A070826 -id:A070826 - cp's OEIS frontend

A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42

Offset: 0

Alois P. Heinz, Jul 16 2021

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

			A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
Square array A(n,k) begins:
   1,  1,   2,    5,   15,    52,    203,     877,    4140, ...
   1,  2,   5,   15,   52,   203,    877,    4140,   21147, ...
   2,  4,  11,   36,  135,   566,   2610,   13082,   70631, ...
   3,  7,  21,   74,  296,  1315,   6393,   33645,  190085, ...
   5, 12,  38,  141,  592,  2752,  13960,   76464,  448603, ...
   7, 19,  64,  250, 1098,  5317,  28009,  158926,  963913, ...
  11, 30, 105,  426, 1940,  9722,  52902,  309546, 1933171, ...
  15, 45, 165,  696, 3281, 16972,  95129,  572402, 3670878, ...
  22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)

A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).

A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 2, 15, 15, 9, 5, 2, 52, 52, 31, 16, 7, 3, 203, 203, 120, 59, 25, 10, 4, 877, 877, 514, 244, 100, 38, 14, 5, 4140, 4140, 2407, 1112, 442, 161, 56, 19, 6, 21147, 21147, 12205, 5516, 2134, 750, 249, 80, 25, 8, 115975, 115975, 66491, 29505, 11147, 3799, 1213, 372, 111, 33, 10

Offset: 0

Alois P. Heinz, Jul 21 2021

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1) into distinct factors; A(3,1) = 5: 2*3*4, 4*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

			A(2,2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.
Square array A(n,k) begins:
  1,  1,   2,   5,   15,    52,   203,    877,    4140, ...
  1,  2,   5,  15,   52,   203,   877,   4140,   21147, ...
  1,  3,   9,  31,  120,   514,  2407,  12205,   66491, ...
  2,  5,  16,  59,  244,  1112,  5516,  29505,  168938, ...
  2,  7,  25, 100,  442,  2134, 11147,  62505,  373832, ...
  3, 10,  38, 161,  750,  3799, 20739, 121141,  752681, ...
  4, 14,  56, 249, 1213,  6404, 36332, 220000, 1413937, ...
  5, 19,  80, 372, 1887, 10340, 60727, 379831, 2516880, ...
  6, 25, 111, 539, 2840, 16108, 97666, 629346, 4288933, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000009, A036469, A346822, A346823, A346824, A346825, A346826, A346827, A346828, A346829, A346830.
Rows n=0+1,2-10 give: A000110, A087648, A346813, A346814, A346815, A346816, A346817, A346818, A346819, A346820.
Main diagonal gives A346519.
Antidiagonal sums give A346521.
Cf. A000040, A000079, A045778, A048993, A070826, A346426.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j]*b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jul 31 2021, after Alois P. Heinz *)

A(n,k) = A045778(A000079(n)*A070826(k+1)).

A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).

A068068 Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 19 2002

Shadow transform of triangular numbers.
a(n) is the number of primitive Pythagorean triangles with inradius n. For the smallest inradius of exactly 2^n primitive Pythagorean triangles see A070826.
Number of primitive Pythagorean triangles with leg 4n. For smallest (even) leg of exactly 2^n PPTs, see A088860. - Lekraj Beedassy, Jul 12 2006
As shown by Chi and Killgrove, a(n) is the total number of primitive Pythagorean triples satisfying area = n * perimeter, or equivalently 2 raised to the power of the number of distinct, odd primes contained in n. - Ant King, Mar 15 2011
This is the case k=0 of the sum over the k-th powers of the odd unitary divisors of n, which is multiplicative with a(2^e)=1 and a(p^e)=1+p^(e*k), p>2, and has Dirichlet g.f. zeta(s)*zeta(s-k)*(1-2^(k-s))/( zeta(2s-k)*(1-2^(k-2*s)) ). - R. J. Mathar, Jun 20 2011
Also the number of odd squarefree divisors of n: a(n) = Sum_{k = 1..A034444(k)} (A077610(n,k) mod 2) = Sum_{k = 1..A034444(k)} (A206778(n,k) mod 2). - Reinhard Zumkeller, Feb 12 2012
a(n) is also the number of even unitary divisors of 2*n. - Amiram Eldar, Jan 28 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henjin Chi and Raymond Killgrove, Problem 1447, Crux Math 15(5), May 1989.
Henjin Chi and Raymond Killgrove, Solution to Problem 1447, Crux Math 16(7), September 1990.
L. J. Gerstein, Pythagorean triples and inner products, Math. Mag. 78 (2005), 205-213.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012.
OEIS Wiki, Shadow transform.
Neville Robbins, On the number of primitive Pythagorean triangles with a given inradius, Fibonacci Quart. 44(4) (2006), 368-369.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A001221, A001620, A024361, A034444, A056901, A068067, A008683, A323239.

