A363061 -id:A363061 - cp's OEIS frontend

A379746 a(1)=1. For n>1 if a(n-1)=A002110(k), a(n)=prime(k+1). Otherwise a(n) is the smallest novel number whose prime factors have already occurred as previous terms.

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 7, 14, 21, 28, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180

Offset: 1

David James Sycamore, Jan 01 2025

Equivalent definition: Lexicographically earliest infinite sequence of distinct positive integers such that a(n) is the smallest novel number having prime power factorization Product_p_i^e_i where p_i is the least nondivisor prime of at most e_i distinct terms a(j); 1<=j<=n-1.

A permutation of the positive integers with prime powers q^k appearing in order (k>=1), and whose underlying sequence of least nondivisor primes is a permutation of A053669. Also, for distinct x, y; x

No multiple m*p (m>1) of a prime p can occur before p itself is a term.

From Michael De Vlieger, Jan 02 2025: (Start)

Efficient method of generating the sequence:

Define row k to be a(A363061(k)+1..A363061(k+1)).

Define R(i) to be { m <= i : rad(m) | i } = tensor product of prime power factor ranges of i that do not exceed i.

Then row k contains R(A002110(k+1)) \ R(A002110(k)).

Row 0 is R(1) = {1}.

Row 1 is R(2)\R(1) = {1, 2} \ {1} = {2},

i.e., {row 2 of A162306} \ {row 1 of A162306}

= {first A363061(1) terms of A000079} \ {1}.

Row 2 is R(6)\R(2) = {1, 2, 3, 4, 6} \ {1, 2} = {3, 4, 6},

where R(6) = row 6 of A162306 = first A363061(2) terms of A003586.

Row 3 is R(30)\R(6)

= {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30} \ {1, 2, 3, 4, 6}

= {5, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30},

where R(30) = row 30 of A162306 = first A363061(3) terms of A051037, etc.

Therefore, for k > 1, within each row, terms strictly increase from prime(k) to primorial A002110(k).

Furthermore, a(1..A363061(k)) is a permutation of R(A002110(k)), hence the sequence is infinite and a permutation of natural numbers. (End)

			a(1) = 1 = A002110(0) therefore a(2) = A053669(1) = 2.
a(2) = 2 = A002110(1) therefore a(3) = A053669(2) = 3.
a(3) = 3 is not a primorial term so a(4)=4 = 2^2 is the smallest novel number whose prime factors do not exceed 3.
Using the second definition we have a(1,2,3,4)=1,2,3,4
                  with least nondivisor primes 2,3,2,3 respectively. Therefore a(5)=2^1*3^1=6, the smallest novel number whose prime factors (2,3) are nondivisor primes of the first 4 terms, and whose exponents do not exceed the number of times these primes have occurred in the underlying sequence of least nondivisor primes.

Michael De Vlieger, Table of n, a(n) for n = 1..19985 (to include A002110(8).)
Michael De Vlieger, Efficient Wolfram program to generate this sequence. (Note: pay attention to dataset size in A363061).
Michael De Vlieger, Log log scatterplot of a(n), n = 1..19985.
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, 12X vertical exaggeration, with a color function representing m = 1 in black, m = 2 in red, m = 3 in orange, ..., highest value of m in the dataset in magenta.

Cf. A000027, A000040, A002110, A007947, A053669, A162306, A351495, A363061, A368108, A377566.

nn = 120; kk = 12;
c[] := False; m[] := 0; h = 0; q = j = 1; u = 2;
f[x_] := f[x] = FactorInteger[x][[All, 1]];
MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[kk]]];
{1}~Join~Reap[Do[
    If[j == P[h],
      If[h == kk, Break[]]; k = Prime[h + 1]; h++; q = Prime[h],
      k = u; While[Or[c[k], ! AllTrue[f[k], # <= q &]], k++]];
    j = Sow[k]; c[k] = True; If[k == u, While[c[u], u++] ],
{n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Jan 02 2025 *)

From Michael De Vlieger, Jan 02 2025: (Start)
a(A363061(k)) = A002110(k).
a(A363061(k)+1) = prime(k).
Seen as a table T(j,k), k = 1..A363061(j)-A363061(j-1) for j > 0, row 0 = {1},
  row j = {row A002110(j) of A162306} \ {row A002110(j-1) of A162306}. (End)

