A299701 -id:A299701 - cp's OEIS frontend

A301830 Number of factorizations of n into factors (greater than 1) of two kinds.

1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 20, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 36, 6, 6, 6, 46, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 76, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 65, 6, 22, 2, 16, 6, 22, 2, 108, 2, 6, 16, 16, 6

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6).
The a(12) = 16 factorizations:
  ()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),
  ()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),
  ()*(12), (12)*().

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Index to sequences related to prime signature

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301829.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[facs[d]]*Length[facs[n/d]],{d,Divisors[n]}],{n,100}]

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={MultEulerT(vector(n, i, 2))} \\ Andrew Howroyd, Nov 18 2018

Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by Ilya Gutkovskiy, Dec 14 2020]

a(p^n) = A000712(n) for prime p. - Andrew Howroyd, Nov 18 2018

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 02 2020

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 5 4 5 5 5
  2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
  2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7

Row lengths are A011782.
Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Golomb rulers are counted by A169942 and ranked by A333222.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702
Knapsack compositions are counted by A325676 and ranked by A333223.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Knapsack compositions are counted by A334268 and ranked by A334967.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Total/@Subsets[stc[n]]]],{n,0,100}]

a(n) = A299701(A333219(n)).

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 29 2018

nonn

A rational number q is a nonempty-subset-average of an integer partition y if there exists a nonempty submultiset of y with average q.

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

			The a(42) = 7 subset-averages of (4,2,1) are 1, 3/2, 2, 7/3, 5/2, 3, 4.
The a(72) = 7 subset-averages of (2,2,1,1,1) are 1, 5/4, 4/3, 7/5, 3/2, 5/3, 2.

Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299701, A299702, A301899, A301957, A304793, A316313.

One less than A316398.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Mean/@Rest[Subsets[primeMS[n]]]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A316314(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = v056239[d]/bigomega(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316398(n) - 1.

More terms from Antti Karttunen, Sep 23 2018

A366741 Number of semi-sums of strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 6, 9, 13, 21, 26, 37, 48, 63, 86, 108, 139, 175, 223, 274, 350, 422, 527, 638, 783, 939, 1146, 1371, 1648, 1957, 2341, 2770, 3285, 3867, 4552, 5353, 6262, 7314, 8529, 9924, 11511, 13354, 15423, 17825, 20529, 23628, 27116, 31139, 35615

Offset: 0

Gus Wiseman, Nov 05 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The strict partitions of 9 and their a(9) = 13 semi-sums:
    (9) ->
   (81) -> 9
   (72) -> 9
   (63) -> 9
  (621) -> 3,7,8
   (54) -> 9
  (531) -> 4,6,8
  (432) -> 5,6,7

The non-strict non-binary version is A304792.
The non-binary version is A365925.
The non-strict version is A366738.
A000041 counts integer partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365543 counts partitions with a subset summing to k, complement A046663.
A365661 counts strict partitions w/ subset summing to k, complement A365663.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366739 counts semi-sums of prime indices, firsts A367097.
Cf. A008967, A122768, A299701, A364272, A364350, A365541, A365832, A366753.

Table[Total[Length[Union[Total/@Subsets[#, {2}]]]&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n,0,30}]

A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299702 (knapsack) in having 525: {2,3,3,4}.
First differs from A325778 in lacking 462: {1,2,4,5}.
These are the Heinz numbers of partitions whose parts are disjoint from their own non-singleton subset-sums.

Partitions of this type are counted by A237667, strict A364349.
The binary version is A364462, complement A364461.
The complement is A364532, counted by A237668.
A000005 counts divisors, nonprime A033273, composite A055212.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, counted by A108917, complement A299729.
A363260 counts partitions disjoint from differences, complement A364467.
Cf. A002865, A236912, A237113, A320340, A326083, A363225, A364347.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
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.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 5, 3, 3, 8, 4, 9, 1, 17, 6, 16, 1, 2, 24, 7, 33, 4, 9, 46, 11, 52, 3, 18, 1, 4, 64, 12, 91, 6, 38, 3, 15, 1, 1, 107, 17, 138, 9, 68, 2, 28, 2, 12, 0, 2, 147, 19, 219, 12, 117, 6, 56, 3, 34, 2, 9, 0, 3

