A355200 -id:A355200 - cp's OEIS frontend

A016945 a(n) = 6*n+3.

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 321, 327

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(37).
Continued fraction expansion of tanh(1/3).
If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
Leaves of the Odd Collatz-Tree: a(n) has no odd predecessors in all '3x+1' trajectories where it occurs: A139391(2*k+1) <> a(n) for all k; A082286(n)=A006370(a(n)). - Reinhard Zumkeller, Apr 17 2008
Let random variable X have a uniform distribution on the interval [0,c] where c is a positive constant. Then, for positive integer n, the coefficient of determination between X and X^n is (6n+3)/(n+2)^2, that is, A016945(n)/A000290(n+2). Note that the result is independent of c. For the derivation of this result, see the link in the Links section below. - Dennis P. Walsh, Aug 20 2013
Positions of 3 in A020639. - Zak Seidov, Apr 29 2015
a(n+2) gives the sum of 6 consecutive terms of A004442 starting with A004442(n). - Wesley Ivan Hurt, Apr 08 2016
Numbers k such that Fibonacci(k) mod 4 = 2. - Bruno Berselli, Oct 17 2017
Also numbers k such that t^k == -1 (mod 7), where t is a member of A047389. - Bruno Berselli, Dec 28 2017

Friedrich L. Bauer, Der (ungerade) Collatz-Baum, Informatik Spektrum 31 (Springer, April 2008), pp. 379-384.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Dennis P. Walsh, The correlation for a power curve on nonnegative support.
Eric Weisstein's World of Mathematics, Collatz Problem.
Index entries for sequences related to 3x+1 (or Collatz) problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Third row of A092260.
Cf. A004442, A008588, A010503, A016921, A016933, A016957, A016969, A019679, A115754.
Subsequence of A061641; complement of A047263; bisection of A047241.
Cf. A000225. - Loren Pearson, Jul 02 2009
Cf. A020639. - Zak Seidov, Apr 29 2015
Odd numbers in A355200.

List([0..60], n-> 3*(1+2*n)); # G. C. Greubel, Sep 18 2019

a016945 = (+ 3) . (* 6)
a016945_list = [3, 9 ..]
-- Wesley Ivan Hurt, Sep 29 2013

[6*n+3 : n in [0..60]]; // Wesley Ivan Hurt, Sep 29 2013

seq(6*n+3, n=0..60); # Dennis P. Walsh, Aug 20 2013
A016945:=n->6*n+3; # Wesley Ivan Hurt, Sep 29 2013

Range[3, 350, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6n+3, {n, 0, 60}] (* Wesley Ivan Hurt, Sep 29 2013 *)
LinearRecurrence[{2, -1}, {3, 9}, 55] (* Ray Chandler, Jul 17 2015 *)
CoefficientList[Series[3(1+x)/(1-x)^2, {x, 0, 60}], x] (* Robert G. Wilson v, Dec 14 2016 *)

makelist(6*n+3, n, 0, 60); /* Wesley Ivan Hurt, Sep 29 2013 */

{a(n) = 6*n + 3} \\ Wesley Ivan Hurt, Sep 29 2013

x='x+O('x^60); Vec(3*(1+x)/(1-x)^2) \\ Altug Alkan, Apr 08 2016

[3*(1+2*n) for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 3*(2*n + 1) = 3*A005408(n), odd multiples of 3.
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A103333(n+1). - Reinhard Zumkeller, Feb 24 2009
a(n) = 12*n - a(n-1) for n>0, a(0)=3. - Vincenzo Librandi, Nov 20 2010
G.f.: 3*(1+x)/(1-x)^2. - Mario C. Enriquez, Dec 14 2016
E.g.f.: 3*(1 + 2*x)*exp(x). - G. C. Greubel, Sep 18 2019
Sum_{n>=0} (-1)^n/a(n) = Pi/12 (A019679). - Amiram Eldar, Dec 10 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)/2 (A010503).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(3/2) (A115754). (End)
a(n) = (n+2)^2 - (n-1)^2. - Alexander Yutkin, Mar 15 2025

