A342050 -id:A342050 - cp's OEIS frontend

A257993 Least gap in the partition having Heinz number n; index of the least prime not dividing n.

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3

Offset: 1

Emeric Deutsch, May 18 2015

nonn

The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
Sum of least gaps of all partitions of m = A022567(m).
From Antti Karttunen, Aug 22 2016: (Start)
Index of the least prime not dividing n. (After a formula given by Heinz.)
Least k such that A002110(k) does not divide n.
One more than the number of trailing zeros in primorial base representation of n, A049345.
(End)
The least gap is also called the mex (minimal excludant) of the partition. - Gus Wiseman, Apr 20 2021

			a(18) = 3 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having least gap equal to 3.

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics).
Index entries for sequences related to primorial base.

Cf. A215366, A002110, A022567, A049345, A053669, A055396, A064648, A276086, A276094, A276152.
Positions of 1's are A005408.
Positions of 2's are A047235.
The number of gaps is A079067.
The version for crank is A257989.
The triangle counting partitions by this statistic is A264401.
One more than A276084.
The version for greatest difference is A286469 or A286470.
A maximal instead of minimal version is A339662.
Positions of even terms are A342050.
Positions of odd terms are A342051.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339737 counts partitions by sum and greatest gap.
Cf. A001223, A001511, A005117, A026794, A029707, A072233, A079068, A098743, A124010, A279945, A325351, A325352.

with(numtheory): a := proc (n) local B, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: for q while member(q, B(n)) = true do  end do: q end proc: seq(a(n), n = 1 .. 150);
# second Maple program:
a:= n-> `if`(n=1, 1, (s-> min({$1..(max(s)+1)} minus s))(
        {map(x-> numtheory[pi](x[1]), ifactors(n)[2])[]})):
seq(a(n), n=1..100);  # Alois P. Heinz, May 09 2016
# faster:
A257993 := proc(n) local p, c; c := 1; p := 2;
while n mod p = 0 do p := nextprime(p); c := c + 1 od: c end:
seq(A257993(n), n=1..100); # Peter Luschny, Jun 04 2017

A053669[n_] := For[p = 2, True, p = NextPrime[p], If[CoprimeQ[p, n], Return[p]]]; a[n_] := PrimePi[A053669[n]]; Array[a, 100] (* Jean-François Alcover, Nov 28 2016 *)
Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; k, {n, 100}] (* Michael De Vlieger, Jun 22 2017 *)

a(n) = forprime(p=2,, if (n % p, return(primepi(p)))); \\ Michel Marcus, Jun 22 2017

from sympy import nextprime, primepi
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def a(n): return primepi(a053669(n)) # Indranil Ghosh, May 12 2017

(define (A257993 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) i (loop (/ (- n d) p) (+ 1 i))))))
;; Antti Karttunen, Aug 22 2016

a(n) = A000720(A053669(n)). - Alois P. Heinz, May 18 2015
From Antti Karttunen, Aug 22-30 2016: (Start)
a(n) = 1 + A276084(n).
a(n) = A055396(A276086(n)).
A276152(n) = A002110(a(n)).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} 1/A002110(k) = 1.705230... (1 + A064648). - Amiram Eldar, Jul 23 2022
a(n) << log n/log log n. - Charles R Greathouse IV, Dec 03 2022

A simpler description added to the name by Antti Karttunen, Aug 22 2016

A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 2, 1, 4, 6, 4, 1, 7, 8, 5, 2, 8, 11, 8, 3, 12, 15, 10, 4, 1, 14, 20, 15, 6, 1, 21, 26, 19, 9, 2, 24, 35, 27, 12, 3, 34, 45, 34, 17, 5, 41, 58, 47, 23, 6, 1, 55, 75, 59, 31, 10, 1, 66, 96, 79, 41, 13, 2

Offset: 0

Emeric Deutsch, Nov 21 2015

The "least gap" or "mex" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
Sum of entries in row n is A000041(n).
T(n,1) = A002865(n).
Sum_{k>=1} k*T(n,k) = A022567(n).

