A036035 -id:A036035 - cp's OEIS frontend

A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64, 8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128, 9, 16, 21, 24, 25, 28, 36, 40, 45, 48, 60, 64, 72, 80, 81, 96, 108, 128, 144, 192, 256

Offset: 0

Amiram Eldar, Jun 16 2022

First differs from A074139 at the 8th row.
The n-th row begins with n+1, which corresponds to powers of primes, and ends with 2^n, which corresponds to squarefree numbers.
The n-th row contains the distinct values of the n-th row of A238963.

			Table begins:
  1;
  2;
  3, 4;
  4, 6, 8;
  5, 8, 9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64;
  8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128;
  ...
Numbers k with Omega(k) = 2 are either of the form p^2 with p prime, or of the form p1*p2 with p1 and p2 being distinct primes. The corresponding numbers of divisors are 3 and 4, respectively. Therefore the second row is {3, 4}.

Amiram Eldar, Table of n, a(n) for n = 0..18645 (rows 0..32, flattened)

Cf. A000005, A001222, A036035, A063008, A074139, A238963, A355027 (row lengths).

row[n_] := Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[row, 9, 0] // Flatten

row(n) = { my (m=Map()); forpart(p=n, mapput(m,prod(k=1, #p, 1+p[k]),0)); Vec(m) } \\ Rémy Sigrist, Jun 17 2022

A035796 Words over signatures (derived from multisets and multinomials).

Original entry on oeis.org

1, 1, 2, 2, 3, 18, 4, 48, 6, 5, 36, 100, 144, 6, 200, 180, 600, 7, 450, 900, 294, 24, 300, 1800, 8, 882, 7200, 448, 1200, 1470, 4410, 9, 1568, 22050, 648, 7200, 3136, 1800, 9408, 10, 14700, 2592, 16200, 1960, 56448, 900, 29400, 6048, 22050, 18144

Offset: 1

Alford Arnold

nonn

A reordering of A049009(n)=A049009(p(n)): distribution of words by numeric partition where the partition sequence: p(n)=[1],[2],[1,1],[3],[2,1],[1,1,1],[4],[3,1],[2,2],[2,1,1],... (A036036) is encoded by prime factorization ([P1,P2,P3,...] with P1 >= P2 >= P3 >= ... is encoded as 2^P1 * 3^P2 * 5^P3 *...): ep(n)=2,4,6,8,12,30,16,24,36,60, ... (A036035(n)) and then sorted: s(m)=2,4,6,8,12,16,24,30,32,36,48,60,... (A025487(m)). Hence A035796(n) = A049009(s(m)).

			27 = a(5) + a(6) + a(9) since a8(4) = 3, a12(5) = 18, a30(8) = 6; 256 = a(7) + a(8) + a(11) + a(13) + a(22) = 4 + 48 + 36 + 144 + 24
27 = a(5) + a(6) + a(9) = A049009(4) + A049009(5) + A049009(6) = 3 + 18 + 6 since A036035(4) = 8 = A025487(4+1), A036035(5) = 12 = A025487(5+1), A036035(6) = 30 = A025487(8+1);...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000312, A025487, A019575, A001700, A005651, A036035, A036036, A025487, A049009.

\\ here P is A025487 as vector and C is A049009 by partition.
GenS(lim)={my(L=List(), S=[1]); forprime(p=2, oo, listput(L, S); my(pp=vector(logint(lim, p), i, p^i)); S=concat([k*pp[1..min(if(k>1, my(f=factor(k)[, 2]); f[#f], oo), logint(lim\k, p))] | k<-S]); if(!#S, return(Set(concat(L)))) )}
P(n)={my(lim=1, v=[1]); while(#vt==S[k], sig))!) * prod(k=1, #sig, sig[k]!))}
seq(n)={[C(factor(t)[,2]) | t<-P(n)]} \\ Andrew Howroyd, Oct 18 2020

a(n) = A049009(p) where p is such that A036035(p) = A025487(n). [Corrected by Andrew Howroyd and Sean A. Irvine, Oct 18 2020]

More terms and additional comments from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001

a(1)=1 inserted by Andrew Howroyd and Sean A. Irvine, Oct 18 2020

A085643 Refines sequence A078812 using permutations and products of exponents on least prime signatures.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 6, 4, 6, 1, 5, 8, 12, 9, 12, 8, 1, 6, 10, 16, 9, 12, 36, 8, 12, 24, 10, 1, 7, 12, 20, 24, 15, 48, 36, 27, 16, 72, 32, 15, 40, 12, 1, 8, 14, 24, 30, 16, 18, 60, 48, 72, 54, 20, 96, 144, 54, 16, 20, 120, 80, 18, 60, 14, 1

Offset: 1

Alford Arnold, Aug 15 2003

The shape sequence for this table is A000041. Row sums in the example are A000079, A006906 and A001906.

