A182850 -id:A182850 - cp's OEIS frontend

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 7, 2, 0, 0, 1, 0, 1, 12, 1, 0, 0, 0, 1, 0, 1, 17, 2, 1, 0, 0, 0, 1, 0, 1, 24, 4, 0, 0, 0, 0, 0, 1, 0, 1, 33, 5, 1, 1, 0, 0, 0, 0, 1, 0, 1, 44, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 57, 14, 3, 0, 1

Offset: 0

Gus Wiseman, Apr 18 2019

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. The omicron of the partition is 0 if the omega-sequence is empty, 1 if it is a singleton, and otherwise the second-to-last part. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), and its omicron is 2.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  5  0  0  1
  0  1  7  2  0  0  1
  0  1 12  1  0  0  0  1
  0  1 17  2  1  0  0  0  1
  0  1 24  4  0  0  0  0  0  1
  0  1 33  5  1  1  0  0  0  0  1
  0  1 44  9  1  0  0  0  0  0  0  1
  0  1 57 14  3  0  1  0  0  0  0  0  1
  0  1 76 20  3  0  0  0  0  0  0  0  0  1
Row n = 8 counts the following partitions.
  (8)  (44)       (431)  (2222)  (11111111)
       (53)       (521)
       (62)
       (71)
       (332)
       (422)
       (611)
       (3221)
       (3311)
       (4211)
       (5111)
       (22211)
       (32111)
       (41111)
       (221111)
       (311111)
       (2111111)

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

Row sums are A000041. Column k = 2 is A325267.
Cf. A181819, A181821, A304634, A304636, A323014, A323023, A325250, A325273, 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).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==k&]],{n,0,10},{k,0,n}]

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

A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 3, 4, 3, 0, 0, 1, 1, 4, 8, 1, 0, 0, 1, 3, 6, 10, 2, 0, 0, 0, 1, 2, 8, 13, 6, 0, 0, 0, 0, 1, 3, 11, 20, 7, 0, 0, 0, 0, 0, 1, 1, 11, 29, 14, 0, 0, 0, 0, 0, 0, 1, 5, 19, 31, 20, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
  1
  1  1
  1  1  1
  1  2  1  1
  1  1  2  3  0
  1  3  4  3  0  0
  1  1  4  8  1  0  0
  1  3  6 10  2  0  0  0
  1  2  8 13  6  0  0  0  0
  1  3 11 20  7  0  0  0  0  0
  1  1 11 29 14  0  0  0  0  0  0
  1  5 19 31 20  1  0  0  0  0  0  0
  1  1 17 50 30  2  0  0  0  0  0  0  0
  1  3 25 64 37  5  0  0  0  0  0  0  0  0
  1  3 29 74 62  7  0  0  0  0  0  0  0  0  0
Row n = 8 counts the following partitions:
  (8)  (44)        (53)    (332)      (4211)
       (2222)      (62)    (422)      (32111)
       (11111111)  (71)    (611)
                   (431)   (3221)
                   (521)   (5111)
                   (3311)  (22211)
                           (41111)
                           (221111)
                           (311111)
                           (2111111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000041.
Column k = 1 is A032741.
Column k = 2 is A329745.
A similar invariant is frequency depth; see A323014, A325280.
The version for compositions is A329744.
The version for binary words is A329767.
Cf. A098504, A182850, A225485, A242882, A318928, A325410, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

\\ rr(p) gives runs resistance of partition.
rr(p)={my(r=0); 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)); p=Vec(L); r++); r}
row(n)={my(v=vector(n)); forpart(p=n, v[1+rr(Vec(p))]++); v}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

A305563 Number of reducible integer partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 7, 15, 16, 27, 30, 56, 56, 100, 105, 157, 188, 287, 303, 470, 524, 724, 850, 1197, 1339, 1856, 2135, 2814, 3305, 4360, 4951, 6532, 7561, 9563, 11195, 14165, 16328, 20631, 23866, 29471, 34320, 42336, 48672, 59872, 69139, 83625, 96911, 117153

Offset: 1

Gus Wiseman, Jun 05 2018

nonn

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

			The a(6) = 7 reducible integer partitions are (6), (51), (411), (321), (3111), (21111), (111111). Missing from this list are (42), (33), (222), (2211).

Cf. A007916, A071625, A181819, A182850, A182857, A275870, A304465, A304660, A304687, A304818, A305564, A305565, A305566.

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

A325249 Sum of the omega-sequence of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 4, 3, 5, 1, 8, 1, 5, 5, 5, 1, 8, 1, 8, 5, 5, 1, 9, 3, 5, 4, 8, 1, 7, 1, 6, 5, 5, 5, 7, 1, 5, 5, 9, 1, 7, 1, 8, 8, 5, 1, 10, 3, 8, 5, 8, 1, 9, 5, 9, 5, 5, 1, 12, 1, 5, 8, 7, 5, 7, 1, 8, 5, 7, 1, 10, 1, 5, 8, 8, 5, 7, 1, 10, 5, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Apr 16 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 omega-sequence of 180 is (5,3,2,2,1) with sum 13, so a(180) = 13.

Positions of m's are A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A118914, A181819, A182850, A323023, A325274.
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).

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

a(n) = A056239(A325248(n)).

a(n!) = A325274(n).

A225486 Maximal frequency depth for the partitions of n.

Original entry on oeis.org

0, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Clark Kimberling, May 08 2013

nonn

See A225485 for the definition of frequency depth.

The frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). Differs from A325282 at a(0) and a(1). - Gus Wiseman, Apr 19 2019

			(See A225485.)

Run lengths are A325258, i.e., first differences of Levine's sequence A011784.