			Row n=5 is 2,3,2; indeed, the least gaps of [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] are 1, 2, 1, 2, 3, 3, and 2, respectively (i.e., two 1s, three 2s, and two 3s).
Triangle begins:
   1
   0   1
   1   1
   1   1   1
   2   2   1
   2   3   2
   4   4   2   1
   4   6   4   1
   7   8   5   2
   8  11   8   3
  12  15  10   4   1
  14  20  15   6   1
  21  26  19   9   2

Alois P. Heinz, Rows n = 0..1000, flattened
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Row sums are A000041.
Row lengths are A002024.
Column k = 1 is A002865.
Column k = 2 is A027336.
The strict case is A343348.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A257993 gives the least gap of the partition with Heinz number n.
A339564 counts factorizations with a selected factor.
A342050 ranks partitions with even least gap.
A342051 ranks partitions with odd least gap.
Cf. A003242, A022567, A048004, A083710, A098743, A130689, A338470, A343341.

g := (sum(t^j*x^((1/2)*j*(j-1))*(1-x^j), j = 1 .. 80))/(product(1-x^i, i = 1 .. 80)): gser := simplify(series(g, x = 0, 23)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, [1, 0],
      [0, x]), `if`(i<1, 0, (p-> [0, p[2] +p[1]*x^i])(
      b(n, i-1)) +add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=1..degree(p)))(b(n, n+1)[2]):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 29 2015

Needs["Combinatorica`"]; {1, 0}~Join~Flatten[Table[Count[Map[If[# == {}, 0, First@ #] &@ Complement[Range@ n, #] &, Combinatorica`Partitions@ n], n_ /; n == k], {n, 17}, {k, n}] /. 0 -> Nothing] (* Michael De Vlieger, Nov 21 2015 *)
mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];Table[Length[Select[IntegerPartitions[n],mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}] (* Gus Wiseman, Apr 19 2021 *)
b[n_, i_] := b[n, i] = If[n == 0, If[i == 0, {1, 0}, {0, x}], If[i<1, {0, 0}, {0, #[[2]] + #[[1]]*x^i}&[b[n, i-1]] + Sum[b[n-i*j, i - 1], {j, 1, n/i}]]];
T[n_] := CoefficientList[b[n, n + 1], x][[2]] // Rest;
T /@ Range[0, 20] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

G.f.: G(t,x) = Sum_{j>=1} (t^j*x^{j(j-1)/2}*(1-x^j))/Product_{i>=1}(1-x^i).

A342193 Numbers with no prime index dividing all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299

Offset: 1

Gus Wiseman, Apr 11 2021

nonn

Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.
First differs from A339562 in having 45.
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.
Also 1 and Heinz numbers of integer partitions with smallest part not dividing all the others (counted by A338470). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}         105: {2,3,4}      201: {2,19}
     15: {2,3}      119: {4,7}        203: {4,10}
     33: {2,5}      123: {2,13}       205: {3,13}
     35: {3,4}      135: {2,2,2,3}    207: {2,2,9}
     45: {2,2,3}    141: {2,15}       209: {5,8}
     51: {2,7}      143: {5,6}        215: {3,14}
     55: {3,5}      145: {3,10}       217: {4,11}
     69: {2,9}      153: {2,2,7}      219: {2,21}
     75: {2,3,3}    155: {3,11}       221: {6,7}
     77: {4,5}      161: {4,9}        225: {2,2,3,3}
     85: {3,7}      165: {2,3,5}      231: {2,4,5}
     91: {4,6}      175: {3,3,4}      245: {3,4,4}
     93: {2,11}     177: {2,17}       247: {6,8}
     95: {3,8}      187: {5,7}        249: {2,23}
     99: {2,2,5}    195: {2,3,6}      253: {5,9}

The complement is counted by A083710 (strict: A097986).
The complement with no 1's is A083711 (strict: A098965).
These partitions are counted by A338470 (strict: A341450).
The squarefree case is A339562, with squarefree complement A339563.
The case with maximum prime index not divisible by all others is A343338.
The case with maximum prime index divisible by all others is A343339.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices, row sums of A112798.
A299702 lists Heinz numbers of knapsack partitions.
A339564 counts factorizations with a selected factor.
Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]

A342051 Numbers k which have an even number of trailing zeros in their primorial base representation A049345(k).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 102, 103