A354591 Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 170, 172

Offset: 1

Wesley Ivan Hurt, Aug 18 2022

nonn

All terms are even. - Robert Israel, Aug 31 2022
Is it true that a(n) = 2*A080671(n)? - Michel Marcus, Sep 01 2022 (True for n <= 10000. - N. J. A. Sloane, Sep 01 2022)
This is true.  In other words, k is in the sequence if and only if k is even and divisible by 3, 4 or 5.  Proof: the positive integer solutions of 1/a + 1/b + 1/c + 1/d = 1 can be enumerated explicitly, and each contains at least one even number and at least one divisible by 3, 4 or 5.  Of course k = k/a + k/b + k/c + k/d if and only if 1 = 1/a + 1/b + 1/c + 1/d, and this writes k as the sum of 4 divisors of k if k is divisible by a,b,c, and d.  If k is even and divisible by 3, we can use 1 = 1/3 + 1/3 + 1/6 + 1/6; if divisible by 4, 1 = 1/4 + 1/4 + 1/4 + 1/4; if even and divisible by 5, 1 = 1/2 + 1/5 + 1/5 + 1/10. - Robert Israel, Sep 01 2022
The asymptotic density of this sequence is 11/30. - Amiram Eldar, Aug 08 2023

			20 is in the sequence since 20 = 10+5+4+1 = 5+5+5+5 where each summand divides 20.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), this sequence (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

Cf. A080671.

F:= proc(x,S,j) option remember;
      local s,k;
      if j = 0  then return(x = 0) fi;
      if S = [] or x > j*S[-1] then return false fi;
      s:= S[-1];
      for k from 0 to min(j,floor(x/s)) do
        if procname(x-k*s, S[1..-2],j-k) then return true fi
      od;
      false
end proc:
select(t -> F(t, sort(convert(numtheory:-divisors(t),list)),4), [$1..200]); # Robert Israel, Aug 31 2022

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[200], q[#, 4] &] (* Amiram Eldar, Aug 19 2022 *)
CoefficientList[Series[2 (2 - x + 2*x^2 - x^3 + 2*x^4 + x^6 + 2*x^8 + x^10 + 2*x^12 + x^14 + 2*x^16 - x^17 + 2*x^18 - x^19 + 2*x^20)/((x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(1 + x + x^5 + x^6 + x^7 + x^8 + x^9 + x^2 + x^4 + x^3 + x^10)*(x - 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jul 17 2025 *)

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [4,4]); \\ Michel Marcus, Aug 19 2022

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 2*a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - 2*a(n-16) + 2*a(n-17) - 2*a(n-18) + 2*a(n-19) - 2*a(n-20) + 2*a(n-21) - a(n-22). - Wesley Ivan Hurt, Jun 29 2024

G.f.: 2*x*(2 - x + 2*x^2 - x^3 + 2*x^4 + x^6 + 2*x^8 + x^10 + 2*x^12 + x^14 + 2*x^16 - x^17 + 2*x^18 - x^19 + 2*x^20)/((x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(1 + x + x^5 + x^6 + x^7 + x^8 + x^9 + x^2 + x^4 + x^3 + x^10)*(x - 1)^2). - Wesley Ivan Hurt, Jul 17 2025

A355641 Numbers k that can be written as the sum of 5 divisors of k (not necessarily distinct).

Original entry on oeis.org

5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 78, 80, 81, 84, 85, 88, 90, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 120, 125, 126, 128, 130, 132, 135, 136, 138

Offset: 1

Wesley Ivan Hurt, Aug 18 2022

nonn

Numbers that are divisible by at least one of 5, 6, 8, 9, 14 and 21. For proof see link. - Robert Israel, Sep 01 2022

The asymptotic density of this sequence is 17/35. - Amiram Eldar, Aug 08 2023

			9 is in the sequence since 9 = 3+3+1+1+1, where each summand divides 9.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Proof that A355641 consists of all numbers divisible by at least one of 5, 6, 8, 9, 14, 21.

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), this sequence (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

F:= proc(x,S,j) option remember;
      local s,k;
      if j = 0  then return(x = 0) fi;
      if S = [] or x > j*S[-1]  or x < j*S[1] then return false fi;
      s:= S[-1];
      for k from 0 to min(j,floor(x/s)) do
        if procname(x-k*s, S[1..-2],j-k) then return true fi
      od;
      false
end proc:
select(t -> F(t, sort(convert(numtheory:-divisors(t),list)),5), [$1..200]); # Robert Israel, Aug 31 2022

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[140], q[#, 5] &] (* Amiram Eldar, Aug 19 2022 *)

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [5,5]); \\ Michel Marcus, Aug 19 2022

A356609 Numbers k that can be written as the sum of 6 divisors of k (not necessarily distinct).