a068068 = length . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

A068068 := proc(n) local a,f; a :=1 ; for f in ifactors(n)[2] do if op(1,f) > 2 then a := a*2 ; end if; end do: a ; end proc: # R. J. Mathar, Apr 16 2011

a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]]
a[n_] := 2^(PrimeNu[n]+Mod[n, 2]-1); Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := If[p == 2, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, (d%2)*(gcd(d, n/d)==1)); \\ Michel Marcus, May 13 2014

a(n) = 2^omega(n>>valuation(n,2)) \\ Charles R Greathouse IV, May 14 2014

a(n) = A034444(2n)/2. If n is even, a(n) = 2^(omega(n)-1); if n is odd, a(n) = 2^omega(n). Here omega(n) = A001221(n) is the number of distinct prime divisors of n.
Multiplicative with a(2^e) = 1, a(p^e) = 2, p>2. - Christian G. Bower May 18 2005
a(n) = A024361(4n). - Lekraj Beedassy, Jul 12 2006
Dirichlet g.f.: zeta^2(s)/ ( zeta(2*s)*(1+2^(-s)) ). Dirichlet convolution of A034444 and A154269. - R. J. Mathar, Apr 16 2011
a(n) = Sum_{d|n} mu(2*d)^2. - Ridouane Oudra, Aug 11 2019
Sum_{k=1..n} a(k) ~ 4*n*((log(n) + 2*gamma - 1 + log(2)/3) / Pi^2 - 12*zeta'(2) / Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 18 2020
a(n) = Sum_{d divides n, d odd} mu(d)^2. - Peter Bala, Feb 01 2024

Edited by Dean Hickerson, Jun 08 2002

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42

Offset: 1

Jean-François Alcover, Nov 26 2015

If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021

			Triangle begins:
1, 2;                        squarefree and 2-smooth
1, 2, 3, 6;                  squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...

Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number

Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336

Offset: 0

Gus Wiseman, Jul 20 2022

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
Counting sequences instead of multisets gives A066843.
The integers themselves are the rows of A131818 (shifted).
For prime indices we have A355733, only prime factors A355744.
For prime factors instead of divisors we have A355746, factors A355537.
A000005 counts divisors.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]

from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

a(n) = A355733(A070826(n)).

a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

a(15)-a(21) from Michael S. Branicky, Aug 03 2022

a(22)-a(23) from Michael S. Branicky, Aug 08 2022

A147516 List giving least odd integer of each prime signature.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 81, 105, 135, 225, 243, 315, 405, 675, 729, 945, 1155, 1215, 1575, 2025, 2187, 2835, 3375, 3465, 3645, 4725, 6075, 6561, 8505, 10125, 10395, 10935, 11025, 14175, 15015, 17325, 18225, 19683, 23625, 25515, 30375, 31185, 32805

Offset: 1

Will Nicholes, Nov 05 2008

nonn

All numbers of the form 3^k2*5^k3*...*p_n^k_n, where k2 >= k3 >= ... >= k_n, sorted.