Offset: 0

Gus Wiseman, Nov 07 2023

			The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
    0   1
    1   1
    2   2   1
    5   3   3
    8   4   9   1
   17   6  16   1   2
   24   7  33   4   9
   46  11  52   3  18   1   4
   64  12  91   6  38   3  15   1   1
  107  17 138   9  68   2  28   2  12   0   2
  147  19 219  12 117   6  56   3  34   2   9   0   3
Row n = 4 counts the following partitions:
  (8)     (44)        (431)      (221111)
  (71)    (3311)      (422)
  (62)    (2222)      (4211)
  (611)   (11111111)  (41111)
  (53)                (3221)
  (521)               (32111)
  (5111)              (311111)
  (332)               (22211)
                      (2111111)

Row sums w/o the first column are A002219, ranks A357976, strict A237258.
Column k = 0 is A006827.
Row sums are A058696.
Column k = 1 is A108917.
The corresponding rank statistic is A357879 (without empty rows).
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, ranks A366321.
A182616 counts partitions of 2n with at least one odd part, ranks A366530.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sums of partitions, rank statistic A299701.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.

t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]

T(n,1) = A108917(n).

A301854 Number of positive special sums of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 13, 25, 40, 67, 100, 158, 220, 336, 452, 649, 862, 1228, 1553, 2155, 2738, 3674, 4612, 6124, 7497, 9857, 12118, 15524, 18821, 24152, 28863, 36549, 44002, 54576, 65125, 80943, 95470, 117991, 139382, 169389, 199144, 242925, 283353, 342139, 400701, 479001

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

A positive special sum of an integer partition y is a number n > 0 such that exactly one submultiset of y sums to n.

			The a(4) = 13 special positive subset-sums:
1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111),
1<=(211),  3<=(211),  4<=(211),
1<=(31),   3<=(31),   4<=(31),
2<=(22),   4<=(22),
4<=(4).

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854, A301855, A301856.

uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A301854(n): return sum(sum(1 for r in Counter(sum(q) for l in range(1,len(p)+1) for q in multiset_combinations(p,l)).values() if r==1) for p in (tuple(Counter(x).elements()) for x in partitions(n))) # Chai Wah Wu, Sep 26 2023

a(21)-a(35) from Alois P. Heinz, Apr 08 2018

a(36)-a(43) from Chai Wah Wu, Sep 26 2023

A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).

Original entry on oeis.org

90, 180, 210, 270, 360, 420, 450, 462, 525, 540, 550, 630, 720, 810, 840, 858, 900, 910, 924, 990, 1050, 1080, 1100, 1155, 1170, 1260, 1326, 1350, 1386, 1440, 1470, 1530, 1575, 1620, 1650, 1666, 1680, 1710, 1716, 1800, 1820, 1848, 1870, 1890, 1911, 1938, 1980

Offset: 1

Gus Wiseman, Nov 05 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.
From Robert Israel, Nov 06 2023: (Start)
Positive integers divisible by the product of four primes, prime(i)*prime(j)*prime(k)*prime(l), i < j <= k < l, with i + l = j + k.
All positive multiples of terms are terms. (End)

			The semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5), which are not all different, so 90 is in the sequence.
The terms together with their prime indices begin:
    90: {1,2,2,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   270: {1,2,2,2,3}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   450: {1,2,2,3,3}
   462: {1,2,4,5}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
   550: {1,3,3,5}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}

The complement is too dense.
For all divisors instead of just semiprimes we have A299729, strict A316402.
Distinct semi-sums of prime indices are counted by A366739.
Partitions of this type are counted by A366753, non-binary A366754.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts distinct subset-sums of prime indices, positive A304793.
A299702 ranks knapsack partitions, counted by A108917, strict A275972.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A365541, A365920, A366737, A366738, A366741, A367093, A367095, A367097.

N:= 10^4: # for terms <= N
P:= select(isprime, [$1..N]): nP:= nops(P):
R:= {}:
for i from 1 while P[i]*P[i+1]^2*P[i+2] < N do
  for j from i+1 while P[i]*P[j]^2 * P[j+1] < N do
    for k from j do
      l:= j+k-i;
      if l <= k or l > nP then break fi;
      v:= P[i]*P[j]*P[k]*P[l];
      if v <= N then
        R:= R union {seq(t,t=v..N,v)};
      fi
od od od:
sort(convert(R,list)); # Robert Israel, Nov 06 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!UnsameQ@@Total/@Union[Subsets[prix[#],{2}]]&]

These are numbers k such that A086971(k) > A366739(k).

cp's OEIS Frontend

A301830 Number of factorizations of n into factors (greater than 1) of two kinds.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A366741 Number of semi-sums of strict integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

Views

Author

Keywords

Examples

Crossrefs

Programs