A364225 a(n) = number of k <= m such that rad(k) | m, where m = A025487(n) and rad(n) = A007947(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 5, 11, 18, 6, 14, 15, 26, 7, 18, 20, 36, 8, 23, 44, 25, 68, 26, 49, 9, 29, 58, 31, 96, 32, 65, 10, 35, 76, 38, 131, 39, 83, 84, 11, 88, 42, 156, 43, 97, 45, 174, 46, 104, 106, 12, 111, 50, 283, 206, 51, 121, 53, 228, 54, 130, 133, 13, 138, 58

Offset: 1

Michael De Vlieger, Oct 24 2023

nonn

Not a permutation of natural numbers: a(4) = a(7) = 5.
Let S_rad(m) be the sequence { k : rad(k) | rad(m) }. This sequence gives the number of k <= rad(m). Seen another way, this sequence gives the position of m in S_rad(m).
The number m appears after its factors in S_rad(m). If k < sqrt(m) then k^2 also appears before m.
Scatterplot exhibits trajectories according to t = omega(m) = A001221(A025487(n)). The first term in each trajectory is A002110(t).

			a(1) = 1 since 1 is the only number k that does not exceed 1 such that rad(k) | 1.
a(2) = 2 since k in {1, 2} are such that rad(k) | 2.
a(3) = 3 since k in {1, 2, 4} are such that rad(k) | 4.
a(4) = 5 since k in {1, 2, 3, 4, 6} are such that rad(k) | 6, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..1809 (all terms a(n) <= A002110(10))
Michael De Vlieger, Log log scatterplot of a(n), n = 1..1809, accentuating primorial A025487(n) in red, noting a(n) above in black, and labeling n in blue italic below for the first 24 terms.

Cf. A001221, A002110, A007947, A010846, A025487, A363061.

rad[x_] := Times @@ FactorInteger[#][[All, 1]];
Map[Function[{n, r},
Count[Range[n], _?(Divisible[r, rad[#]] &)]] @@ {#, rad[#]} &,
  {1}~Join~Select[Range[Times @@ Prime@ Range[6]],
     # == Transpose@ {Prime@ Range[Length[#]], ReverseSort[#[[All, -1]] ]} &@
     FactorInteger[#] &] ]

a(n) = A010846(A025487(n)).

A380483 a(n) = least k such that A010846(6*prime(k)) = n, or -1 if no solution exists.

Original entry on oeis.org

3, 4, -1, -1, 5, 6, -1, 7, 8, 9, -1, 10, -1, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 29, 31, 32, 35, 38, 40, 44, 48, 54, 55, 62, 67, 69, 73, 77, 84, 93, 98, 106, 119, 124, 130, 136, 151, 165, 173, 184, 191, 211, 219, 232, 243, 270, 296, 310, 328, 343, 378, 399, 422

Offset: 18

Michael De Vlieger, Jul 08 2025

sign

Conjecture: n = 30 is the largest number for which a(n) = -1.

			Let f = A010846.
For k >= 3 and m = 6*prime(k), f(k) = [k=3] + f(6) + Sum_{j=0..floor(log_3 m)} floor(log_2 m/3^j), with Iverson brackets. (This, since for k = 3, prime(k)^2 < 2*3*prime(k) < prime(k)^3, but for k > 3, 2*3*prime(k) < prime(k)^2.)
a(18) = 3 since 2*3*prime(3) = 30 is the smallest sphenic number k, and f(30) = A363061(3) = 18. This is to say that row 30 of A162306 = {m : rad(m) | 30} = {1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30} has cardinality 18.
a(19) = 4 since 2*3*prime(4) = 42 and f(42) = 19. This is to say that row 30 of A162306 = {m : rad(m) | 42} = {1,2,3,4,6,7,8,9,12,14,16,18,21,24,27,28,32,36,42} has cardinality 19.
a(m) = -1, m = 20..21, since for k >= 3, there exists no solution to f(6*prime(k)) = m.
a(22) = 5 since 2*3*prime(5) = 66 and f(66) = 22.
a(23) = 6 since 2*3*prime(6) = 78 and f(78) = 23.
a(24) = = -1 since for k >= 3, there exists no solution to f(6*prime(k)) = 24.
a(25) = 7 since 2*3*prime(7) = 102 and f(102) = 25.
a(26) = 8 since 2*3*prime(8) = 114 and f(114) = 26.
a(27) = 9 since 2*3*prime(9) = 138 and f(138) = 27.
a(28) = = -1 since for k >= 3, there exists no solution to f(6*prime(k)) = 28.
a(29) = 10 since 2*3*prime(10) = 174 and f(174) = 29.
a(30) = = -1 since for k >= 3, there exists no solution to f(6*prime(k)) = 30.
Note: 2*3*prime(11) = 186; f(186) = f(174) = 29.
a(31) = 12 since 2*3*prime(12) = 222 and f(222) = 31, etc.