Original entry on oeis.org

6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186, 190

Offset: 1

Wesley Ivan Hurt, Aug 18 2022

nonn

Numbers divisible by at least one of 6, 8, 10, 14, 44, 52. For proof see link. - Robert Israel, Sep 02 2022

The asymptotic density of this sequence is 483/1430. - Amiram Eldar, Aug 08 2023

			18 is in the sequence since 18 = 9+2+2+2+2+1, where each summand divides 18.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Proof that A356609 consists of all numbers divisible by at least one of 6, 8, 10, 14, 44, 52.

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), this sequence (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

filter:= n-> ormap(t -> n mod t = 0, [6,8,10,14,44,52]):
select(filter, [$1..200]); # Robert Israel, Sep 02 2022

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[200], q[#, 6] &] (* Amiram Eldar, Aug 19 2022 *)

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [6,6]); \\ Michel Marcus, Aug 19 2022

A356635 Numbers k that can be written as the sum of 7 divisors of k (not necessarily distinct).

Original entry on oeis.org

7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 119, 120, 126, 128, 130

Offset: 1

Wesley Ivan Hurt, Aug 18 2022

nonn

If k is in the sequence then so is k*m for positive m. - David A. Corneth, Aug 19 2022
Numbers that are divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138.  For proof, see link. - Robert Israel, Sep 02 2022
The asymptotic density of this sequence is 17819629/37182145 = 0.479252... . - Amiram Eldar, Aug 08 2023

			10 is in the sequence since 10 = 2+2+2+1+1+1+1, where each summand divides 10.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Proof that A356635 consists of all numbers divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138.

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), this sequence (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

filter:= n -> ormap(t -> n mod t = 0, [7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138]):
select(filter, [$1..200]); # Robert Israel, Sep 02 2022

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[130], q[#, 7] &] (* Amiram Eldar, Aug 19 2022 *)

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [7,7]); \\ Michel Marcus, Aug 19 2022

A356657 Numbers k that can be written as the sum of 8 divisors of k (not necessarily distinct).

Original entry on oeis.org

8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176

Offset: 1

Wesley Ivan Hurt, Aug 20 2022

nonn

Terms are even. Proof by contradiction. Suppose m = a(n) is odd. Then each divisor is odd. Adding 8 odd numbers gives an even number. A contradiction. - David A. Corneth, Sep 02 2022

			14 is in the sequence since 14 = 2+2+2+2+2+2+1+1, where each summand divides 14.

David A. Corneth, Table of n, a(n) for n = 1..10000

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), this sequence (j=8), A356659 (j=9), A356660 (j=10).

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [8,8]); \\ Michel Marcus, Aug 21 2022

is(n) = if(n % 2 == 1, return(0)); my(d = divisors(n)); forvec(x = vector(8, i, [1, #d-1]), s=sum(i=1, #x, d [x[i]]); if(n == s, print(vector(#x, j, d[x[j]]));return(1)), 1); 0 \\ David A. Corneth, Aug 21 2022

A356659 Numbers k that can be written as the sum of 9 divisors of k (not necessarily distinct).

Original entry on oeis.org

9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 125

Offset: 1

Wesley Ivan Hurt, Aug 20 2022

nonn

If k is in the sequence then so is k*m. - David A. Corneth, Oct 08 2022

			14 is in the sequence since 14 = 2+2+2+2+2+1+1+1+1, where each summand divides 14.

David A. Corneth, Table of n, a(n) for n = 1..10000

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), this sequence (j=9), A356660 (j=10).

upto(n) = { my(v = vector(n,i,-1), t = 0); for(i = 1, n, if(v[i] == -1, print1(i", "); v[i] = is(i, 9); if(v[i] == 1, for(j = 2, n \ i, v[i*j] = 1; ) ) ); ); select(x->x >= 1, v, 1); }
is(n, {qd = 10}) = { my(d = divisors(n)); d = d[^#d]; forvec(x = vector(qd-1, i, [1, #d]), s = sum(i = 1, qd-1, d[x[i]]); if(n - s >= d[x[qd - 1]], if(n % (n - s) == 0, return(1); ) ) , 1 ); 0 } \\ David A. Corneth, Oct 08 2022

a(n + t) = a(n) + s for some finite t and s. - David A. Corneth, Oct 08 2022

A356660 Numbers k that can be written as the sum of 10 divisors of k (not necessarily distinct).