Cf. A008284, A116608, A181819, A182850, A182857, A225485, A323014, A323023, A325239, A325242, A325254, A325282, A325283.

c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
t[s_] := t[s] = Length[f[s]]
u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], k]],
    {k, 1, Length[IntegerPartitions[n]]}];
Prepend[Table[Max[u[n]], {n, 2, 10}], 0]
(* second program *)
grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
Join@@MapIndexed[ConstantArray[#2[[1]]-1,#1]&,Length[#]-Last[#]&/@NestList[grw,{1,1},6]] (* Gus Wiseman, Apr 19 2019 *)

a(n) = number of terms in row n of the array in A225485, for n > 0.

More terms from Gus Wiseman, Apr 19 2019

A317245 Number of supernormal integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 1, 3, 3, 4, 2, 4, 5, 6, 6, 10, 7, 10, 9, 9, 10, 11, 12, 12, 21, 12, 18, 17, 21, 19, 28, 23, 28, 26, 27, 24, 32, 29, 36, 34, 46, 42, 55, 48, 65, 65, 74, 70, 88, 81, 83, 103, 112, 129, 153, 157, 190, 205, 210, 242, 283, 276, 321

Offset: 0

Gus Wiseman, Jul 24 2018

nonn

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

			The a(10) = 4 supernormal integer partitions are (4321), (33211), (322111), (1111111111).
The a(21) = 10 supernormal integer partitions:
  (654321),
  (4443321),
  (44432211), (44333211), (44332221),
  (4432221111), (4333221111), (4332222111),
  (433322211),
  (111111111111111111111).

Cf. A000041, A181819, A182850, A182857, A275870, A304660, A305563, A317081, A317086, A317088, A317246.

supnrm[q_]:=Or[q=={}||Union[q]=={1},And[Union[q]==Range[Max[q]],supnrm[Sort[Length/@Split[q],Greater]]]];
Table[Length[Select[IntegerPartitions[n],supnrm]],{n,0,30}]

A325248 Heinz number of the omega-sequence of n.

Original entry on oeis.org

1, 2, 2, 6, 2, 18, 2, 10, 6, 18, 2, 90, 2, 18, 18, 14, 2, 90, 2, 90, 18, 18, 2, 126, 6, 18, 10, 90, 2, 50, 2, 22, 18, 18, 18, 42, 2, 18, 18, 126, 2, 50, 2, 90, 90, 18, 2, 198, 6, 90, 18, 90, 2, 126, 18, 126, 18, 18, 2, 630, 2, 18, 90, 26, 18, 50, 2, 90, 18, 50

Offset: 1

Gus Wiseman, Apr 16 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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The omega-sequence of 180 is (5,3,2,2,1) with Heinz number 990, so a(180) = 990.

Positions of squarefree terms are A325247.
Positions of normal numbers (A055932) are A325251.
First positions of each distinct term are A325238.
Cf. A056239, A070175, A112798, A118914, A181819, A181821, A323023, A325249, A325275, 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).

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

A001222(a(n)) = A323014(n).
A061395(a(n)) = A001222(n).
A304465(n) = A055396(a(n)/2).
A325249(n) = A056239(a(n)).
a(n!) = A325275(n).

A317491 Number of fully normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 30, 33, 46, 50, 68, 77, 100, 112, 146, 167, 201, 234, 290, 326, 400, 456, 545, 622, 744, 845, 1004, 1153, 1351, 1551, 1819, 2103, 2434, 2808, 3248, 3735, 4304, 4943, 5661, 6506, 7446, 8499, 9657, 11070, 12505, 14325, 16183

Offset: 0

Gus Wiseman, Jul 30 2018

nonn

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

			The a(6) = 6 fully normal partitions are (6), (51), (42), (411), (321), (111111). Missing from this list are (33), (3111), (222), (2211), (21111).

Cf. A181819, A182850, A182857, A275870, A304660, A305563, A317081, A317086, A317088, A317246, A317492, A317493.

fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],fulnrmQ]],{n,0,30}]

a(n) = A317245(n) iff n is 1 or a prime number.

A182853 Squarefree composite integers and powers of squarefree composite integers.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Matthew Vandermast, Jan 04 2011

Numbers that require exactly three iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.

Numbers such that A001221(n) > 1 and A071625(n) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 3. See also A182854, A182855.
Subsequence of A072774 and A182851.
Cf. A120944.

isoka(n) = (omega(n) > 1) && issquarefree(n); \\ A120944
isok(n) = isoka(n) || (ispower(n,,&k) && isoka(k)); \\ Michel Marcus, Jun 24 2017

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A182853(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

(define A182853 (MATCHING-POS 1 1 (lambda (n) (= 3 (A182850 n))))) ;; After the alternative definition of the sequence given by the original author. Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 05 2016

A316496 Number of totally strong integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668

Offset: 0

Gus Wiseman, Jul 29 2018

nonn

An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.

			The a(1) = 1 through a(8) = 12 totally strong partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.

Cf. A181819, A182850, A182857, A304660, A305563, A316597.
The Heinz numbers of these partitions are A316529.
The version for compositions is A332274.
The dual version is A332275.
The version for reversed partitions is (also) A332275.
The narrowly normal version is A332297.
The alternating version is A332339 (see also A317256).
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.

totincQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]

Updated with corrected terminology by Gus Wiseman, Mar 07 2020

cp's OEIS Frontend

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

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A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

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A305563 Number of reducible integer partitions of n.

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A325249 Sum of the omega-sequence of n.

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A225486 Maximal frequency depth for the partitions of n.

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A317245 Number of supernormal integer partitions of n.

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A325248 Heinz number of the omega-sequence of n.

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A317491 Number of fully normal integer partitions of n.

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