A025157 -id:A025157 - cp's OEIS frontend

A373411 Sum of the n-th maximal antirun of squarefree numbers differing by more than one.

1, 2, 8, 6, 17, 24, 14, 72, 22, 78, 30, 64, 34, 72, 38, 80, 42, 89, 263, 58, 120, 127, 66, 136, 70, 144, 151, 78, 161, 168, 86, 360, 94, 293, 102, 208, 106, 216, 110, 224, 114, 233, 241, 379, 130, 264, 271, 138, 280, 142, 288, 600, 312, 158, 648, 166, 510, 351

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this antirun is given by A373127.

An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

			Row-sums of:
   1
   2
   3  5
   6
   7 10
  11 13
  14
  15 17 19 21
  22
  23 26 29
  30
  31 33
  34
  35 37
  38
  39 41
  42
  43 46
  47 51 53 55 57

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A173143.
Functional neighbors: A007674, A373127 (firsts A373128, sorted firsts A373200), A373404, A373405, A373408, A373412, A373413.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A061398, A077641, A077643, A112925, A112926, A143658, A372683, A373123, A373197, A373198.

Total/@Split[Select[Range[100],SquareFreeQ],#1+1!=#2&]//Most

A350841 Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 63, 68, 76, 80, 84, 88, 92, 99, 100, 104, 112, 116, 117, 124, 126, 132, 136, 140, 148, 152, 153, 156, 160, 164, 168, 171, 172, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 244, 248, 252, 260, 261

Offset: 1

Gus Wiseman, Jan 26 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The terms together with their prime indices begin:
   20: (3,1,1)
   28: (4,1,1)
   40: (3,1,1,1)
   44: (5,1,1)
   52: (6,1,1)
   56: (4,1,1,1)
   63: (4,2,2)
   68: (7,1,1)
   76: (8,1,1)
   80: (3,1,1,1,1)
   84: (4,2,1,1)
   88: (5,1,1,1)
   92: (9,1,1)
   99: (5,2,2)

Heinz number rankings are in parentheses below.
Taking just one condition gives (A073492) and (A065201), counted by A239955.
These partitions are counted by A350839.
A000041 = integer partitions, strict A000009.
A034296 = partitions with no gaps (A073491), strict A001227 (A073485).
A090858 = partitions with a single gap of size 1 (A325284).
A116931 = partitions with no successions (A319630), strict A003114.
A116932 = partitions with no successions or gaps of size 1, strict A025157.
A350842 = partitions with no gaps of size 1, strict A350844, sets A005314.
Cf. A000070, A024619, A026424, A055932, A183558, A277103, A305148, A321440, A350838.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],(Min@@Differences[Reverse[primeMS[#]]]<-1)&&(Min@@Differences[conj[primeMS[#]]]<-1)&]

A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.

Original entry on oeis.org

6, 8, 2, 3, 2, 7, 8, 0, 3, 8, 2, 8, 0, 1, 9, 3, 2, 7, 3, 6, 9, 4, 8, 3, 7, 3, 9, 7, 1, 1, 0, 4, 8, 2, 5, 6, 8, 9, 1, 1, 8, 8, 5, 8, 1, 8, 9, 7, 9, 9, 8, 5, 7, 7, 8, 0, 3, 7, 2, 8, 6, 0, 6, 6, 3, 9, 8, 9, 6, 6, 7, 8, 6, 8, 6, 9, 9, 8, 0, 2, 1, 0, 8, 1, 7, 3, 2, 0, 4, 3, 7, 8, 6, 2, 0, 5, 1, 2, 8, 2, 9, 5, 5, 9, 3, 3, 1, 8, 7, 6

Offset: 0

Paul D. Hanna, Oct 24 2015

Constant from Narayana's cows sequence: Limit A000930(n)/A000930(n+1) = r.

Reciprocal of constant described by A092526.

			0.682327803828019327369483739711048256891188581897998577803728606639896...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017. See Corollary 4.22. on p. 24.
Index entries for algebraic numbers, degree 3

Cf. A000930, A025157, A076725, A092526, A274112.