Original entry on oeis.org

10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162

Offset: 1

Wesley Ivan Hurt, Aug 20 2022

nonn

From David A. Corneth, Oct 08 2022: (Start)
All terms are even. Proof: suppose a term is odd. Then all divisors are odd. Adding 10 odd numbers gives an even number. A contradiction.
If k is a term then so is k*m for m >= 1. Proof: Multiply each divisor in this sum of 10 divisors that give k with m. Then each term is a divisor of k*m and their sum is k*m. (End)

			14 is in the sequence since 14 = 2+2+2+2+1+1+1+1+1+1, where each summand divides 14.

David A. Corneth, Table of n, a(n) for n = 1..10000

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), this sequence (j=10).

upto(n) = { my(v = vector(n,i,-1), t = 0); forstep(i = 2, n, 2, if(v[i] == -1, v[i] = is(i); if(v[i] == 1, for(j = 2, n \ i, v[i*j] = 1; ) ) ); ); select(x->x >= 1, v, 1); }
is(n, {qd = 10}) = { my(d = divisors(n), res = 0); d = d[^#d]; forvec(x = vector(qd-1, i, [1, #d]), s = sum(i = 1, qd-1, d[x[i]]); if(n - s >= d[x[qd - 1]], if(n % (n - s) == 0,  return(1); ) ) , 1 ); 0 } \\ David A. Corneth, Oct 08 2022

from sympy import divisors
def t_sum_of_n_div(n, target):
    out, p = [], divisors(n)[::-1][1:]
    def dfs(t, divs,  index_s, kk):
        if len(out)!=0 or kk>target:return
        if kk == target and t == 0:
            out.append(divs)
            return
        for i in range(index_s, len(p)):
            if t >= p[i]:
                temp_divs = divs.copy()
                temp_divs.append(p[i])
                dfs(t-p[i], temp_divs, i, kk+1)
    dfs(n, [], 0, 0)
    return out
terms = [i for i in range(2, 200) if len(t_sum_of_n_div(i,10))!=0]
print(terms) # Gleb Ivanov, Sep 02 2022

A383402 Smallest number whose largest odd divisor is its n-th divisor.

Original entry on oeis.org

1, 3, 6, 15, 18, 36, 30, 105, 60, 120, 90, 315, 816, 1360, 180, 700, 450, 360, 720, 1008, 420, 1540, 630, 900, 840, 1080, 1620, 1680, 2160, 1800, 1890, 5280, 1260, 3240, 3150, 17325, 7200, 29120, 5670, 9072, 2520, 3960, 10296, 18144, 3780, 20020, 5040, 7920, 10800

Offset: 1

Omar E. Pol, May 14 2025

nonn

From Peter Munn, May 15 2025 and May 20 2025: (Start)
A038547 is easily seen to be an upper bound for the sequence and a term equals this upper bound if and only if it is odd. Moreover, if a(n) = 2m with m odd, then the largest odd divisor of 2m is m, its second largest divisor, and a(n) = 2 * A038547((n+1)/2). It follows that 1 is the only term not divisible by 4 or by a nonunit term of A038547.
a(8) = 105 is the last squarefree term. (This is a corollary to lemma: prime p > 9 cannot be a divisor of a squarefree term. Proof of lemma: Let p divide squarefree k. If 3p is also divisor, set m = 9k/p, otherwise set m = 3k/p. Then k is not a term as m is a smaller number whose largest odd divisor is in the same position in the divisor list.)
(End)
If a(n) = m then m has at least n divisors. - David A. Corneth, May 16 2025
Every term a(n) = t > 1 is divisible by 2 or 3. Proof: Suppose it is not. Then it is odd and n is the number of divisors of t (cf. A000005). But t is not the smallest number that has n odd divisors that is odd. Setting every prime factor p to the largest prime < p and then multiplying gives a smaller odd number that has n divisors (cf. A064989). - David A. Corneth, May 17 2025

			The divisors of 18 are [1, 2, 3, 6, 9, 18] and the largest odd divisor is 9 and 9 is its 5th divisor, so a(5) = 18 because 18 the smallest number having that property.

