A182850 -id:A182850 - cp's OEIS frontend

A317258 Heinz numbers of integer partitions that are not totally nonincreasing.

18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 180, 198, 234, 242, 245, 250, 252, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 396, 414, 450, 468, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 612, 630, 648, 650, 666, 684

Offset: 1

Gus Wiseman, Jul 25 2018

nonn

An integer partition is totally nonincreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly decreasing and are themselves a totally nonincreasing integer partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of all integer partitions that are not totally nonincreasing begins: (221), (331), (2221), (332), (3221), (441), (22211), (4221), (442), (3321), (22221), (32211), (5221), (6221), (551), (443), (3331), (42211), (32221), (4421), (33211), (7221), (222211), (661), (8221), (4331), (552), (3332), (42221), (52211), (9221), (33221).

Cf. A056239, A071365, A100883, A112769, A181819, A182850, A242031, A296150, A305733, A317256, A317257.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totincQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totincQ[Reverse[Length/@Split[q]]]]];
Select[Range[1000],!totincQ[Reverse[primeMS[#]]]&]

A325264 Numbers whose omega-sequence sums to 7.

Original entry on oeis.org

30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The sequence of terms together with their prime indices and omega-sequences begins:
   30: {1,2,3} (3,3,1)
   36: {1,1,2,2} (4,2,1)
   42: {1,2,4} (3,3,1)
   64: {1,1,1,1,1,1} (6,1)
   66: {1,2,5} (3,3,1)
   70: {1,3,4} (3,3,1)
   78: {1,2,6} (3,3,1)
  100: {1,1,3,3} (4,2,1)
  102: {1,2,7} (3,3,1)
  105: {2,3,4} (3,3,1)
  110: {1,3,5} (3,3,1)
  114: {1,2,8} (3,3,1)
  130: {1,3,6} (3,3,1)
  138: {1,2,9} (3,3,1)
  154: {1,4,5} (3,3,1)
  165: {2,3,5} (3,3,1)
  170: {1,3,7} (3,3,1)
  174: {1,2,10} (3,3,1)
  182: {1,4,6} (3,3,1)
  186: {1,2,11} (3,3,1)
  190: {1,3,8} (3,3,1)
  195: {2,3,6} (3,3,1)
  196: {1,1,4,4} (4,2,1)

Positions of 7's in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],Total[omseq[#]]==7&]

A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129, 132

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also numbers whose adjusted frequency depth is one plus their number of prime factors counted with multiplicity. The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose adjusted frequency depth is equal to their length plus 1. The enumeration of these partitions by sum is given by A127002.

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
   6:     {1,2} (2,2,1)
  10:     {1,3} (2,2,1)
  12:   {1,1,2} (3,2,2,1)
  14:     {1,4} (2,2,1)
  15:     {2,3} (2,2,1)
  18:   {1,2,2} (3,2,2,1)
  20:   {1,1,3} (3,2,2,1)
  21:     {2,4} (2,2,1)
  22:     {1,5} (2,2,1)
  26:     {1,6} (2,2,1)
  28:   {1,1,4} (3,2,2,1)
  33:     {2,5} (2,2,1)
  34:     {1,7} (2,2,1)
  35:     {3,4} (2,2,1)
  38:     {1,8} (2,2,1)
  39:     {2,6} (2,2,1)
  44:   {1,1,5} (3,2,2,1)
  45:   {2,2,3} (3,2,2,1)
  46:     {1,9} (2,2,1)
  50:   {1,3,3} (3,2,2,1)
  51:     {2,7} (2,2,1)
  52:   {1,1,6} (3,2,2,1)
  55:     {3,5} (2,2,1)
  57:     {2,8} (2,2,1)
  58:    {1,10} (2,2,1)
  60: {1,1,2,3} (4,3,2,2,1)

Cf. A056239, A112798, A118914, A127002, A181819, A323023, A325246, A325259, A325266, A325270, A325277, A325282.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
Select[Range[100],fdadj[#]==PrimeOmega[#]+1&]

A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 0, 1, 1, 3, 3, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 6, 0, 0, 0

Offset: 0

Gus Wiseman, May 01 2019

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  0  1
  0  0  1  0  2  0
  0  0  1  2  1  0  0
  0  0  1  0  3  1  0  0
  0  0  1  0  3  2  0  0  0
  0  0  1  1  3  3  0  0  0  0
  0  0  1  1  5  3  0  0  0  0  0
  0  0  1  0  8  3  0  0  0  0  0  0
  0  0  1  2  6  6  0  0  0  0  0  0  0
  0  0  1  0 13  4  0  0  0  0  0  0  0  0
  0  0  1  0 12  8  1  0  0  0  0  0  0  0  0
  0  0  1  2 14  7  3  0  0  0  0  0  0  0  0  0
  0  0  1  0 17 11  3  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 22  7  8  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  2 17 16 10  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 28 10 15  0  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  1 29 13 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0
Row 15 counts the following partitions:
  111111111111111  54321       433221          333321        4322211
                   2222211111  443211          3332211       4332111
                               3322221         33222111      43221111
                               22222221        322221111
                               32222211        332211111
                               33321111        432111111
                               222222111       321111111111
                               3222111111
                               3321111111
                               22221111111
                               32211111111
                               222111111111
                               2211111111111
                               21111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000009.
Column k = 3 is A325334.
Column k = 4 is A325335.
Cf. A181819, A182850, A317246, A320348, A323014, A325280, A325372.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==k&]],{n,0,30},{k,0,n}]

depth(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L; r++); r)}
isok(p)={if(#p, for(i=1, #p, if(p[i]-1 > if(i>1, p[i-1], 0), return(0)))); 1}
row(n)={my(v=vector(1+n)); forpart(p=n, if(isok(p), v[1+depth(Vec(p))]++)); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 02 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.

