A325351 -id:A325351 - cp's OEIS frontend

A325355 One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.

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

Offset: 1

Gus Wiseman, Apr 23 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The fixed points of A325351 are the Heinz numbers of hooks A093641.

			Repeatedly applying A325351 starting with 78 gives 78 -> 66 -> 42 -> 30 -> 18 -> 12, and 12 is a fixed point, so a(78) = 6.

Positions of 2's are A325359.

Cf. A056239, A093641, A112798, A130091, A289509, A325351, A325352, A325366, A325389, A325394, A325395, A325396.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpi-pi))); if(f[i, 2]>1, f[i, 2]--, i--)); if(pi, listput(diffs,pi)); Vec(diffs); };
A325351(n) = factorback(apply(prime,augdiffs(n)));
A325355(n) = { my(u=A325351(n)); if(u==n,1,1+A325355(u)); }; \\ Antti Karttunen, Nov 16 2019

More terms from Antti Karttunen, Nov 16 2019

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

Original entry on oeis.org

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

A325352 Heinz number of the differences plus one of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 6, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 9, 1, 16, 7, 17, 3, 12, 1, 19, 11, 20, 1, 15, 1, 22, 6, 23, 1, 24, 2, 10, 13, 26, 1, 12, 5, 28, 17, 29, 1, 18, 1, 31, 10, 32, 7, 21, 1, 34, 19, 15, 1, 24, 1, 37, 6, 38

Offset: 1

Gus Wiseman, Apr 23 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The only fixed point is 1 because otherwise the sequence decreases omega (A001222) by one.

			The partition (3,2,2,1) with Heinz number 90 has differences plus one (2,1,2) with Heinz number 18, so a(90) = 18.

Positions of m's are A008578 (m = 1), A001248 (m = 2), A006094 (m = 3), A030078 (m = 4), A090076 (m = 5).

Cf. A007294, A049988, A056239, A093641, A112798, A240026, A320466, A325328, A325351, A325360, A325361, A325368, A325405.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
db[n_]:=Times@@Prime/@(1+Differences[primeMS[n]]);
Table[db[n],{n,100}]

A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99, 101, 102, 103

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325349.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   29: {10}
   31: {11}

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

Positions of squarefree numbers in A325351.

Cf. A056239, A093641, A112798, A130091, A325349, A325355, A325367, A325368, A325389, A325394, A325395, A325396, A325405.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A325349 Number of integer partitions of n whose augmented differences are distinct.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 5, 7, 7, 12, 10, 13, 15, 21, 21, 31, 34, 38, 45, 55, 60, 71, 80, 84, 103, 119, 134, 152, 186, 192, 228, 263, 292, 321, 377, 399, 454, 514, 565, 618, 709, 752, 840, 958, 1050, 1140, 1297, 1402, 1568, 1755, 1901, 2080, 2343, 2524, 2758, 3074

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325366.

			The a(1) = 1 through a(11) = 10 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (22)  (41)  (33)  (43)   (44)   (54)   (55)   (65)
                  (31)        (42)  (52)   (62)   (63)   (64)   (83)
                              (51)  (61)   (71)   (72)   (73)   (92)
                                    (421)  (422)  (81)   (82)   (A1)
                                           (431)  (522)  (91)   (443)
                                           (521)  (621)  (433)  (641)
                                                         (442)  (722)
                                                         (541)  (731)
                                                         (622)  (821)
                                                         (631)
                                                         (721)
For example, (4,4,3) has augmented differences (1,2,3), which are distinct, so (4,4,3) is counted under a(11).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..440

Cf. A000837, A049988, A098859, A325324, A325325, A325328, A325351, A325366, A325404.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[Append[#,1]]&]],{n,0,30}]

A383512 Heinz numbers of conjugate Wilf partitions.

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, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
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.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

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

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

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

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

A383513 Heinz numbers of non conjugate Wilf partitions.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A381433 in having 65.
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.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}

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

Partitions of this type are counted by A336866.
The conjugate version is A130092, complement A130091.
Including differences of 0 gives complement of A325367, counted by A325324.
The strict case is the complement of A325388, counted by A320348.
The complement is A383512, counted by A098859.
Also forbidding distinct multiplicities gives A383531, counted by A383530.
These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

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

A342192 Heinz numbers of partitions of crank 0.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 100, 106, 118, 122, 134, 140, 142, 146, 158, 166, 178, 194, 196, 202, 206, 214, 218, 220, 226, 254, 260, 262, 274, 278, 298, 300, 302, 308, 314, 326, 334, 340, 346, 358, 362, 364, 380, 382, 386, 394, 398

Offset: 1

Gus Wiseman, Apr 05 2021

nonn

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.

See A257989 or the program for a definition of crank of a partition.

			The sequence of terms together with their prime indices begins:
      6: {1,2}        106: {1,16}       218: {1,29}
     10: {1,3}        118: {1,17}       220: {1,1,3,5}
     14: {1,4}        122: {1,18}       226: {1,30}
     22: {1,5}        134: {1,19}       254: {1,31}
     26: {1,6}        140: {1,1,3,4}    260: {1,1,3,6}
     34: {1,7}        142: {1,20}       262: {1,32}
     38: {1,8}        146: {1,21}       274: {1,33}
     46: {1,9}        158: {1,22}       278: {1,34}
     58: {1,10}       166: {1,23}       298: {1,35}
     62: {1,11}       178: {1,24}       300: {1,1,2,3,3}
     74: {1,12}       194: {1,25}       302: {1,36}
     82: {1,13}       196: {1,1,4,4}    308: {1,1,4,5}
     86: {1,14}       202: {1,26}       314: {1,37}
     94: {1,15}       206: {1,27}       326: {1,38}
    100: {1,1,3,3}    214: {1,28}       334: {1,39}

Indices of zeros in A257989.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Cf. A000726, A008965, A056239, A112798, A124010, A130091, A325351, A325352.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[primeMS[#]]==0&]

A325394 Heinz numbers of integer partitions whose augmented differences are weakly increasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325356.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

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

Cf. A056239, A093641, A112798, A240026, A325351, A325356, A325360, A325362, A325366, A325389, A325395, A325396, A325400.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

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

cp's OEIS Frontend

A325355 One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.

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

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A325352 Heinz number of the differences plus one of the integer partition with Heinz number n.

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A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

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A325349 Number of integer partitions of n whose augmented differences are distinct.

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A383512 Heinz numbers of conjugate Wilf partitions.

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A383513 Heinz numbers of non conjugate Wilf partitions.

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A342192 Heinz numbers of partitions of crank 0.