A257993 -id:A257993 - cp's OEIS frontend

A339562 Squarefree numbers with no prime index dividing all the other prime indices.

1, 15, 33, 35, 51, 55, 69, 77, 85, 91, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 203, 205, 209, 215, 217, 219, 221, 231, 247, 249, 253, 255, 265, 285, 287, 291, 295, 299, 301, 309, 323, 327, 329, 335, 341, 345, 355, 357, 377, 381

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

First differs from A342193 in lacking 45.
Alternative name: 1 and squarefree numbers with smallest prime index not dividing all the other prime indices.
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 strict integer partitions with smallest part not dividing all the others (counted by A341450). 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: {}         141: {2,15}     219: {2,21}
     15: {2,3}      143: {5,6}      221: {6,7}
     33: {2,5}      145: {3,10}     231: {2,4,5}
     35: {3,4}      155: {3,11}     247: {6,8}
     51: {2,7}      161: {4,9}      249: {2,23}
     55: {3,5}      165: {2,3,5}    253: {5,9}
     69: {2,9}      177: {2,17}     255: {2,3,7}
     77: {4,5}      187: {5,7}      265: {3,16}
     85: {3,7}      195: {2,3,6}    285: {2,3,8}
     91: {4,6}      201: {2,19}     287: {4,13}
     93: {2,11}     203: {4,10}     291: {2,25}
     95: {3,8}      205: {3,13}     295: {3,17}
    105: {2,3,4}    209: {5,8}      299: {6,9}
    119: {4,7}      215: {3,14}     301: {4,14}
    123: {2,13}     217: {4,11}     309: {2,27}

The squarefree complement is A339563.
These partitions are counted by A341450.
The not necessarily squarefree version is A342193.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A005117 lists squarefree numbers.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices (row sums of A112798).
A083710 counts partitions with a dividing part (strict: A097986).
Cf. A253249, A264401, A257993, A338470, A343337, A343338, A343339, A343379, A343380, A343382.

Select[Range[100],#==1||SquareFreeQ[#]&&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 *)

A304531 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least unitary divisor of a(n-1) not already present, or (if all unitary divisors already used), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 36, 9, 18, 90, 5, 10, 30, 15, 60, 20, 180, 45, 360, 8, 24, 120, 40, 1080, 27, 54, 270, 135, 540, 108, 2700, 25, 50, 150, 75, 300, 100, 900, 225, 450, 3150, 7, 14, 42, 21, 84, 28, 252, 63, 126, 630, 35, 70, 210, 105, 420, 140, 1260, 315, 2520, 56, 168, 840, 280, 7560, 189, 378, 1890, 945, 3780, 756, 18900, 175, 350, 1050, 525, 2100, 700

Offset: 1

Antti Karttunen, May 14 2018

nonn

The greedy algorithm which constructs the sequence is easiest to grasp in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition. The choices for constructing the next partition are: either remove some parts from the partition, but with the condition that if any summand k is removed, then all copies of k present in partition must be removed in toto. One may remove all copies of several distinct summands as well. If by such a removal of parts we can find any smaller partitions that have not yet occurred in the sequence, then we choose the one which has the smallest Heinz encoding value. On the other hand, if all partitions obtained by such removals have already occurred in the sequence, then one adds to the current partition the least number of copies of the least positive integer that is not yet a part of the partition (see A257993), until a partition is found which is not yet in the sequence. This process also implies that one never removes the summand(s) that was/were just added in the previous step.
It has not yet been rigorously proved that all partitions can be reached this way, i.e., that this sequence is a permutation of natural numbers.
Each a(n+1) is always either a divisor or a multiple of a(n).
No two successive descending terms, that is, a(n) > a(n+1) > a(n+2) never occurs.
For n > 1, if a(n) is odd then a(n-1) = 2^h * k * a(n) and a(n+1) = 2^j * a(n) for some h, k and j, that is, odd terms occur between two larger even numbers.
If a(n) < a(n+1) then (a(n+1) / a(n)) is a divisor of a(n+2). This follows because clearly (in case A) when a(n) < a(n+1) < a(n+2) then (a(n+1) / a(n)) is a divisor of a(n+2) because on ascending subsections each successive term is obtained by multiplying by some prime (or its power) not already present. But it is also true (in case B) when a(n) < a(n+1) > a(n+2), as:
In contrast to A303751, this permutation is specified with an additional constraint that gcd(a(n+1), a(n)/a(n+1)) = 1, whenever a(n) > a(n+1). From this then follows that also when a(n) < a(n+1) > a(n+2) then (a(n+1) / a(n)) is guaranteed to be a divisor of a(n+2). It also follows from this that also the squarefree version A304537(n) = A019565(A052331(a(1+n))) satisfies the divisor-or-multiple property.
Odd numbers occur at A304530.
Primes occur at : 2, 4, 11, 42, 237, 1798, 7192, 69611, 431203, 2401568, ...
Primorials (A002110) occur at: 1, 2, 3, 13, 54, 290, 2087, 11333, 118777, 934737, ...

