A238354 -id:A238354 - cp's OEIS frontend

A238353 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n.

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0, 2, 3, 1, 1, 0, 0, 4, 3, 2, 1, 1, 0, 0, 2, 6, 3, 2, 1, 1, 0, 0, 4, 6, 6, 2, 2, 1, 1, 0, 0, 3, 10, 6, 5, 2, 2, 1, 1, 0, 0, 4, 11, 11, 6, 4, 2, 2, 1, 1, 0, 0, 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 0, 0, 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 0, 0, 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Reversed rows and also the columns converge to A002865 (setting A002865(0)=0).
Column k=0 is A000005 (n>=1), column k=1 is A237665.
Row sums are A000041.
Sum_{i=0..k} T(n,i) for k=0-9 gives: A000005, A034296, A224956, A238863, A238864, A238865, A238866, A238867, A238868, A238869.

			Triangle starts:
00:  1;
01:  1,  0;
02:  2,  0,  0;
03:  2,  1,  0,  0;
04:  3,  1,  1,  0,  0;
05:  2,  3,  1,  1,  0,  0;
06:  4,  3,  2,  1,  1,  0, 0;
07:  2,  6,  3,  2,  1,  1, 0, 0;
08:  4,  6,  6,  2,  2,  1, 1, 0, 0;
09:  3, 10,  6,  5,  2,  2, 1, 1, 0, 0;
10:  4, 11, 11,  6,  4,  2, 2, 1, 1, 0, 0;
11:  2, 16, 13, 10,  5,  4, 2, 2, 1, 1, 0, 0;
12:  6, 17, 19, 12,  9,  4, 4, 2, 2, 1, 1, 0, 0;
13:  2, 24, 24, 18, 11,  8, 4, 4, 2, 2, 1, 1, 0, 0;
14:  4, 27, 34, 22, 17, 10, 7, 4, 4, 2, 2, 1, 1, 0, 0;
15:  4, 35, 39, 33, 20, 15, 9, 7, 4, 4, 2, 2, 1, 1, 0, 0;
...
The 7 partitions of 5 and their maximal ascents are:
1:  [ 1 1 1 1 1 ]   0
2:  [ 1 1 1 2 ]   1
3:  [ 1 1 3 ]   2
4:  [ 1 2 2 ]   1
5:  [ 1 4 ]   3
6:  [ 2 3 ]   1
7:  [ 5 ]   0
There are 2 rows with 0 ascents, 3 with 1 ascent, 1 for ascents 2 and 3, giving row 5 of the triangle.

Joerg Arndt and Alois P. Heinz, Rows 0..140, flattened

Cf. A238354 (partitions by minimal ascent).

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, t)+`if`(i>n, 0, (p->
      `if`(t=0 or t-i=0, p, add(coeff(p, x, j)*x^
      max(j, t-i), j=0..degree(p))))(b(n-i, i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, k), k=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, Function[{p}, If[t == 0 || t-i == 0, p, Sum[Coefficient[p, x, j]*x^ Max[j, t-i], {j, 0, Exponent[p, x]}]]][b[n-i, i, i]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, k], {k, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *)

G.f. for column k>=1: sum(j>=1, q^j/(1-q^j) * (prod(i=1..j-1, (1-q^((k+1)*i))/(1-q^i) ) - prod(i=1..j-1, (1-q^(k*i))/(1-q^i) ) ) ), see the comment about the g.f. in A238863.

A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 3, 6, 1, 0, 0, 7, 4, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 6, 9, 0, 0, 0, 10, 0, 0, 3, 1, 0, 0, 7, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 12, 0, 0, 4, 13, 8

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A077017 w/o the first term.
Positions of terms > 0 are A120944.
Positions of zeros are A130091.
Triangle A238353 counts m such that A056239(m) = n and a(m) = k.
For maximal difference we have A286470 or A355526.
Positions of terms > 1 are A325161.
If singletons (k) have minimal difference k we get A355525.
Positions of 1's are A355527.
Prepending 0 to the prime indices gives A355528.
A115720 and A115994 count partitions by their Durfee square.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A066312, A238354, A238710, A286469, A351294, A352822, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],0,Min@@Differences[primeMS[n]]],{n,2,100}]

A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 0, 6, 1, 1, 0, 7, 0, 8, 0, 2, 1, 9, 0, 0, 1, 0, 0, 10, 1, 11, 0, 2, 1, 1, 0, 12, 1, 2, 0, 13, 1, 14, 0, 0, 1, 15, 0, 0, 0, 2, 0, 16, 0, 2, 0, 2, 1, 17, 0, 18, 1, 0, 0, 3, 1, 19, 0, 2, 1, 20, 0, 21, 1, 0, 0, 1, 1, 22, 0, 0, 1, 23

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The 0-prepended prime indices of 9842 are {0,1,4,8,12}, with differences (1,3,4,4), so a(9842) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of positive terms are A005117, complement A013929.
A similar statistic is counted by A238353.
The maximal version is A286469, without prepending A355526.
Without prepending we have A355524 or A355525.
Positions of ones are A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A112798 lists prime indices, with sum A056239.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A064428, A066312, A091602, A120944, A238354, A286470, A325161, A352822, A355527, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min@@Differences[Prepend[primeMS[n],0]],{n,2,100}]

