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).
For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).
This gives an estimate of 67.499... and the actual value of a(n)=67.

cp's OEIS Frontend

User: Jamie Morken

A343496 First point of the straight lines in A340649.

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A335334 Sum of the integers in the reduced residue system of A002110(n).

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A335261 Irregular triangle S(n,k) = denominators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

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A335260 Irregular triangle S(n,k) = numerators of k*A002110(n)/A005867(n) for 1 <= k <= A005867(n).

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A329815 Number of distinct terms in the first difference sequence of the reduced residue system of the n-th primorial.

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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)*k-A038110(n)*A286941(n,k).

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A307964 Irregular triangle read by rows: T(n,k) = A308121(A024556(n),k).

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A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k).

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).

A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).