A243055 -id:A243055 - cp's OEIS frontend

A325200 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.

1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0, 0, 3, 2, 0, 2, 0, 1, 0, 6, 2, 0, 2, 0, 0, 4, 3, 4, 2, 0, 2, 0, 0, 6, 2, 6, 4, 2, 0, 2, 0, 0, 4, 9, 5, 4, 4, 2, 0, 2, 0, 1, 0, 15, 6, 8, 4, 4, 2, 0, 2, 0, 0, 5, 12, 12, 9, 6, 4, 4, 2, 0, 2, 0, 0, 10, 6, 21, 10, 12, 6, 4, 4, 2, 0, 2, 0

Offset: 0

Gus Wiseman, Apr 11 2019

			Triangle begins:
  1
  1  0
  0  2  0
  1  0  2  0
  0  3  0  2  0
  0  3  2  0  2  0
  1  0  6  2  0  2  0
  0  4  3  4  2  0  2  0
  0  6  2  6  4  2  0  2  0
  0  4  9  5  4  4  2  0  2  0
  1  0 15  6  8  4  4  2  0  2  0
  0  5 12 12  9  6  4  4  2  0  2  0
  0 10  6 21 10 12  6  4  4  2  0  2  0
  0 10 12 20 18 13 10  6  4  4  2  0  2  0
  0  5 27 20 23 16 16 10  6  4  4  2  0  2  0
  1  0 38 22 32 22 19 14 10  6  4  4  2  0  2  0
  0  6 34 38 34 35 20 22 14 10  6  4  4  2  0  2  0
  0 15 22 57 44 40 34 23 20 14 10  6  4  4  2  0  2  0
  0 20 20 71 55 54 45 32 26 20 14 10  6  4  4  2  0  2  0
  0 15 43 70 71 66 60 44 35 24 20 14 10  6  4  4  2  0  2  0
  0  6 74 64 99 83 70 65 42 38 24 20 14 10  6  4  4  2  0  2  0
Row n = 9 counts the following partitions (empty columns not shown):
  (432)   (333)    (54)      (63)      (72)       (81)        (9)
  (3321)  (441)    (621)     (6111)    (711)      (21111111)  (111111111)
  (4221)  (522)    (22221)   (222111)  (2211111)
  (4311)  (531)    (51111)   (411111)  (3111111)
          (3222)   (321111)
          (5211)
          (32211)
          (33111)
          (42111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Row sums are A000041. Column k = 1 is A325191. Column k = 2 is A325199.
T(n,k) = A325189(n,k) - A325188(n,k).
Cf. A046660, A065770, A071724, A243055, A325169, A325178, A325195, A325196, A325197, A366157, A368986.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]-otb[#]==k&]],{n,0,20},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(b=#p,c=0); for(i=1, #p, my(x=#p-i+p[i]); b=min(b,x); c=max(c,x)); r[c-b+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A366157(n) - A368986(n). - Andrew Howroyd, Jan 13 2024

A359682 Least positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 8, 10, 15, 12, 16, 18, 20, 26, 24, 28, 50, 36, 40, 46, 48, 52, 56, 62, 68, 74, 88, 76, 107, 86, 92, 94, 131, 106, 136, 118, 124, 122, 152, 134, 173, 142, 164, 146, 193, 158, 199, 166, 188, 178, 229, 194, 239, 202, 236, 206, 263, 214, 271, 218

Offset: 0

Gus Wiseman, Jan 15 2023

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 weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			The 5 numbers with weighted sum of prime indices 12, together with their prime indices:
  20: {1,1,3}
  27: {2,2,2}
  33: {2,5}
  37: {12}
  49: {4,4}
Hence a(12) = 20.

The version for standard compositions is A089633, zero-based A359756.
First position of n in A304818, reverse A318283.
The greatest instead of least is A359497, reverse A359683.
The sorted zero-based version is A359675, reverse A359680.
The zero-based version is A359676, reverse A359681.
The reverse version is A359679.
The sorted version is  A359755, reverse A359754.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359360.

nn=20;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[primeMS[n]],{n,1,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]

A372437 (Least binary index of n) minus (least prime index of n).

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, May 06 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
Is 0 the only integer not appearing in the data?

