A286469 -id:A286469 - 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

A286470 a(n) = maximal gap between indices of successive primes in the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 1, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 2, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 1, 0, 10, 2, 0, 3, 3, 0, 6, 7, 2, 0, 1, 0, 11, 1, 7, 1, 4, 0, 2, 0, 12, 0, 2, 4, 13, 8, 4, 0, 1, 2, 8, 9, 14, 5, 1, 0, 3, 3, 2, 0, 5, 0, 5, 1, 15, 0, 1, 0, 2, 10, 3, 0, 6, 6, 9, 4, 16, 3, 1

Offset: 1

Antti Karttunen, May 13 2017

nonn

			For n = 70 = 2*5*7 = prime(1)*prime(3)*prime(4), the largest index difference occurs between prime(1) and prime(3), thus a(70) = 3-1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001221, A032742, A055396, A073490, A243055, A286455, A286471, A286472.
Cf. A286469 (version which considers the index of the smallest prime as the initial gap).
Cf. A000961 (positions of zeros).
Differs from A242411 for the first time at n=70, where a(70) = 2, while A242411(70) = 1.

Table[If[Or[n == 1, PrimeNu@ n == 1], 0, Max@ Differences@ PrimePi[FactorInteger[n][[All, 1]]]], {n, 120}] (* Michael De Vlieger, May 16 2017 *)

from sympy import primepi, isprime, primefactors, divisors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def x(n): return 1 if n==1 else divisors(n)[-2]
def a(n): return 0 if n==1 or len(primefactors(n))==1 else max(a055396(x(n)) - a055396(n), a(x(n))) # Indranil Ghosh, May 17 2017

(define (A286470 n) (cond ((or (= 1 n) (= 1 (A001221 n))) 0) (else (max (- (A055396 (A032742 n)) (A055396 n)) (A286470 (A032742 n))))))

a(1) = 0, for n > 1, if A001221(n) = 1 [when n is a prime power], a(n) = 0, otherwise a(n) = max((A055396(A032742(n))-A055396(n)), a(A032742(n))).

For all n >= 1, a(n) <= A243055(n).

Definition corrected by Zak Seidov, May 16 2017

A287352 Irregular triangle T(n,k) = A112798(n,1) followed by first differences of A112798(n).

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, May 23 2017

Irregular triangle T(n,k) = first differences of indices of prime divisors p of n.
Row lengths = (big) Omega(n) = A001222(n).
Row sums = A061395(n).
Row maxima = A286469(n).
We can concatenate the rows 1 <= n <= 28 as none of the values of k in this range exceed 9: {0, 1, 2, 10, 3, 11, 4, 100, 20, 12, 5, 101, 6, 13, 21, 1000, 7, 110, 8, 102, 22, 14, 9, 1001, 30, 15, 200, 103}; a(29) = {10}, which would require a digit greater than 9.
a(1) = 0 by convention.
a(0) is not defined (i.e., null set). a(n) is defined for positive nonzero n.
a(p) = A000720(p) for p prime.
a(p^e) = A000720(p) followed by (e - 1) zeros.
a(Product(p^e)) is the concatenation of the a(p^e) of the unitary prime power divisors p^e of n, sorted by the prime p (i.e. the function a(n) mapped across the terms of row n of A141809).
a(A002110(n)) = an array of n 1s.
T(n,k) could be used to furnish A054841(n). We read data in row n of T(n,k). If T(n,1) = 0, then write 0. If T(n,1) > 0, then increment the k-th place from the right. For k > 1, increment the k-th place to the right of the last-incremented place.
T(n,k) can be used to render n in decimal. If T(n,1) = 0, then write 1. If T(n,1) > 0, then multiply 1 by A000720(T(n,1)). For k > 1, multiply the previous product by pi(x) = A000720(x) of the running total of T(n,k) for each k.
Ignoring zeros in row n > 1 and decoding the remaining values of T(n,k) as immediately above yields the squarefree kernel of n = A007947(n).
Leading zeros of a(n) are trimmed, but as in decimal notation numbers that include leading zeros symbolize the same n as without them. Zeros that precede nonzero values merely multiply implicit 1 by itself until we encounter nonzero values. Thus, {0,0,2} = 1*1*pi(2) = 3, as {2} = pi(2) = 3. Because of this no row n > 1 has 0 for k = 1 of T(n,k).
Interpreting n written in binary as a row of a(n) yields A057335(n).

			a(1) = {0} by convention.
a(2) = {pi(2)} = {1}.
a(4) = {pi(2), pi(2) - pi(2)}, = {1, 0} since 4 = 2 * 2.
a(6) = {pi(2), pi(3) - pi(2)} = {1, 1} since 6 = 2 * 3.
a(12) = {pi(2), pi(2) - pi(2), pi(3) - pi(2) - pi(2)} = {1, 0, 1}, since 12 = 2 * 2 * 3.
The triangle starts:
   1:  0;
   2:  1;
   3:  2;
   4:  1, 0;
   5:  3;
   6:  1, 1;
   7:  4;
   8:  1, 0, 0;
   9:  2, 0;
  10:  1, 2;
  11:  5;
  12:  1, 0, 1;
  13:  6;
  14:  1, 3;
  15:  2, 1;
  16:  1, 0, 0, 0;
  17:  7;
  18:  1, 1, 0;
  19:  8;
  20:  1, 0, 2;
       ...