A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

Original entry on oeis.org

6, 15, 30, 35, 42, 66, 70, 77, 78, 102, 105, 114, 138, 143, 154, 165, 174, 186, 195, 210, 221, 222, 231, 246, 255, 258, 282, 285, 286, 318, 323, 330, 345, 354, 366, 385, 390, 402, 426, 429, 435, 437, 438, 442, 455, 462, 465, 474, 498, 510, 534, 546, 555, 570

Offset: 1

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
    6: {1,2}
   15: {2,3}
   30: {1,2,3}
   35: {3,4}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   77: {4,5}
   78: {1,2,6}
  102: {1,2,7}
  105: {2,3,4}
  114: {1,2,8}
  138: {1,2,9}
  143: {5,6}
  154: {1,4,5}
  165: {2,3,5}
  174: {1,2,10}
  186: {1,2,11}
  195: {2,3,6}
  210: {1,2,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For minimal difference <= 1 we have A055932.
For maximal instead of minimal difference = 1 we have A066312.
For minimal difference > 1 we have A325160.
If zero is considered a prime index we get A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 or A355525 give minimal difference between prime indices.
Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[primeMS[#]]==1&]

Intersection of A005117 (squarefree) and A104210 (has consecutive primes).

A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

Original entry on oeis.org

2, 6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 42, 46, 58, 62, 66, 70, 74, 77, 78, 82, 86, 94, 102, 105, 106, 110, 114, 118, 122, 130, 134, 138, 142, 143, 146, 154, 158, 165, 166, 170, 174, 178, 182, 186, 190, 194, 195, 202, 206, 210, 214, 218, 221, 222, 226, 230

Offset: 1

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
   2: {1}
   6: {1,2}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  22: {1,5}
  26: {1,6}
  30: {1,2,3}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  42: {1,2,4}
  46: {1,9}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}
  70: {1,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For maximal instead of minimal difference we have A055932 or A066312.
Not prepending zero gives A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A056239 adds up prime indices.
A238352 counts partitions by fixed points, rank statistic A352822.
A279945 counts partitions by number of distinct differences.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 gives minimal difference if singletons go to 0, to index A355525.
Cf. A000005, A000040, A120944, A238354, A286469, A286470, A325160, A325161, A355526, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[Prepend[primeMS[#],0]]==1&]

Equals A005117 /\ (A005843 \/ A104210).

A238708 Number of strict partitions of n that include a pair of consecutive integers.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 15, 19, 23, 28, 33, 41, 48, 58, 68, 81, 95, 113, 131, 154, 179, 209, 241, 281, 323, 374, 429, 494, 565, 649, 739, 845, 961, 1095, 1241, 1411, 1595, 1807, 2039, 2303, 2593, 2923, 3283, 3691, 4139, 4643, 5195

Offset: 0

Clark Kimberling, Mar 03 2014

Column 2 of the array at A238709.

Column k=1 of A238354. - Alois P. Heinz, May 02 2014

			a(10) = 4 counts these partitions: 721, 541, 532, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A238709, A000009, A238354.

PartitionsQ[Range[50]] - Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 50}]

a(n) = A000009(n) - A003114(n). - George Beck, Jan 01 2022

A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.

Original entry on oeis.org

2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 6, 6, 2, 2, 1, 1, 3, 10, 6, 5, 2, 2, 1, 1, 4, 11, 11, 6, 4, 2, 2, 1, 1, 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1

Offset: 2

Gus Wiseman, Jul 08 2022

The triangle starts with n = 2, and k ranges from 0 to n - 2.

			Triangle begins:
  2
  2  1
  3  1  1
  2  3  1  1
  4  3  2  1  1
  2  6  3  2  1  1
  4  6  6  2  2  1  1
  3 10  6  5  2  2  1  1
  4 11 11  6  4  2  2  1  1
  2 16 13 10  5  4  2  2  1  1
  6 17 19 12  9  4  4  2  2  1  1
  2 24 24 18 11  8  4  4  2  2  1  1
  4 27 34 22 17 10  7  4  4  2  2  1  1
  4 35 39 33 20 15  9  7  4  4  2  2  1  1
  5 39 56 39 30 19 14  8  7  4  4  2  2  1  1
For example, row n = 8 counts the following reversed partitions:
  (8)         (233)      (35)      (125)    (26)    (116)  (17)
  (44)        (1223)     (134)     (11114)  (1115)
  (2222)      (11123)    (224)
  (11111111)  (11222)    (1124)
              (111122)   (1133)
              (1111112)  (111113)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Leading terms are A000005.
Row sums are A000041.
Counts m such that A056239(m) = n and A286470(m) = k.
This is a trimmed version of A238353, which extends to k = n.
For minimum instead of maximum we have A238354.
Ignoring singletons entirely gives A238710.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A115720 and A115994 count partitions by their Durfee square.
A279945 counts partitions by number of distinct differences.
Cf. A064428, A091602, A179254, A238352, A239455, A286469, A325404, A355524, A355526, A355532.

Table[Length[Select[Reverse/@IntegerPartitions[n], If[Length[#]==1,0,Max@@Differences[#]]==k&]],{n,2,15},{k,0,n-2}]

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A238353 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n.

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A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

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A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

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A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

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A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

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A238708 Number of strict partitions of n that include a pair of consecutive integers.

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A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.

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