Positions of first appearances are A174090.
For sum instead of minimum we have A372428, zeros A372427.
For maximum instead of minimum we have A372442, zeros A372436.
For length instead of minimum we have A372441, zeros A071814.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A061712, A243055, A304818, A355536, A359495, A359402, A372429-A372432, A372588-A372591.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min[bix[n]]-Min[prix[n]],{n,2,100}]

a(2n) = A001511(n).
a(2n + 1) = -A038802(n).
a(n) = A001511(n) - A055396(n).

A243056 If n is the i-th prime, p_i = A000040(i), then a(n) = i, otherwise the difference between the indices of the smallest and the largest prime dividing n: for n = p_i * ... * p_k, where p_i <= ... <= p_k, a(n) = (k-i); a(1) = 0 by convention.

Original entry on oeis.org

0, 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, 2, 11, 0, 3, 6, 1, 1, 12, 7, 4, 2, 13, 3, 14, 4, 1, 8, 15, 1, 0, 2, 5, 5, 16, 1, 2, 3, 6, 9, 17, 2, 18, 10, 2, 0, 3, 4, 19, 6, 7, 3, 20, 1, 21, 11, 1, 7, 1, 5, 22, 2, 0, 12, 23, 3, 4, 13, 8, 4, 24, 2

Offset: 1

Antti Karttunen, May 31 2014

nonn

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

Useful when computing A243057 or A243059.
A025475 (prime powers that are not primes) gives the positions of zeros.
Differs from A241917 for the first time at n=18.
Cf. A243055, A241919, A242411, A049084, A027748, A055396, A061395.

(define (A243056 n) (if (prime? n) (A049084 n) (A243055 n)))

a(1) = 0, for n>1, if n = A000040(i), a(n) = i, otherwise a(n) = A061395(n) - A055396(n) = A243055(n).

A359497 Greatest positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 25, 29, 35, 49, 55, 77, 121, 91, 143, 169, 187, 221, 289, 247, 323, 361, 391, 437, 539, 605, 847, 1331, 715, 1001, 1573, 1183, 1859, 2197, 1547, 2431, 2873, 3179, 3757, 4913, 3553, 4199, 5491, 4693, 6137, 6859, 9317, 14641

Offset: 0

Gus Wiseman, Jan 15 2023

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 weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   25: {3,3}
   29: {10}
   35: {3,4}
   49: {4,4}
   55: {3,5}
   77: {4,5}
The 5 numbers with weighted sum of prime indices 12, together with their prime indices:
  20: {1,1,3}
  27: {2,2,2}
  33: {2,5}
  37: {12}
  49: {4,4}
Hence a(12) = 49.

Andrew Howroyd, Table of n, a(n) for n = 0..500

First position of n in A304818, reverse A318283.
The least instead of greatest is given by A359682, reverse A359679.
The reverse version is A359683.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359043, A359676, A359681, A359755.

nn=10;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[primeMS[n]],{n,1,2^nn}];
Table[Position[seq,k][[-1,1]],{k,0,nn}]

a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)),
  my(z=0); for(j=1, min(m, (r-k*(k-1)/2)\k), z=max(z, self()(r-k*j, k-1, j)*prime(j))); z));
  if(n==0, 1, vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n, k, n))));
} \\ Andrew Howroyd, Jan 21 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 21 2023

A359679 Least number with weighted sum of reversed (weakly decreasing) prime indices (A318283) equal to n.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 8, 12, 19, 18, 16, 24, 27, 36, 43, 32, 48, 59, 61, 67, 71, 64, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 0

Gus Wiseman, Jan 14 2023

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 weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			12 has reversed prime indices (2,1,1), with weighted sum 7, and no number < 12 has the same weighted sum of reversed prime indices, so a(7) = 12.

The version for standard compositions is A089633, zero-based A359756.
First position of n in A318283, unreversed A304818.
The unreversed zero-based version is A359676.
The sorted zero-based version is A359680, unreversed A359675.
The zero-based version is A359681.
The unreversed version is A359682.
The greatest instead of least is A359683, unreversed A359497.
The sorted version is A359754, unreversed A359755.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A053632, A055932, A231204, A243055, A359043, A358194, A359360, A359677.

nn=20;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[Reverse[primeMS[n]]],{n,1,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]

A324515 Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 40, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223

Offset: 1

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of the integer partitions enumerated by A324516. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  23: {9}
  29: {10}
  31: {11}
  37: {12}
  40: {1,1,1,3}
  41: {13}
  43: {14}
  45: {2,2,3}

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

Cf. A001221, A001222, A006141, A046660, A047993, A055396, A056239, A061395, A106529, A112798, A243055.