Michael De Vlieger, Table of n, a(n) for n = 18..280
Michael De Vlieger, Plot of terms k = p^a*q^b*r^c, primes p < q < r, in row a(n) of A162306, with n = 18..58 and a(n) > -1, at (x,y,z) = (a,b,c). For a(n) there are n blocks in each diagram.

Cf. A007304, A010846, A363061.

k = 3; Table[While[Set[t, Boole[k == 3] + 5 + Sum[Floor[1 + Log2[#/3^j]], {j, 0, Floor[Log[3, #]]}] &[6*Prime[k] ] ] < n, k++]; If[t == n, k, -1], {n, 18, 30}]

A383180 Irregular table T(n,k) = A010846(A019565(2^n + k)).

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 5, 18, 2, 6, 5, 19, 5, 20, 16, 68, 2, 7, 6, 22, 5, 21, 18, 77, 5, 22, 17, 79, 16, 74, 60, 283, 2, 7, 6, 23, 5, 23, 18, 80, 5, 22, 18, 82, 16, 78, 62, 295, 5, 24, 19, 87, 16, 82, 64, 315, 15, 80, 62, 316, 55, 290, 226, 1161

Offset: 0

Michael De Vlieger, May 09 2025

			Triangle begins:
  0: 1;
  1: 2;
  2: 2, 5;
  3: 2, 6, 5, 18;
  4: 2, 6, 5, 19, 5, 20, 16, 68;
  5: 2, 7, 6, 22, 5, 21, 18, 77, 5, 22, 17, 79, 16, 74, 60, 283;
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows n = 0..14, flattened)
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^14.
Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..14, with a color function showing smaller numbers in reds and oranges, larger numbers in blues and magentas.

Cf. A005117, A010846, A019565, A363061.

(* Load the "theta" program at the Mathematica link in A369609, then: *)
f[x_] := Times @@ Prime@ Position[Reverse@ IntegerDigits[x, 2], 1][[All, 1]]; Table[theta[f[2^n + k] ], {n, 0, 7}, {k, 0, 2^n - 1}]

T(0,0) = 1.
T(n,0) = 2.
T(n,2^(n-1)-1) = A363061(n).

A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).

Original entry on oeis.org

16, 72, 540, 6300, 92400, 1681680, 36756720, 921470550, 27886608750, 970453984500, 37905932634570

Offset: 1

Michael De Vlieger, Jun 22 2023

Let R = r(P(n)) = A010846(A002110(n)) = A363061(n).
Let S(n) be the sorted tensor product of prime power ranges {p(i)^e : i<=n, e>=0}, e.g., S(1) = A000079, S(2) = A003586, S(3) = A051037, etc.
Let T(n) = A002110(n)*S(n). Note that S(1) = T(1) since omega(A002110(1)) = 1.
Let S(n,i) be the i-th term in S(n).
Then a(n) is the smallest S(n,i), i >= R, such that S(n,i) is also in T. Equivalently, a(n) is the smallest S(n,i), i >= R, such that rad(S(n,i)) = A002110(n), where rad(n) = A007947(n).

			a(1) = 16 since r(2^4) = 5 and r(6) = 5; numbers in row 16 of A162306 are its divisors {1, 2, 4, 8, 16}, while row 6 of A162306 is {1, 2, 3, 4, 6}.
a(2) = 72 = A003586(18) since r(72) = r(30) = 18. 72 is the 8th term in A003586 that is not in A000961.
a(3) = 540 since r(540) = 69 which exceeds r(210) = 68.
a(4) = 6300 since r(6300) = 290 which exceeds r(2310) = 283, etc.
Table showing the relationship of a(n) to r(P(n)) = A363061(n), with p(n) = prime(n), P(n+1) = A002110(n+1), r(a(n)) = A010846(a(n)), and j the index such that S(r(a(n))) = T(j) = a(n). a(n) = m*P(n).
   n p(n)        P(n+1)          a(n)  r(P(n))  r(a(n))   j    m
  --------------------------------------------------------------
   1   2             6            16        5        5    4    8
   2   3            30            72       18       18    8   12
   3   5           210           540       68       69   13   18
   4   7          2310          6300      283      290   22   30
   5  11         30030         92400     1161     1165   29   40
   6  13        510510       1681680     4843     4848   42   56
   7  17       9699690      36756720    19985    19994   53   72
   8  19     223092870     921470550    83074    83435   68   95
   9  23    6469693230   27886608750   349670   351047   89  125
  10  29  200560490130  970453984500  1456458  1457926  107  150