Ray Chandler, Table of n, a(n) for n = 1..10000
David Ryan, Mathematical Harmony Analysis, arXiv preprint arXiv:1603.08904 [cs.SD], 2016-2017.

Multiplicative closure of A001147.

Cf. A025487, A070826.

PrimeExponents[n_] := FactorInteger[n][[All, 2]]; lpe = {}; A147516 = {1}; Do[pe = PrimeExponents[n] // Sort; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[A147516, n]], {n, 3, 40000, 2}]; A147516 (* Jean-François Alcover, Jan 27 2015, after Robert G. Wilson v *)

is(n)=my(k=oo, t); forprime(p=3,, t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1))) \\ Charles R Greathouse IV, Aug 20 2015

Sum_{n>=1} 1/a(n) = Product_{n>=2} 1/(1 - 1/A070826(n)) = 1.6241170949... - Amiram Eldar, Oct 20 2020

Edited and extended by Ray Chandler, Jul 29 2010

A061346 Odd numbers that are neither primes nor prime powers.

Original entry on oeis.org

15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 123, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205, 207, 209, 213, 215, 217, 219, 221

Offset: 1

N. J. A. Sloane, Jun 08 2001

Odd numbers with at least two distinct prime factors. - N. J. A. Sloane, Oct 15 2022
Odd leg of more than one primitive Pythagorean triangles. For smallest odd leg common to 2^n PPTs, see A070826. - Lekraj Beedassy, Jul 12 2006
Numbers that can be factored by Shor's algorithm. - Charles R Greathouse IV, Mar 05 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A225375 is a subsequence.

Cf. A061345.

for k := 3 to 253 by 2 do ar := factorlist(k); if ar[0] < ar[length(ar)-1] then write(k," ") end; end;

select(t -> nops(ifactors(t)[2]) > 1, [seq(2*i+1,i=1..1000)]); # Robert Israel, Dec 14 2014

Select[Range[1, 249, 2], Length[FactorInteger[#]] > 1 &] (* Alonso del Arte, Jan 30 2012 *)
Select[ Range[1, 475, 2], PrimeNu@# > 1 &] (* Robert G. Wilson v, Dec 12 2014 *)

is(n)=ispower(n,,&n);n%2&&!isprime(n)&&n>1 \\ Charles R Greathouse IV, Jan 30 2012

is(n)=n%2 && !isprimepower(n) && n>1 \\ Charles R Greathouse IV, May 06 2016

count(x)=if(x<9, 0, (x\=1) - sum(k=1,logint(x,3), primepi(sqrtnint(x,k)) - 1) - x\2 - 1) \\ Charles R Greathouse IV, Mar 06 2018

a(n) ~ 2n. - Charles R Greathouse IV, Aug 20 2012

More terms from Klaus Brockhaus, Jun 10 2001

A355746 Number of different multisets that can be obtained by choosing a prime index (or a prime factor) of each integer from 2 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 20, 20, 20, 26, 26, 36, 58, 116, 116, 140, 140, 280, 280, 384, 384, 536, 536, 536, 844, 1688, 2380, 2716, 2716, 5432, 8484, 10152, 10152, 13308, 13308, 18064, 21616, 43232, 43232, 47648, 47648, 54656, 84480, 114304, 114304

Offset: 1

Gus Wiseman, Jul 20 2022

nonn

			The a(n) multisets for n = 2, 6, 10, 12:
  {1}  {1,1,1,2,3}  {1,1,1,1,1,2,2,3,4}  {1,1,1,1,1,1,2,2,3,4,5}
       {1,1,2,2,3}  {1,1,1,1,2,2,2,3,4}  {1,1,1,1,1,2,2,2,3,4,5}
                    {1,1,1,1,2,2,3,3,4}  {1,1,1,1,1,2,2,3,3,4,5}
                    {1,1,1,2,2,2,3,3,4}  {1,1,1,1,2,2,2,2,3,4,5}
                                         {1,1,1,1,2,2,2,3,3,4,5}
                                         {1,1,1,2,2,2,2,3,3,4,5}