			When the signatures (A025487) are listed in A036035 order the number of permutations of the exponents begin
1, 1 1, 1 2 1, 1 2 1 3 1, 1 2 2 3 3 4 1, ... b(n) and the products begin
1, 2 1, 3 2 1, 4 3 4 2 1, 5 4 6 3 4 2 1, ... c(n) thus we can write
1, 2 1, 3 4 1, 4 6 4 6 1, 5 8 12 9 12 8 1, ... a(n) = b(n)*c(n)

Cf. A078812.

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 32, 12, 64, 24, 36, 128, 48, 72, 256, 96, 144, 30, 512, 192, 216, 288, 60, 1024, 384, 432, 576, 120, 2048, 768, 864, 180, 1152, 240, 1296, 4096, 1536, 1728, 360, 2304, 480, 2592, 8192, 3072, 3456, 720, 900, 4608, 960, 5184, 1080, 16384

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

The number of second signatures represented by the divisors of A181800(n) equals the number of prime signatures represented among the divisors of a(n). Cf. A212172, A212644.

A permutation of A025487.

			6 (whose prime factorization is 2*3) is the largest squarefree divisor of 144 (whose prime factorization is 2^4*3^2). Since 144 = A181800(10), and 144/6 = 24, a(10) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A003557, A007947, A085082, A181800, A212172, A212642, A212644.

Other permutations of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.

a(n) = A003557(A181800(n)).

A238955 Maximal level size of arcs in divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 6, 1, 3, 4, 7, 12, 1, 3, 5, 8, 11, 18, 30, 1, 3, 5, 6, 8, 12, 15, 19, 24, 38, 60, 1, 3, 5, 7, 8, 13, 16, 19, 20, 30, 37, 46, 58, 90, 140, 1, 3, 5, 7, 8, 8, 13, 17, 20, 23, 20, 31, 36, 43, 52, 47, 66, 80, 100, 122, 185, 280

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  1, 3, ;
  1, 3, 4, 7, 12;
  1, 3, 5, 8, 11, 18, 30;
  1, 3, 5, 6,  8, 12, 15, 19, 24, 38, 60;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A238946 in graded colexicographic order.

Cf. A036035, A238968.

\\ here b(n) is A238946.
b(n)={if(n==1, 0, my(v=vector(bigomega(n))); fordiv(n, d, if(d>1, v[bigomega(d)] += omega(d))); vecmax(v))}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Apr 25 2020

T(n,k) = A238946(A036035(n,k)).

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 25 2020

A238957 The number of nodes at even level in divisor lattice in graded colexicographic order.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 3, 5, 6, 8, 9, 12, 16, 4, 6, 8, 8, 10, 12, 14, 16, 18, 24, 32, 4, 7, 9, 10, 12, 15, 16, 18, 20, 24, 27, 32, 36, 48, 64, 5, 8, 11, 12, 13, 14, 18, 20, 23, 24, 24, 30, 32, 36, 41, 40, 48, 54, 64, 72, 96, 128

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  1;
  1;
  2, 2;
  2, 3, 4;
  3, 4, 5, 6,  8;
  3, 5, 6, 8,  9, 12, 16;
  4, 6, 8, 8, 10, 12, 14, 16, 18, 24, 32;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A038548 in graded colexicographic order.

Cf. A036035, A074139, A238958, A238970.

\\ here b(n) is A038548.
b(n)={ceil(numdiv(n)/2)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020

T(n,k) = A038548(A036035(n,k)).
From Andrew Howroyd, Apr 01 2020: (Start)
T(n,k) = A074139(n,k) - A238958(n,k).
T(n,k) = ceiling(A074139(n,k)/2). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 01 2020

A238958 The number of nodes at odd level in divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 2, 4, 4, 6, 8, 3, 5, 6, 8, 9, 12, 16, 3, 6, 7, 8, 10, 12, 13, 16, 18, 24, 32, 4, 7, 9, 10, 12, 15, 16, 18, 20, 24, 27, 32, 36, 48, 64, 4, 8, 10, 12, 12, 14, 18, 20, 22, 24, 24, 30, 32, 36, 40, 40, 48, 54, 64, 72, 96, 128

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 3, 4;
  2, 4, 4, 6,  8;
  3, 5, 6, 8,  9, 12, 16;
  3, 6, 7, 8, 10, 12, 13, 16, 18, 24, 32;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A056924 in graded colexicographic order.