Cf. A324516, A324517, A324519, A324521, A324522, A324560, A324562.

filter:= proc(n) local F, Inds, t;
  if isprime(n) then return true fi;
  F:= ifactors(n)[2];
  Inds:= map(numtheory:-pi, F[..,1]);
  max(Inds) - min(Inds) = add(t[2],t=F) - nops(F)
end proc:
select(filter, [$2..300]); # Robert Israel, Nov 19 2023

Select[Range[2,100],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]-PrimePi[f[[1,1]]]==Total[Last/@f]-Length[f]]&]

A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

A324516 Number of integer partitions of n > 0 where the maximum part minus the minimum part equals the length minus the number of distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 5, 2, 8, 6, 6, 10, 14, 12, 20, 27, 23, 40, 40, 51, 62, 82, 88, 123, 135, 173, 197, 253, 285, 350, 419, 497, 594, 708, 855, 978, 1195, 1395, 1648, 1915, 2313, 2625, 3170, 3625, 4336, 4948, 5900, 6751, 7970, 9180, 10704, 12337, 14436, 16517

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324515.

			The a(8) = 5 through a(14) = 14 integer partitions:
  (8)      (9)      (A)       (B)       (C)        (D)        (E)
  (332)    (32211)  (433)     (443)     (4422)     (544)      (554)
  (3311)            (3331)    (33221)   (33321)    (43222)    (4442)
  (32111)           (4222)    (44111)   (422211)   (52222)    (5333)
  (41111)           (32221)   (422111)  (5211111)  (422221)   (43322)
                    (33211)   (431111)  (6111111)  (433111)   (44411)
                    (421111)                       (442111)   (442211)
                    (511111)                       (4321111)  (443111)
                                                   (5221111)  (551111)
                                                   (5311111)  (4322111)
                                                              (5222111)
                                                              (5411111)
                                                              (62111111)
                                                              (71111111)

Cf. A006141, A039900, A046660, A047993, A064174, A243055.

Cf. A324515, A324518, A324520.

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Length[#]-Length[Union[#]]&]],{n,30}]

A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

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

			The partition (3,3) has Heinz number 25 and diagram
  o o o
  o o o
containing maximal triangular partition
  o o
  o
and contained in minimal triangular partition
  o o o o
  o o o
  o o
  o
so a(25) = 4 - 2 = 2.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325183, A325188, A325189, A325191, A325196, A325197, A325199, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[otbmax[primeptn[n]]-otb[primeptn[n]],{n,100}]

A359675 Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 16, 20, 24, 30, 32, 36, 40, 48, 52, 56, 72, 80, 92, 96, 100, 104, 112, 124, 136, 148, 152, 172, 176, 184, 188, 212, 214, 236, 244, 248, 262, 268, 272, 284, 292, 304, 316, 328, 332, 346, 356, 376, 386, 388, 398, 404, 412, 428, 436, 452, 458

Offset: 1

Gus Wiseman, Jan 13 2023

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 zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The terms together with their prime indices begin:
   1: {}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}

Positions of first appearances in A359674.
The unsorted version A359676.
The reverse version is A359680, unsorted A359681.
The reverse one-based version is A359754, unsorted A359679.
The one-based version is A359755, unsorted A359682.
The version for standard compositions is A359756, one-based A089633.
A053632 counts compositions by zero-based weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A124757 gives zero-based weighted sum of standard compositions, rev A231204.
A304818 gives weighted sum of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359360, A359497, A359677, A359683.

nn=100;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[primeMS[n]],{n,1,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

cp's OEIS Frontend

A325200 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.

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A359682 Least positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

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A372437 (Least binary index of n) minus (least prime index of n).

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A243056 If n is the i-th prime, p_i = A000040(i), then a(n) = i, otherwise the difference between the indices of the smallest and the largest prime dividing n: for n = p_i * ... * p_k, where p_i <= ... <= p_k, a(n) = (k-i); a(1) = 0 by convention.

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A359497 Greatest positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

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A359679 Least number with weighted sum of reversed (weakly decreasing) prime indices (A318283) equal to n.

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A324515 Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

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A324516 Number of integer partitions of n > 0 where the maximum part minus the minimum part equals the length minus the number of distinct parts.

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