			a(64) = 280 = 2^3 * 5 * 7 = prime(1)^3 * prime(3) * prime(4), which by Heinz-encoding corresponds to integer partition {1+1+1+3+4}. We try to remove all 1's (to get {3+4}, i.e. prime(3)*prime(4) = 35, but that has already been used as a(52)), or to remove either 3 or 4 or both, but also 8, 40 and 56 have already been used, and if we remove all 1's and either 3 or 4, then also prime(3) and prime(4), 5 and 7 have already been used. So we must add one or more copies of 2 (the least missing part) to find a partition that has not already been used. And it turns out we need to add three copies, to get {1+1+1+2+2+2+3+4} to obtain value prime(1)^3 * prime(2)^3 * prime(3) * prime(4) = 7560 not used before, so a(65) = 7560.
For the next partition, we remove all 1's and the sole 3, to get {2+2+2+4}, with Heinz-encoding prime(2)^3 * prime(4) = 27 * 7 = 189 to obtain the smallest not yet present in sequence, thus a(66) = 189. Note that the partition {1+1+1+2+2} would give even a smaller Heinz-code 2^3 * 3^2 = 72, which also has not been used before, but 72 is not a unitary divisor of 7560, which can be seen from the fact that just one 2 (but not all 2's) was removed from the partition {1+1+1+2+2+2+3+4} that corresponds to 7560. At this point A303751 selects 72 as it has no unitary divisor constraint.

Antti Karttunen, Table of n, a(n) for n = 1..8447
Index entries for sequences that are permutations of the natural numbers

Cf. A304532 (inverse).
Cf. A304530 (positions of odd terms).
Cf. A053669, A215366, A257993, A304533, A304535, A304536, A304537.
Cf. A113552, A282291, A303751 for other variants.
Differs from A303751 for the first time at n=66, where a(66) = 189, while A303751(66) = 72.

up_to = 2^12;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v304531 = vector(up_to);
m304532 = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m304532,d) && (1==gcd(d, prev/d)),v304531[n] = d;mapput(m304532,d,n);break)); if(!v304531[n], p = A053669(prev); while(mapisdefined(m304532,prev), prev *= p); v304531[n] = prev; mapput(m304532,prev,n)); prev = v304531[n]);
A304531(n) = v304531[n];
A304532(n) = mapget(m304532,n);

Observed patterns:
For n = 2 .. 2+0, a(n) = 2*a(n-1).
For n = 4 .. 4+0, a(n) = 3*a(n-3).
For n = 11 .. 11+7, a(n) = 5*a(n-10).
For n = 42 .. 42+38, a(n) = 7*a(n-41).
For n = 237 .. 237+64, a(n) = 11*a(n-236).
For n = 1798 .. 1798+336, a(n) = 13*a(n-1797).
For n = 7192 .. 7192+1255, a(n) = 17*a(n-7191).
For n = 69611 .. 69611+4820, a(n) = 19*a(n-69610).
For n = 431203 .. 431203+41802, a(n) = 23*a(n-431202).
For n = 2401568 .. 2401568+131366, a(n) = 29*a(n-2401567).
Derived sequences. For all n >= 1:
A052331(a(n)) = A304533(n-1).
A064547(a(n)) = A304536(n-1).

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}]

A303751 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least divisor of a(n-1) not already present, or (if all divisors already used), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 36, 9, 18, 90, 5, 10, 30, 15, 60, 20, 180, 45, 360, 8, 24, 120, 40, 1080, 27, 54, 270, 135, 540, 108, 2700, 25, 50, 150, 75, 300, 100, 900, 225, 450, 3150, 7, 14, 42, 21, 84, 28, 252, 63, 126, 630, 35, 70, 210, 105, 420, 140, 1260, 315, 2520, 56, 168, 840, 280, 7560, 72, 1800, 200, 600, 4200