Cf. A036035, A074139, A238957, A238971.

\\ here b(n) is A056924.
b(n)={numdiv(n)\2}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020

T(n,k) = A056924(A036035(n,k)).
From Andrew Howroyd, Apr 01 2020: (Start)
T(n,k) = A074139(n,k) - A238957(n,k).
T(n,k) = floor(A074139(n,k)/2). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 01 2020

A238959 The number of arcs from even to odd level vertices in divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 6, 2, 5, 6, 10, 16, 3, 7, 9, 14, 17, 26, 40, 3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96, 4, 10, 14, 16, 22, 30, 32, 38, 46, 58, 68, 88, 102, 152, 224, 4, 11, 16, 19, 20, 26, 36, 41, 48, 52, 56, 74, 80, 93, 108, 112, 140, 162, 208, 240, 352, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 4,  6;
  2, 5,  6, 10, 16;
  3, 7,  9, 14, 17, 26, 40;
  3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A238950 in graded colexicographic order.

Cf. A036035, A238953, A238960, A238972.

T(n,k) = A238950(A036035(n,k)).
From Andrew Howroyd, Apr 25 2020: (Start)
T(n,k) = ceiling(A238953(n,k)/2).
T(n,k) = A238953(n,k) - A238960(n,k). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 25 2020

A238960 The number of arcs from odd to even level vertices in divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 6, 2, 5, 6, 10, 16, 2, 6, 8, 14, 16, 26, 40, 3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96, 3, 9, 13, 15, 22, 29, 32, 37, 46, 58, 67, 88, 102, 152, 224, 4, 11, 16, 19, 20, 26, 36, 41, 48, 52, 56, 74, 80, 93, 108, 112, 140, 162, 208, 240, 352, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  0;
  1, 2;
  1, 3,  6;
  2, 5,  6, 10, 16;
  2, 6,  8, 14, 16, 26, 40;
  3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014, Table A.1 entry |E_O(S)|.

Cf. A238951 in graded colexicographic order.

Cf. A036035, A238953, A238959, A238973.

T(n,k) = A238951(A036035(n,k)).
From Andrew Howroyd, Apr 25 2020: (Start)
T(n,k) = floor(A238953(n,k)/2).
T(n,k) = A238953(n,k) - A238959(n,k). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 25 2020

A122401 Subsequence of A074139 omitting values derived from partitions with a part of size 1.

Original entry on oeis.org

1, 3, 4, 5, 9, 6, 12, 7, 15, 16, 27, 8, 18, 20, 36, 9, 21, 24, 25, 45, 48, 81, 10, 24, 28, 30, 54, 60, 64, 108, 11, 27, 32, 35, 63, 36, 72, 75, 80, 135, 144, 243, 12, 30, 36, 40, 72, 42, 84, 90, 96, 162, 100, 180, 192, 324, 13, 33, 40, 45, 81, 48, 96, 49, 105, 112, 189, 108, 120, 216, 125, 225, 240, 405, 256, 432, 729, 14, 36, 44, 50, 90, 54

Offset: 0

Alford Arnold, Sep 01 2006

When viewed as a table, row sums are given by sequence A079274.

Corresponds to members of A036035 which are also powerful numbers (A001694).

			The two cyclic partitions of five are 5 and 3+2 yielding (5+1)=6 and (3+1)*(2+1) = 4*3 = 12
The array begins
1
(empty)
3
4
5 9
6 12
7 15 16 27
8 18 20 36

Cf. A122172 A079274 A122402, A316532.

A122401_row := proc(n)
    local e, a,L;
    L := [] ;
    for e in ListTools[Reverse](partition(n)) do
        if member(1,e) then
            ;
        else
            a := 1;
            for p in e do
                a := a*(p+1) ;
            end do:
            L := [op(L),a] ;
        end if;
    end do:
    L ;
end proc:
seq(A122401_row(i), i=0..15); # R. J. Mathar, Aug 28 2018

Extended by R. J. Mathar, Aug 28 2018

cp's OEIS Frontend

A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

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A035796 Words over signatures (derived from multisets and multinomials).

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A085643 Refines sequence A078812 using permutations and products of exponents on least prime signatures.

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A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

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A238955 Maximal level size of arcs in divisor lattice in graded colexicographic order.

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A238957 The number of nodes at even level in divisor lattice in graded colexicographic order.

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A238958 The number of nodes at odd level in divisor lattice in graded colexicographic order.

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A238959 The number of arcs from even to odd level vertices in divisor lattice in graded colexicographic order.

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