Michael De Vlieger, Table of n, a(n) for n = 1..15568 (rows 1 <= n <= 5000).

Cf. A000720, A001222, A007947, A054841, A057335, A061395, A112798, A141809, A286469.

Table[Prepend[Differences@ #, First@ #] & Flatten[FactorInteger[n] /. {p_, e_} /; p > 0 :> ConstantArray[PrimePi@ p, e]], {n, 41}] // Flatten (* Michael De Vlieger, May 23 2017 *)

T(n,1) = A117798(n,1); T(n,k) = A117798(n,k) - A117798(n, k - 1) for 2 <= k <= A001222(n).

A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 60, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 120, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 150, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 180, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212

Offset: 1

Amiram Eldar, Feb 26 2021

Numbers k such that A276084(k) is odd.
All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * Sum_{k=1..m}(-1)^k/A002110(k) = 1, 2, 11, 76, 837, 10880, 184961, ...
The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/A002110(k) = 0.362306... (A132120).
Also Heinz numbers of partitions with even 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
Numbers k such that A000720(A053669(k)) is even. Differences from the related A353531 seem to be terms that are multiples of 210, but not all of them, for example primorial 30030 (= 143*210) is in neither sequence. Consider also A038698. - Antti Karttunen, Apr 25 2022

			2 is a term since A049345(2) = 10 has 1 trailing zero.
4 is a term since A049345(2) = 20 has 1 trailing zero.
30 is a term since A049345(2) = 1000 has 3 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}             46: {1,9}             90: {1,2,2,3}
      4: {1,1}           50: {1,3,3}           92: {1,1,9}
      8: {1,1,1}         52: {1,1,6}           94: {1,15}
     10: {1,3}           56: {1,1,1,4}         98: {1,4,4}
     14: {1,4}           58: {1,10}           100: {1,1,3,3}
     16: {1,1,1,1}       60: {1,1,2,3}        104: {1,1,1,6}
     20: {1,1,3}         62: {1,11}           106: {1,16}
     22: {1,5}           64: {1,1,1,1,1,1}    110: {1,3,5}
     26: {1,6}           68: {1,1,7}          112: {1,1,1,1,4}
     28: {1,1,4}         70: {1,3,4}          116: {1,1,10}
     30: {1,2,3}         74: {1,12}           118: {1,17}
     32: {1,1,1,1,1}     76: {1,1,8}          120: {1,1,1,2,3}
     34: {1,7}           80: {1,1,1,1,3}      122: {1,18}
     38: {1,8}           82: {1,13}           124: {1,1,11}
     40: {1,1,1,3}       86: {1,14}           128: {1,1,1,1,1,1,1}
     44: {1,1,5}         88: {1,1,1,5}        130: {1,3,6}
(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)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A002110, A049345, A053669, A132120, A276084.
Complement of A342051.
A099800 is subsequence.
Analogous sequences: A001950 (Zeckendorf representation), A036554 (binary), A145204 (ternary), A217319 (base 4), A232745 (factorial base).
The version for reversed binary expansion is A079523.
Positions of even 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.
A333214 lists positions of adjacent unequal prime gaps.
A339662 gives greatest gap in prime indices.
Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function).
Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.

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

A353525(n) = { for(i=1,oo,if(n%prime(i),return((i+1)%2))); }
isA342050(n) = A353525(n);
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n,", "))); \\ Antti Karttunen, Apr 25 2022

More terms added (to differentiate from A353531) by Antti Karttunen, Apr 25 2022

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}]

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}]

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)).

A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

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

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) = 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.
Positions of first appearances are 4 followed by A000040.
Positions of 0's are A025475, minimal version A013929.
Positions of 1's are 2 followed by A066312, minimal version A355527.
Triangle A238710 counts m such that A056239(m) = n and a(m) = k.
Prepending 0 to the prime indices gives A286469, minimal version A355528.
See also A286470, minimal version A355524.
The minimal version is A355525, triangle A238709.
The augmented version is A355532.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A047966, A091602, A238353, A279945, A325160, A325161, A352822, A351294, A355530.

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

A355525 Minimal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

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

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 4 followed by A000040.
Positions of 0's are A013929, see also A130091.
Triangle A238709 counts m such that A056239(m) = n and a(m) = k.
For maximal instead of minimal difference we have A286470.
Positions of terms > 1 are A325160, also A325161.
See also A355524, A355528.
Positions of 1's are A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A238352 counts partitions by fixed points, rank statistic A352822.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A066312, A120944, A238353, A279945, A286469, A325351, A325352, A351294, A355526, A355530, A355531, A355532.

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

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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A286470 a(n) = maximal gap between indices of successive primes in the prime factorization of n.

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A287352 Irregular triangle T(n,k) = A112798(n,1) followed by first differences of A112798(n).

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A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

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

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