Offset: 1

Amiram Eldar, Feb 26 2021

Numbers k such that A276084(k) is even.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * (1 - Sum_{k=1..m}(-1)^k/A002110(k)) = 1, 4, 19, 134, 1473, 19150, 325549 ...
The asymptotic density of this sequence is Sum_{k>=0} (-1)^k/A002110(k) = 0.637693... = 1 - A132120.
Also Heinz numbers of partitions with odd least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021

			1 is a term since A049345(1) = 1 has 0 trailing zero.
6 is a term since A049345(6) = 100 has 2 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
     1: {}           25: {3,3}          51: {2,7}
     3: {2}          27: {2,2,2}        53: {16}
     5: {3}          29: {10}           54: {1,2,2,2}
     6: {1,2}        31: {11}           55: {3,5}
     7: {4}          33: {2,5}          57: {2,8}
     9: {2,2}        35: {3,4}          59: {17}
    11: {5}          36: {1,1,2,2}      61: {18}
    12: {1,1,2}      37: {12}           63: {2,2,4}
    13: {6}          39: {2,6}          65: {3,6}
    15: {2,3}        41: {13}           66: {1,2,5}
    17: {7}          42: {1,2,4}        67: {19}
    18: {1,2,2}      43: {14}           69: {2,9}
    19: {8}          45: {2,2,3}        71: {20}
    21: {2,4}        47: {15}           72: {1,1,1,2,2}
    23: {9}          48: {1,1,1,1,2}    73: {21}
    24: {1,1,1,2}    49: {4,4}          75: {2,3,3}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Cf. A002110, A049345, A132120, A276084.
Complement of A342050.
A099788 is subsequence.
Analogous sequences: A000201 (Zeckendorf representation), A003159 (binary), A007417 (ternary), A232744 (factorial base).
The version for reversed binary expansion is A121539.
Positions of odd terms in A257993.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A339662 gives greatest gap in prime indices.
Cf. A001223, A005408, A026794, A029707, A047235, A079068, A079523, A286469, A286470, A325351.

seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100],OddQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 0, 1, 2, 0, 5, 0, 0, 1, 1, 4, 0, 8, 0, 0, 0, 1, 2, 4, 0, 10, 0, 0, 0, 2, 1, 2, 7, 0, 16, 0, 0, 0, 0, 2, 1, 3, 8, 0, 20, 0, 0, 0, 0, 2, 2, 2, 4, 12, 0, 31, 0, 0, 0, 0, 0, 2, 2, 2, 5, 14, 0

Offset: 0

Gus Wiseman, Sep 30 2023

			The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  0  1  2  0
   4  0  0  1  2  0
   5  0  0  1  1  4  0
   8  0  0  0  1  2  4  0
  10  0  0  0  2  1  2  7  0
  16  0  0  0  0  2  1  3  8  0
  20  0  0  0  0  2  2  2  4 12  0
  31  0  0  0  0  0  2  2  2  5 14  0
  39  0  0  0  0  0  4  2  2  3  6 21  0
  55  0  0  0  0  0  0  4  2  4  3  9 24  0
  71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
Row n = 8 counts the following partitions:
  (4211)      .  .  .  (521)   (611)  (71)   (8)     .
  (41111)              (5111)         (431)  (62)
  (3311)                                     (53)
  (3221)                                     (44)
  (32111)                                    (422)
  (311111)                                   (332)
  (22211)                                    (2222)
  (221111)
  (2111111)
  (11111111)

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row sums are A000041.
Diagonal k = n-1 is A002865.
Column k = 1 is A126796 (complete partitions), ranks A325781.
Central diagonal n = 2k is A126796 also.
For parts instead of sums we have A339737, rank stat A339662, min A257993.
This is the triangle for the rank statistic A365920.
Latter row sums are A365924 (incomplete partitions), ranks A365830.
Column sums are A366127.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
A366128 gives the least non-subset-sum of prime indices.
Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Max@@Prepend[nmz[#],0]==k&]],{n,0,10},{k,0,n}]

A339662 Greatest gap in the partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
FindStat, Dyson's crank of a partition.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Positions of first appearances are A000040.
Positions of 0's are A055932.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The version for crank is A257989.
The minimal instead of maximal version is A257993.
The version for greatest difference is A286469 or A286470.
Positive integers by Heinz weight and image are counted by A339737.
Positions of 1's are A339886.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[maxgap[primeMS[n]],{n,100}]

a(n) = A000720(A079068(n)).