The version with multiplicity is A181591.
For partitions we have A355383, with multiplicity A339006.
The version for compositions is A355384.
Positions of first appearances are A355386.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 count prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A000712, A022811, A056239, A071625, A118914, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]

A014643 Triangular array starting with {1,1}; then i-th term in a row gives number of i's in next row.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11

Offset: 1

N. J. A. Sloane

The row {2} could be safely prepended to this triangle. - Gus Wiseman, May 13 2018

			Triangle begins:
{1,1},
{1,2},
{1,2,2},
{1,2,2,3,3},
...

Alois P. Heinz, Rows n = 1..8, flattened

Rows converge to A001462.

Cf. A011784, A012257, A014643, A014644, A112798, A181819, A182850-A182858, A296150, A304455.

T:= proc(n) option remember; `if`(n=0, 2, (l->
      seq(i$l[i], i=1..nops(l)))([T(n-1)]))
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, May 17 2018

NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},8] (* Gus Wiseman, May 13 2018 *)

More terms from Patrick De Geest

A305731 Number of irreducible integer partitions of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 4, 0, 6, 3, 12, 0, 21, 1, 30, 19, 43, 10, 82, 20, 103, 68, 152, 58, 236, 102, 301, 196, 413, 205, 653, 310, 788, 580, 1115, 718, 1649, 1006, 2149, 1714, 3018, 2247, 4502, 3389, 6036, 5509, 8647, 7601, 12678, 11310, 17541

Offset: 0

Gus Wiseman, Jun 22 2018

nonn

A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is irreducible if m is of size > 1 and either gcd(m_1, ..., m_k) > 1 or the multiset {y_1, ..., y_k} is irreducible.

			The a(6) = 4 irreducible partitions are (42), (33), (222), (2211).

Cf. A100953, A181819, A182850, A182857, A275870, A304818, A305563, A305733, A305735.

ptnredQ[y_]:=Or[Length[y]==1,And[GCD@@y==1,ptnredQ[Sort[Length/@Split[y],Greater]]]];
Table[Length[Select[IntegerPartitions[n],!ptnredQ[#]&]],{n,20}]

A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

The first term of this sequence that is absent from A112769 is 150.

An integer partition is totally nondecreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly increasing and, taken in reverse order, are themselves a totally nondecreasing integer partition.

			150 is the Heinz number of (3,3,2,1), with multiplicities (1,1,2), which has multiplicities (2,1), which are decreasing, so 150 does not belong to the sequence.

Complement of A316529.

Cf. A056239, A071365, A112769, A181819, A182850, A242031, A296150, A305733, A316496, A317258.

A317590 Heinz numbers of integer partitions that are not uniformly normal.

Original entry on oeis.org

10, 14, 15, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

			Sequence of all non-uniformly normal integer partitions begins: (31), (41), (32), (311), (42), (51), (2111), (61), (411), (52), (71), (43), (81), (62), (3111), (421), (511), (322), (91), (21111), (331).

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317493, A317588, A317589.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],!uninrmQ[primeMS[#]]&]

A320118 a(1) = a(2) = 1; for n > 2, a(n) = A181819(n) * a(A181819(n)).

Original entry on oeis.org

1, 1, 2, 6, 2, 24, 2, 10, 6, 24, 2, 144, 2, 24, 24, 14, 2, 144, 2, 144, 24, 24, 2, 240, 6, 24, 10, 144, 2, 80, 2, 22, 24, 24, 24, 54, 2, 24, 24, 240, 2, 80, 2, 144, 144, 24, 2, 336, 6, 144, 24, 144, 2, 240, 24, 240, 24, 24, 2, 1728, 2, 24, 144, 26, 24, 80, 2, 144, 24, 80, 2, 360, 2, 24, 144, 144, 24, 80, 2, 336, 14, 24, 2, 1728

Offset: 1

Antti Karttunen, Nov 24 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A181819, A182850.

Cf. also A304465, A320016.

Nest[Append[#1, #2 #1[[#2]] ] & @@ {#, Times @@ Prime@ FactorInteger[Length@ # + 1][[All, -1]]} &, {1, 1}, 82] (* Michael De Vlieger, Nov 25 2018 *)

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A320118(n) = if(n<=2,1,A181819(n)*A320118(A181819(n)));

a(1) = a(2) = 1; for n > 2, a(n) = A181819(n) * a(A181819(n)).

cp's OEIS Frontend

A317258 Heinz numbers of integer partitions that are not totally nonincreasing.

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A325264 Numbers whose omega-sequence sums to 7.

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A325281 Numbers of the form a*b, a*a*b, or a*a*b*c where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

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A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

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A355382 Number of divisors d of n such that bigomega(d) = omega(n).

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A014643 Triangular array starting with {1,1}; then i-th term in a row gives number of i's in next row.

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Extensions

A305731 Number of irreducible integer partitions of n.

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A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

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A317590 Heinz numbers of integer partitions that are not uniformly normal.

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A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).