Offset: 1

Antti Karttunen, May 01 2018

nonn

The greedy algorithm which constructs this sequence can be understood also in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition {s1+...+sk} via mapping a(n) = prime(s1) * ... * prime(sk), where s1 .. sk are the summands of an integer partition. The choices for constructing the next partition are: If by removing any parts from the partition we can find any smaller partitions that have not already occurred in the sequence, then we choose the one which has the smallest Heinz encoding value. On the other hand, if all partitions obtained by such removals have already occurred in the sequence, then we add to the current partition the least number of copies of the least positive integer that is not yet a part of the partition (A257993), until a partition is found which is not yet in the sequence.
From Antti Karttunen & David A. Corneth, May 01 - 04 2018: (Start)
No two successive descending terms, that is, a(n) > a(n+1) > a(n+2) never occurs.
For n > 1, if a(n) is odd then a(n-1) = 2^h * k * a(n) and a(n+1) = 2^j * a(n) for some h, k and j, that is, odd terms occur between two larger even numbers.
If a(n) < a(n+1) < a(n+2) then (a(n+1) / a(n)) is a divisor of a(n+2).
However, when a(n) < a(n+1) > a(n+2) then (a(n+1) / a(n)) might not be a divisor of a(n+2). The first such case occurs at n=64..66, as a(64) = 280 = 2^3 * 5 * 7, a(65) = 7560 = 2^3 * 3^3 * 5 * 7, and a(66) = 72 = 2^3 * 3^2. We have 7560/280 = 27, which is not a divisor of 72 (72/27 = 8/3).
In most cases, when a(n+1) < a(n) then gcd(a(n+1), a(n)/a(n+1)) = 1 (about 87% for the first 100000 descents). However, there are many exceptions to this, the first case occurring at a(65) = 7560 = 2^3 * 3^3 * 5 * 7 and a(66) = 72 =  2^3 * 3^2, with gcd(72,7560/72) = 3.
(End)
From David A. Corneth, May 04 2018: (Start)
The sequence can be partitioned into a tabf sequence with rows having the first element odd and the others even. It would give (1, 2, 6), (3, 12, 4, 36), (9, 18, 90), (5, 10, 30), (15, 60, 20, 180), (45, 360, 8, 24, 120, 40, 1080), (27, 54, 270), ...
It turns out that some rows are multiples of others; for example, the row (5, 10, 30) is five times the row (1, 2, 6). (End)
See also "observed scaling patterns" in the Formula section.
A303750 gives the positions of odd terms.
A282291 and A304531 are unitary divisor variants that satisfy the condition gcd(a(n+1), a(n)/a(n+1)) = 1, whenever a(n) > a(n+1).
The primes 2, 3, 5, 7, 11, 13, 19, 23 and 29 occur at positions 2, 4, 11, 42, 176, 1343, 8470, 57949, 302739, 1632898, thus after 7 and except for 13, a little earlier than they occur in variant A304531.

			a(64) = 280 = 2^3 * 5 * 7 = prime(1)^3 * prime(3) * prime(4), which by Heinz-encoding corresponds to integer partition {1+1+1+3+4}. We try to remove all 1's (to get {3+4}, i.e., prime(3)*prime(4) = 35, but that has already been used as a(52)), or to remove either 3 or 4 or both, but also 8, 40 and 56 have already been used, and if we remove all 1's and either 3 or 4, then also prime(3) and prime(4), 5 and 7 have already been used. So we must add one or more copies of 2 (the least missing part) to find a partition that has not already been used. And it turns out we need to add three copies, to get {1+1+1+2+2+2+3+4} to obtain value prime(1)^3 * prime(2)^3 * prime(3) * prime(4) = 7560 not used before, so a(65) = 7560.
For the next partition, we remove two 2's and both 3 and 4, to get {1+1+1+2+2} which gives Heinz-code 2^3 * 3^2 = 72, which is the smallest divisor of 7560 that has not been used before in the sequence, thus a(66) = 72.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A303752 (inverse).
Cf. A053669, A303750.
Cf. A113552, A282291, A304531, A304755 for similarly defined sequences, and also A064736, A207901, A281978, A302350, A302781, A302783, A303771 for other permutations satisfying the divisor-or-multiple property.
Cf. also A303761.
Cf. A304728, A304729 (see their scatter plots for alternative views to this process).
Differs from a variant A304531 for the first time at n = 66, where a(66) = 72, while A304531(66) = 189.

up_to = 2^14;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303751 = vector(up_to);
m303752 = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m303752,d),v303751[n] = d;mapput(m303752,d,n);break)); if(!v303751[n], p = A053669(prev); while(mapisdefined(m303752,prev), prev *= p); v303751[n] = prev; mapput(m303752,prev,n)); prev = v303751[n]);
A303751(n) = v303751[n];
A303752(n) = mapget(m303752,n);