Michael S. Branicky, Table of n, a(n) for n = 1..75

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
The integers themselves are the rows of A131818 (shifted).
Counting sequences instead of multisets: A355537, with multiplicity A327486.
Using prime indices instead of 2..n gives A355744, for sequences A355741.
The version for divisors instead of prime factors is A355747.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A002110, A076610, A355538, A355731, A355733, A355740, A355742, A355745.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[primeMS/@Range[2,n]]]],{n,15}]

from sympy import factorint
from itertools import count, islice
def agen():
    s = {(1,)}
    for n in count(2):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in factorint(n))
print(list(islice(agen(), 53))) # Michael S. Branicky, Aug 03 2022

a(n) = A355744(A070826(n)).

a(p) = a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

a(28) and beyond from Michael S. Branicky, Aug 03 2022

A060389 a(1)=p_1, a(2)=p_1 + p_1p_2, a(3)=p_1 + p_1p_2 + p_1p_2p_3, ... where p_i is the i-th prime.

Original entry on oeis.org

2, 8, 38, 248, 2558, 32588, 543098, 10242788, 233335658, 6703028888, 207263519018, 7628001653828, 311878265181038, 13394639596851068, 628284422185342478, 33217442899375387208, 1955977793053588026278

Offset: 1

Jason Earls, Apr 04 2001

nonn

Partial sums of the primorials A002110 starting from 2. - Charles R Greathouse IV, Feb 05 2014
All terms are even. From a(98) on, all terms are multiples of 523. - Charles R Greathouse IV, Feb 05 2014
The only values of n for which a(n)/2 is prime are: 3, 5, 7, 11, 15, 47, 49. The corresponding 7 primes are: 19, 1279, 271549, 103631759509, 314142211092671239, 826811434211869939736645732264127163964562391958563586838409421490271014424018927729, 41839936239750050346953677118447851613901200239299781782205558511980130628486398081201749. - Amiram Eldar, May 04 2017

			a(4) = 248 because p_1 + p_1*p_2 + p_1*p_2*p_3 + p_1*p_2*p_3*p_4 = 2 + 6 + 30 + 210 = 248.
a(5) = 2558: A002110(3) = 30, A286624(4) = 85, a(2) = 8; 30*85 + 8 = 2558. - _Bob Selcoe_, Oct 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to primorial numbers

Cf. A002110, A070826, A084737, A143293, A203008 [= A327860(a(n))], A276085, A286624, A373158.

for n from 1 to 30 do printf(`%d,`,sum(product(ithprime(i), i=1..j), j=1..n)) od:

Accumulate[Denominator[Accumulate[1/Prime[Range[20]]]]] (* Alonso del Arte, Mar 21 2013 *)
Accumulate@ FoldList[Times, Prime@ Range@ 17] (* Michael De Vlieger, May 04 2017 *)

a(n)=my(s,P=1); forprime(p=2,prime(n),s+=P*=p); s \\ Charles R Greathouse IV, Feb 05 2014

a(n) = A002110(n-2)*A286624(n-1) + a(n-3), n >= 4. - Bob Selcoe, Oct 12 2017

a(n) = A276085(A070826(1+n)) = A084737(2+n)-2 = A373158(A002110(n)). - Antti Karttunen, Feb 06 2024, Oct 28 2024

More terms from James Sellers, Apr 05 2001

cp's OEIS Frontend

A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A068068 Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.

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A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

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A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

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A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

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A147516 List giving least odd integer of each prime signature.

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A060389 a(1)=p_1, a(2)=p_1 + p_1p_2, a(3)=p_1 + p_1p_2 + p_1p_2p_3, ... where p_i is the i-th prime.