Cf. A000079, A000961, A002110, A002473, A003586, A007947, A010846, A051037, A051038, A080197, A080681, A080682, A080683, A162306, A363061.

nn = 6; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; f[x_] := FactorInteger[x][[-1, 1]]; S = Array[Product[Prime[i], {i, #}] &, nn + 1]; Table[Set[{p, q}, Prime[n + {0, 1}]]; r = Count[Range[S[[n + 1]]], _?(f[#] <= q &)]; c = k = 1; While[Or[c < r, rad[k] != S[[n]]], If[f[k] <= p, c++]; k++]; k, {n, nn}]

a(n) >= A363061(n).

A363844 Number of k <= P(n) such that gcd(k,P(n)) > 1, yet there is a prime q | k that does not divide P(n), where P(n) = A002110(n).

Original entry on oeis.org

0, 0, 0, 5, 95, 1548, 23110, 413508, 8020826, 186514437, 5447473481, 169902931273, 6317112341154, 260105450523376, 11228680152402376, 529602052783103298, 28154196548377380922, 1665532558381753842459, 101854713853486313230170, 6839699495691464491151135, 486637286249491454965285898

Offset: 0

Michael De Vlieger, Jun 23 2023

			a(0) = 0 since P(0) = 1; phi(1) = 1 and A010846(1) = 1, hence 1 - 1 - 1 + 1 = 0.
a(1) = 0 since P(1) = 2; phi(2) = 1 and A010846(2) = 2, hence 2 - 1 - 2 + 1 = 0.
a(2) = 0 since P(2) = 6; phi(6) = 2 and A010846(6) = 5, hence 6 - 2 - 5 + 1 = 0.
a(3) = 5 since P(3) = 30; phi(30) = 8 and A010846(6) = 5, hence 30 - 8 - 18 + 1 = 5. We can also look at this as the cardinality of the set {1..30} \ ({1, 7, 11, 13, 17, 19, 23, 29} U {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30}) = {14, 21, 22, 26, 28}, therefore a(3) = 5.
Table relating a(n) to A002110(n), A363061(n), and A005867(n).
n A002110(n) A363061(n)      a(n) A005867(n)
--------------------------------------------
0         1          1         0          1
1         2          2         0          1
2         6          5         0          2
3        30         18         5          8
4       210         68        95         48
5      2310        283      1548        480
6     30030       1161     23110       5760
7    510510       4843    413508      92160
8   9699690      19985   8020826    1658880
...

Cf. A000010, A002110, A005867, A243823, A363061.

b = Map[Last[ToExpression /@ StringSplit[#]] &, Split[Import["https://oeis.org/A363061/b363061.txt", "Data"]][[2 ;; -1, -1]]]; Array[(If[# == 0, Set[{k, p}, {1, 1}], p *= Prime[#]; k *= (Prime[#] - 1)]; p - k - b[[# + 1]] + 1) &, Length[b], 0]

a(n) = A243823(A002110(n)).
a(n) = P(n) - A000010(P(n)) - A010846(P(n)) + 1, where P(n) = A002110(n).
a(n) = A002110(n) - A005867(n) - A363061(n) + 1.

Showing 1-6 of 6 results.

cp's OEIS Frontend

A379746 a(1)=1. For n>1 if a(n-1)=A002110(k), a(n)=prime(k+1). Otherwise a(n) is the smallest novel number whose prime factors have already occurred as previous terms.

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A364225 a(n) = number of k <= m such that rad(k) | m, where m = A025487(n) and rad(n) = A007947(n).

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A380483 a(n) = least k such that A010846(6*prime(k)) = n, or -1 if no solution exists.

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A383180 Irregular table T(n,k) = A010846(A019565(2^n + k)).

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A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).

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A363844 Number of k <= P(n) such that gcd(k,P(n)) > 1, yet there is a prime q | k that does not divide P(n), where P(n) = A002110(n).

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