David A. Corneth, Table of n, a(n) for n = 1..872
David A. Corneth, PARI program
David A. Corneth, Upper bounds on a(n) for n = 1..10000
Michael De Vlieger, Prime Power Decomposition of A383402(n), n = 1..261.

Row 1 of A383961.
The range of terms is a subset of {1} U A355200.
Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.
See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

With[{t = Table[If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]], {n, 1, 30000}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 14 2025 *)

a(n) = my(k=1); while (select(x->(x==k/2^valuation(k,2)), divisors(k), 1)[1] != n, k++); k; \\ Michel Marcus, May 14 2025

PARI
```
\\ See Corneth link
```

a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - Peter Munn, May 14 2025

More terms from Amiram Eldar, May 14 2025

A157932 Numbers k such that (3^(35k) + 5^(21k) + 7^(15*k)) mod 105 is prime.

Original entry on oeis.org

0, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 68, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 112, 114, 116, 120, 124, 126, 128, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 174

Offset: 1

Kyle D. Balliet, Mar 09 2009

Let b(k) = (3^(35*k) + 5^(21*k) + 7^(15*k)) mod 105, then sequence {b(k)} is 3, repeat (60, 68, 75, 17, 30, 23, 60, 47, 75, 38, 30, 2), with primes 3, 17, 23, 47, 2. First differences of {a(n)} are 4, 2, 2, 4, 4, 2, 2, 4, .... - Michel Marcus, Aug 15 2013
3^(35*k) + 5^(21*k) + 7^(15*k) = (4^k)*(3^k + 5^k + 7^k) mod 105, then by the division algorithm a simple proof yields that only numbers k of the form 24*m, 24*m+4, 24*m+6, 24*m+8, 24*m+12, 24*m+16, 24*m+18, 24*m+20 will be congruent to a prime modulo 105. Thus the pattern 4, 2, 2, 4, 4, 2, 2, ... will repeat infinitely. - Kyle D. Balliet, Jan 01 2014
Even numbers that can be written as the sum of 3 of their divisors, not necessarily distinct (see A355200). Also, numbers k of the form 12*m, 12*m+4, 12*m+6, 12*m+8. - Bernard Schott, Sep 08 2023

			a(4) = 3^(35*4) + 5^(21*4) + 7^(15*4) mod 105 = 17 (prime).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Equals {0} Union (A355200 \ A016945) <=> subsequence of even terms in A355200.

Select[Range[0,180],PrimeQ[Mod[3^(35#)+5^(21#)+7^(15#),105]]&] (* Harvey P. Dale, Oct 10 2017 *)

isok(n) = isprime((3^(35*n)+5^(21*n)+7^(15*n)) % 105); \\ Michel Marcus, Aug 15 2013

a(n)=n\4*12+[-4,0,4,6][n%4+1] \\ Charles R Greathouse IV, Dec 27 2013

is(n)=n%=12;n==0||n==4||n==6||n==8 \\ Charles R Greathouse IV, Dec 27 2013

a(n) = (-6-(-I)^n-I^n+6*n)/2 \\ Colin Barker, Oct 19 2015

concat(0, Vec(2*x^2*(2*x^2-x+2)/((x-1)^2*(x^2+1)) + O(x^100))) \\ Colin Barker, Oct 19 2015

3n - 4 <= a(n) <= 3n - 2. - Charles R Greathouse IV, Dec 27 2013
From Colin Barker, Oct 19 2015: (Start)
a(n) = (-6 - (-i)^n - i^n + 6*n)/2, where i = sqrt(-1).
G.f.: 2*x^2*(2*x^2-x+2) / ((x-1)^2*(x^2+1)). (End)

More terms from Michel Marcus, Aug 15 2013

cp's OEIS Frontend

A016945 a(n) = 6*n+3.

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A354591 Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).

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A355641 Numbers k that can be written as the sum of 5 divisors of k (not necessarily distinct).

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A356609 Numbers k that can be written as the sum of 6 divisors of k (not necessarily distinct).

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A356635 Numbers k that can be written as the sum of 7 divisors of k (not necessarily distinct).

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A356657 Numbers k that can be written as the sum of 8 divisors of k (not necessarily distinct).

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A356659 Numbers k that can be written as the sum of 9 divisors of k (not necessarily distinct).

A157932 Numbers k such that (3^(35k) + 5^(21k) + 7^(15*k)) mod 105 is prime.