Observed scaling patterns:
For n =     2 ..     2 +     0, a(n) =  2*a(n-1).
For n =     4 ..     4 +     0, a(n) =  3*a(n-3).
For n =    11 ..    11 +     7, a(n) =  5*a(n-10).
For n =    42 ..    42 +    23, a(n) =  7*a(n-41).
For n =   176 ..   176 +    80, a(n) = 11*a(n-175).
For n =  1343 ..  1343 +   683, a(n) = 13*a(n-1342).
For n =  8470 ..  8470 +  3610, a(n) = 17*a(n-8469).
For n = 57949 .. 57949 + 18554, a(n) = 19*a(n-57948).

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}]

A328570 Index of the least significant zero digit in the primorial base expansion of n, when the rightmost digit is in the position 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5

Offset: 0

Antti Karttunen, Oct 20 2019

Index of the least prime not dividing A276086(n), where A276086 converts the primorial base expansion of n into its prime product form.

Starting from x = n, repeatedly divide x by prime(1) (discarding the remainder), and set x to the integer quotient floor(x/prime(1)), then divide x with prime(2) (again discarding the remainder, and setting x to the integer quotient), etc., stopping as soon one of the primes is a divisor of the previous integer quotient (leaving zero remainder). a(n) is then the index of that prime, equal to 1 + the number of iterations done.

			For n = 2, we divide it with A000040(1) = 2, and it leaves zero remainder, so we have finished on the first round (needing no actual iterations), and thus a(2) = 1. Note that 2 in primorial base (A049345) is written as "10", and indeed the first zero from the right occurs at the position 1.
For n = 5, we first divide 5 with prime(1) = 2, and discarding the remainder, we are left with floor(5/2) = 2. Then we divide that 2 with prime(2) = 3, leaving floor(2/3) = 0 and remainder 2. And finally we divide 0 with prime(3) = 5, and that doesn't leave any remainder, thus we are finished on the third round, and a(5) = 3. Note that 5 in primorial base is written as "21", and allowing here a leading zero, written as "021", we see that it is in this case the least significant zero occurring at position 3 from the right.
For n = 43, we first divide it with prime(1) = 2, leaving a remainder 1 and integer quotient 21. Then we divide 21 with prime(2) = 3, which doesn't leave any remainder, thus we are finished on the second round, and a(43) = 2. Note that 43 is written as "1201" in primorial base, with the least significant zero occurring in the position 2.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A000720, A002110, A049345, A053669, A055396, A143293, A257993, A276086, A276087, A276088, A326810, A328475, A328476, A328569, A328578, A328580.

A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };

a(n) = A000720(A326810(n)) = A257993(A276086(n)) = A055396(A276087(n)).
For all n >= 0, A002110(a(n)) = A328580(n), a(A276086(n)) = A328578(n).
For all odd n, A000040(a(n)) = A326810(n) > A276088(n).
For all n >= 0, A276086(n)/A002110(a(n)-1) = A328475(n) and A276086(n)-A002110(a(n)-1) = A328476(n).

A328841 Substitute ones for all nonzero digits in primorial base expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 30, 31, 32, 33, 32, 33, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 30, 31, 32, 33, 32, 33, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 30

Offset: 0

Antti Karttunen, Oct 30 2019

Antti Karttunen, Table of n, a(n) for n = 0..30029
Index entries for sequences related to primorial base

Cf. A002110, A007947, A257993, A267263, A276085, A276086, A328570, A328571, A328620, A328842, A328843.
Cf. A276156 (fixed points).
Cf. A276008 for analogous sequence.

A328841(n) = { my(p=2, r=1, s=0); while(n, s += ((!!(n%p))*r); r *= p; n = n\p; p = nextprime(1+p)); (s); };

a(n) = n - A328842(n).
a(n) = A276085(A328571(n)) = A276085(A007947(A276086(n))).
For all n>= 0, a(A276086(n)) = A328843(n).
For all n >= 1, A257993(a(n)) = A257993(n).
For all n >= 0, A328570(a(n)) = A328570(n), A328620(a(n)) = A328620(n), and A267263(a(n)) = A267263(n).

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.

cp's OEIS Frontend

A339562 Squarefree 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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A303751 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least divisor of a(n-1) not already present, or (if all divisors already used), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

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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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A328570 Index of the least significant zero digit in the primorial base expansion of n, when the rightmost digit is in the position 1.

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