RealDigits[ ((Sqrt[93] + 9)/18)^(1/3) - ((Sqrt[93] - 9)/18)^(1/3), 10, 100][[1]] (* G. C. Greubel, May 01 2017 *)

a(n) = my(r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3)); floor(r*10^(n+1))%10
for(n=0,120,print1(a(n),", "))

solve(r=0, 1,  r^3 + r - 1 ) \\ Michel Marcus, Oct 25 2015

r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3).
Constant r satisfies:
(1) 1/(1 - r*i) = (r + r^2*i) where i^2 = -1.
(2) r = real( 1/(1 - r*i) ).
(3) r = norm( 1/(1 - r*i) ).
(4) r = r^2 + r^4.
Equals 1/A092526. - Vaclav Kotesovec, Nov 27 2017

A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 18, 24, 31, 41, 53, 70, 87, 112, 140, 178, 221, 277, 344, 428, 526, 648, 792, 971, 1180, 1436, 1738, 2103, 2533, 3049, 3660, 4387, 5242, 6259, 7450, 8860, 10511, 12453, 14723, 17387, 20489, 24121, 28343, 33269, 38982, 45632, 53327

Offset: 0

Gus Wiseman, Jan 18 2022

nonn

The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (32)     (33)      (43)
                    (31)    (41)     (51)      (52)
                    (1111)  (311)    (222)     (61)
                            (11111)  (411)     (322)
                                     (3111)    (331)
                                     (111111)  (511)
                                               (4111)
                                               (31111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version with quotients >= 2 is A000929, sets A018819.
                           <= 2 is A342094, ranked by A342191.
                           < 2 is A342096, sets A045690, strict A342097.
                           > 2 is A342098, sets A040039.
The sets version (subsets of prescribed maximum) is A045691.
These partitions are ranked by A350838.
The strict case is A350840.
A version for differences is A350842, strict A350844.
The complement is counted by A350846, ranked by A350845.
A000041 = integer partitions.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free partitions, ranked by A320340.
Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.

Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 18 2022

nonn

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

			The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

The version with quotients >= 2 is counted by A000929, sets A018819.
                           <= 2 is A342191, counted by A342094.
                           < 2 is counted by A342096, sets A045690.
                           > 2 is counted by A342098, sets A040039.
The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
For differences instead of quotients we have A350842, strict A350844.
The complement is A350845, counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]

A325161 Nonprime squarefree numbers not divisible by any two consecutive primes.

Original entry on oeis.org

1, 10, 14, 21, 22, 26, 33, 34, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 106, 110, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190, 194, 201

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of non-singleton integer partitions into distinct non-consecutive parts (counted by A003114 minus 1).

			The sequence of terms together with their prime indices begins:
   1: {}
  10: {1,3}
  14: {1,4}
  21: {2,4}
  22: {1,5}
  26: {1,6}
  33: {2,5}
  34: {1,7}
  38: {1,8}
  39: {2,6}
  46: {1,9}
  51: {2,7}
  55: {3,5}
  57: {2,8}
  58: {1,10}
  62: {1,11}
  65: {3,6}
  69: {2,9}
  74: {1,12}
  82: {1,13}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325160, A325162.

Select[Range[100],!PrimeQ[#]&&Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>1&]

A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 7, 8, 10, 13, 17, 19, 22, 25, 30, 35, 43, 52, 60, 70, 81, 93, 106, 122, 142, 166, 190, 216, 249, 287, 325, 371, 420, 479, 543, 617, 695, 784, 888, 1000, 1126, 1266, 1420, 1594, 1792, 2008, 2247, 2514, 2809, 3135, 3496, 3891, 4332

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
  1   2   3   4    5    6    7    8     9     A     B     C     D
              31   32   51   43   53    54    64    65    75    76
                   41        52   62    72    73    74    93    85
                             61   71    81    82    83    A2    94
                                  431   432   91    92    B1    A3
                                        531   532   A1    543   B2
                                              541   641   651   C1
                                                    731   732   643
                                                          741   652
                                                          831   751
                                                                832
                                                                931
                                                                5431

