A025157 -id:A025157 - cp's OEIS frontend

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset: 1

Gus Wiseman, Jan 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
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.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

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

A237979 Number of strict partitions of n such that (least part) > number of parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 28, 32, 35, 40, 45, 50, 56, 63, 70, 78, 87, 96, 107, 118, 131, 144, 160, 175, 194, 213, 235, 257, 284, 310, 342, 373, 410, 447, 491, 534, 585, 637, 696, 756, 826, 896, 977, 1060, 1153, 1250, 1359, 1471, 1597, 1729, 1874, 2026, 2195, 2371, 2565

Offset: 1

Clark Kimberling, Feb 18 2014

Also the number of partitions into distinct parts with minimal part >= 2 and difference between parts >= 3. [Joerg Arndt, Mar 31 2014]

			a(9) = 3 counts these partitions:  9, 63, 54.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A000009, A237976, A096401, A237977, A025157.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

N=66; q='q+O('q^N); Vec(-1+sum(n=0, N, q^(n*(3*n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

G.f. with a(0)=0: sum(n>=0, q^(n*(3*n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Mar 09 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1 + 3*r^2)) * n^(3/4)), where r = A263719 and c = 3*(log(r))^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 15 2022
a(n) ~ A263719 * A025157(n). - Vaclav Kotesovec, Jan 15 2022

A237976 Number of strict partitions of n such that (least part) < number of parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 14, 17, 21, 25, 31, 37, 45, 54, 64, 76, 90, 106, 124, 146, 170, 198, 230, 267, 308, 357, 410, 472, 542, 621, 709, 811, 923, 1051, 1194, 1355, 1534, 1738, 1962, 2215, 2497, 2812, 3161, 3553, 3986, 4469, 5005, 5600, 6258

Offset: 0

Clark Kimberling, Feb 18 2014

			a(8) = 3 counts these partitions:  71, 521, 431.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A237977, A096401, A237979, A025157, A039899.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A025157(n). - Vaclav Kotesovec, Jan 18 2022

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022

A237977 Number of strict partitions of n such that (least part) <= number of parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 29, 36, 42, 51, 60, 72, 84, 100, 117, 137, 160, 187, 216, 251, 290, 334, 385, 442, 507, 581, 664, 757, 864, 982, 1116, 1266, 1435, 1622, 1835, 2069, 2333, 2626, 2954, 3316, 3724, 4172, 4673, 5227, 5844

Offset: 0

Clark Kimberling, Feb 18 2014

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A237976, A096401, A237979, A025157, A039900.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A237979(n). - Vaclav Kotesovec, Jan 18 2022

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022

A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 28, 31, 35, 38, 43, 47, 53, 58, 65, 71, 80, 87, 97, 106, 118, 128, 142, 154, 170, 185, 203, 220, 242, 262, 287, 311, 340, 368, 402, 435, 474, 513, 558, 603, 656, 708, 768, 829, 898, 968, 1048

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n such that if k is the largest part, then each 1,2,...,k-1 occur at least 4 times. Example: a(8)=3 because we have [2,2,1,1,1,1], [2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

			a(8) = 3 because we have [8], [7,1] and [6,2].

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

Cf. A003114, A025157-A025162.

Column k=4 of A194543.

g:=sum(x^(2*k^2-k)/product(1-x^j,j=1..k),k=1..7): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..66); # Emeric Deutsch, Apr 17 2006

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(j*(2*j - 1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 02 2016 *)

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

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*sqrt(Pi*r^3*(1+4*r^3))), where r = 0.72449195900051561158837228218703656578649448135001101727... is the root of the equation r^4 + r = 1 and c = 2*log(r)^2 + polylog(2, 1-r) = 0.50498141294472195442598916817438524920370382784609501495065... . - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 12 2004

A025159 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 26, 29, 32, 35, 38, 42, 46, 50, 55, 60, 66, 72, 79, 86, 95, 103, 113, 123, 135, 146, 160, 173, 189, 204, 222, 239, 260, 280, 303, 326, 353, 379, 410, 440, 475, 510, 550, 590, 636, 682

Offset: 1

Clark Kimberling

nonn

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

Cf. A003114, A025157-A025162.

Column k=5 of A194543.

G.f.: Sum(x^(5/2*k^2-3/2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

a(n) ~ c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(5-4*r)) * n^(3/4)), where r = 0.754877666246692760049508896358528691894606617772793143989... is the root of the equation r^5 + r = 1 and c = 5*log(r)^2/2 + polylog(2, 1-r) = 0.45973143655369174108251201834952526825516678... . - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 12 2004

A025160 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 25, 27, 30, 33, 36, 39, 43, 46, 50, 54, 59, 63, 69, 74, 81, 87, 95, 102, 112, 120, 131, 141, 154, 165, 180, 193, 210, 225, 244, 261, 283, 302, 326, 348, 375, 400, 430, 458, 492

Offset: 1

Clark Kimberling

nonn

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

Cf. A003114, A025157-A025162.

Column k=6 of A194543.

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

More terms from Vladeta Jovovic, Aug 12 2004

A025161 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 14, 15, 17, 19, 21, 23, 26, 28, 31, 34, 37, 40, 44, 47, 51, 55, 59, 63, 68, 73, 78, 84, 90, 97, 104, 112, 120, 130, 139, 150, 161, 174, 186, 201, 215, 232, 248, 267, 285, 307, 327, 351, 374, 401

Offset: 1

Clark Kimberling

nonn

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

Cf. A003114, A025157-A025162.

Column k=7 of A194543.

G.f.: Sum(x^(7/2*k^2-5/2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Naohiro Nomoto, Feb 27 2002

A325162 Squarefree numbers with no two prime indices differing by less than 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122, 123, 127, 129, 131

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

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 into distinct parts, no two differing by less than 3 (counted by A025157).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  34: {1,7}
  37: {12}
  38: {1,8}
  39: {2,6}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325160, A325161.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  if ormap(t -> t[2]>1, F) then return false fi;
  if nops(F) <= 1 then return true fi;
  F:= map(numtheory:-pi,sort(map(t -> t[1],F)));
  min(F[2..-1]-F[1..-2]) >= 3;
end proc:
select(filter, [$1..200]); # Robert Israel, Apr 08 2019

Select[Range[100],Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>2&]

A373574 Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 2024

The unsorted version is A373573.
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:
     4    8
     9   12   16   18   20   24
    28   32   36   40   44
    49
    64   68   72   75
    81   84   88   90   92   96   98
   148  150  152
   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 squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For squarefree antiruns: A373200, firsts of A373127, unsorted A373128.
For composite runs we have A373400, firsts of A176246, unsorted A073051.
For prime antiruns we have A373402, firsts of A027833, unsorted A373401.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
Sorted positions of first appearances in A373409.
The unsorted version is A373573.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A025157, A049094, A061399, A068781, A072284, A077643, A110969, A251092, A294242, A373410, A373412.

t=Length/@Split[Select[Range[100000],!SquareFreeQ[#]&],#1+1!=#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

cp's OEIS Frontend

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

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A237979 Number of strict partitions of n such that (least part) > number of parts.

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A237976 Number of strict partitions of n such that (least part) < number of parts.

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A237977 Number of strict partitions of n such that (least part) <= number of parts.

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Mathematica

PARI

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A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

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A025159 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 5.

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A025160 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 6.

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A025161 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

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