A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 30 2023

nonn

This is the greatest element of {0,...,A056239(n)} that is not equal to A056239(d) for any divisor d|n, d>1. This definition is analogous to the Frobenius number of a numerical semigroup (see link), but it looks only at submultisets of a finite multiset, not all multisets of elements of a set.

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 156 are {1,1,2,6}, with subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, so a(156) = 5.

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

For binary indices instead of sums we have A063250.
Positions of first appearances > 2 are A065091.
Zeros are A325781, nonzeros A325798.
For prime indices instead of sums we have A339662, minimum A257993.
For least instead of greatest non-subset-sum we have A366128.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A079068, A098743, A264401, A286469 or A286470, A339737, A339886.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Max@@Prepend[nmz[prix[n]],0],{n,100}]

A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

Original entry on oeis.org

5, 10, 15, 25, 30, 45, 50, 75, 90, 100, 125, 135, 150, 225, 250, 270, 300, 375, 405, 450, 500, 625, 675, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050, 4500, 5000, 5625, 6075, 6250

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is d-smooth iff its prime divisors are all <= d.

A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).

			The sequence of terms together with their prime indices begins:
       5: {3}           270: {1,2,2,2,3}
      10: {1,3}         300: {1,1,2,3,3}
      15: {2,3}         375: {2,3,3,3}
      25: {3,3}         405: {2,2,2,2,3}
      30: {1,2,3}       450: {1,2,2,3,3}
      45: {2,2,3}       500: {1,1,3,3,3}
      50: {1,3,3}       625: {3,3,3,3}
      75: {2,3,3}       675: {2,2,2,3,3}
      90: {1,2,2,3}     750: {1,2,3,3,3}
     100: {1,1,3,3}     810: {1,2,2,2,2,3}
     125: {3,3,3}       900: {1,1,2,2,3,3}
     135: {2,2,2,3}    1000: {1,1,1,3,3,3}
     150: {1,2,3,3}    1125: {2,2,3,3,3}
     225: {2,2,3,3}    1215: {2,2,2,2,2,3}
     250: {1,3,3,3}    1250: {1,3,3,3,3}

Allowing any number of parts and sum gives A080193, counted by A069905.
The partitions with these Heinz numbers are counted by A325691.
Relaxing the smoothness conditions gives A344291, counted by A110618.
Allowing 3-smoothness gives A344293, counted by A266755.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n whose length is n/2, ranked by A340387.
A051037 lists 5-smooth numbers (complement: A279622).
A056239 adds up prime indices, row sums of A112798.
A257993 gives the least gap of the partition with Heinz number n.
A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]==5&]

Intersection of A080193 and A344291.

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

A132120 Decimal expansion of the constant obtained through Pierce retro-expansion of the prime sequence.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 31 2007

The asymptotic density of numbers which have an odd number of the trailing zeros in their primorial base representation (A342050). - Amiram Eldar, Feb 28 2021

			0.3623062223664980487986263722240934618111798585344209997599510170278418863068...

Paolo P. Lava, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pierce Expansion.

Cf. A064648, A342050.

Digits := 80 : a := 0 : for i from 1 to 100 do a := a+(-1.0)^(i-1)/mul(ithprime(j),j=1..i) ; print(a) ; od:

P=1; -sumalt(n=1,(-1)^n/P*=prime(n)) \\ Charles R Greathouse IV, Oct 03 2016

Equals Sum_{i>=1} (-1)^(i+1)/A002110(i).

Corrected last entry by Paolo P. Lava, May 28 2013

cp's OEIS Frontend

A257993 Least gap in the partition having Heinz number n; index of the least prime not dividing n.

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A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.

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A342193 Numbers with no prime index dividing all the other prime indices.

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A342051 Numbers k which have an even number of trailing zeros in their primorial base representation A049345(k).

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A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

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A339662 Greatest gap in the partition with Heinz number n.

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A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

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A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

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