The version for subsets of prescribed maximum is A045691.
The double-free case is A120641.
The non-strict case is A350837, ranked by A350838.
An additive version (differences) is A350844, non-strict A350842.
The non-strict complement is counted by A350846, ranked by A350845.
Versions for prescribed quotients:
  = 2:  A154402, sets A001511.
  != 2: A350840 (this sequence), sets A045691.
  >= 2: A000929, sets A018819.
  <= 2: A342095, non-strict A342094.
  < 2:  A342097, non-strict A342096, sets A045690.
  > 2:  A342098, sets A040039.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]

A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

Original entry on oeis.org

6, 1, 18, 8, 4, 2, 10, 52, 678

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The sorted version is A373574.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			The maximal antiruns of nonsquarefree numbers begin:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The a(n)-th rows are:
    49
     4    8
   148  150  152
    64   68   72   75
    28   32   36   40   44
     9   12   16   18   20   24
    81   84   88   90   92   96   98
   477  480  484  486  488  490  492  495
  6345 6348 6350 6352 6354 6356 6358 6360 6363

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

For composite runs we have A073051, firsts of A176246, sorted A373400.
For squarefree runs we have the triple (5,3,1), firsts of A120992.
For prime runs we have the triple (1,3,2), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127, sorted A373200.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373401, firsts of A027833, sorted A373402.
For composite antiruns we have the triple (2,7,1), firsts of A373403.
Positions of first appearances in A373409.
The sorted version is A373574.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A025157, A049094, A061399, A068781, A072284, A110969, A251092, A294242, A373410, A373412.

t=Length/@Split[Select[Range[10000],!SquareFreeQ[#]&],#1+1!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A025162 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 29, 32, 35, 38, 41, 45, 48, 52, 56, 60, 64, 69, 73, 78, 83, 89, 94, 101, 107, 115, 122, 131, 139, 150, 159, 171, 182, 196, 208, 224, 238, 256, 272, 292, 310

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003114, A025157-A025162.

Column k=8 of A194543.

G.f.: Sum(x^(4*k^2-3*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Naohiro Nomoto, Feb 27 2002

A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 2, 2, 1, 1, 7, 3, 2, 2, 1, 1, 11, 4, 3, 2, 2, 1, 1, 15, 5, 3, 3, 2, 2, 1, 1, 22, 6, 4, 3, 3, 2, 2, 1, 1, 30, 8, 5, 4, 3, 3, 2, 2, 1, 1, 42, 10, 6, 4, 4, 3, 3, 2, 2, 1, 1, 56, 12, 7, 5, 4, 4, 3, 3, 2, 2, 1, 1, 77, 15, 9, 6, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 0

Alois P. Heinz, Aug 29 2011

T(n,k) = 1 for n >= 0 and k >= n.

In general, column k > 0 is asymptotic to c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(r + k*(1-r))) * n^(3/4)), where r is the smallest real root of the equation r^k + r = 1 and c = k*log(r)^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 02 2016

			T(7,3) = 3: [7], [6,1], [5,2].
T(23,6) = 11: [23], [22,1], [21,2], [20,3], [19,4], [18,5], [17,6], [16,7], [15,8], [15,7,1], [14,8,1].
Triangle begins:
   1;
   1, 1;
   2, 1, 1;
   3, 2, 1, 1;
   5, 2, 2, 1, 1;
   7, 3, 2, 2, 1, 1;
  11, 4, 3, 2, 2, 1, 1;
  15, 5, 3, 3, 2, 2, 1, 1;

Alois P. Heinz, Rows n = 0..140

Columns 0-8 give: A000041, A000009, A003114, A025157, A025158, A025159, A025160, A025161, A025162. T(n,0)-T(n,1) = A047967(n).

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then 1
    else add(b(n-i-j, i+j, k), j=k..n-i)
      fi
    end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[n<0, 0, If[n == 0, 1, Sum[b[n-i-j, i+j, k], {j, k, n-i}]]]; T[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

G.f. of column k: Sum_{j>=0} x^(j*((j-1)*k/2+1))/Product_{i=1..j} (1-x^i).

cp's OEIS Frontend

A373411 Sum of the n-th maximal antirun of squarefree numbers differing by more than one.

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A350841 Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.

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A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.

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A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

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A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

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A325161 Nonprime squarefree numbers not divisible by any two consecutive primes.

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A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

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A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

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