A056239 -id:A056239 - cp's OEIS frontend

A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

1, 1, -1, 1, 2, -3, -6, 1, 3, 8, 24, -6, -120, -30, -20, 1, 720, 15, -5040, 20, 90, 144, 40320, -10, 40, -840, -15, -90, -362880, -120, 3628800, 1, -504, 5760, -420, 45, -39916800, -45360, 3360, 40, 479001600, 630, -6227020800, 504, 210, 403200, 87178291200

Offset: 1

Gus Wiseman, Sep 13 2018

sign

A refinement of Stirling numbers of the first kind.

An unsigned version is A124795.

Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]],{n,100}]

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (-1)^(Sum x_i * y_i - Sum y_i) (Sum x_i * y_i)! / (Product x_i^y_i * Product y_i!).

A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			42 is the Heinz number of (4,2,1) which is strict and knapsack, so is in the sequence. 45 is the Heinz number of (3,2,2) which is knapsack but not strict, so is not in the sequence. 30 is the Heinz number of (3,2,1) which is strict but not knapsack, so is not in the sequence.
Sequence of strict knapsack partitions begins: (), (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (8), (42), (51), (9), (61).

Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301900.

wt[n_]:=If[n===1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
Select[Range[100],SquareFreeQ[#]&&UnsameQ@@wt/@Divisors[#]&]

Intersection of A299702 and A005117.

A365830 Heinz numbers of incomplete integer partitions, meaning not every number from 0 to A056239(n) is the sum of some submultiset.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Gus Wiseman, Sep 26 2023

nonn

First differs from A325798 in lacking 156.
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.
The complement (complete partitions) is A325781.

			The terms together with their prime indices begin:
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
For example, the submultisets of (1,1,2,6) (right column) and their sums (left column) are:
   0: ()
   1: (1)
   2: (2)  or (11)
   3: (12)
   4: (112)
   6: (6)
   7: (16)
   8: (26) or (116)
   9: (126)
  10: (1126)
But 5 is missing, so 156 is in the sequence.

For prime indices instead of sums we have A080259, complement of A055932.
The complement is A325781, counted by A126796, strict A188431.
Positions of nonzero terms in A325799, complement A304793.
These partitions are counted by A365924, strict A365831.
A056239 adds up prime indices, row sums of A112798.
A276024 counts positive subset-sums of partitions, strict A284640
A299701 counts distinct subset-sums of prime indices.
A365918 counts distinct non-subset-sums of partitions, strict A365922.
A365923 counts partitions by distinct non-subset-sums, strict A365545.
Cf. A001223, A046663, A073491, A098743, A304792, A365658, A365919.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Select[Range[100],Length[nmz[prix[#]]]>0&]

A334201 a(n) = A056239(n) - A061395(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 2, 0, 1, 2, 3, 0, 3, 0, 2, 2, 1, 0, 3, 3, 1, 4, 2, 0, 3, 0, 4, 2, 1, 3, 4, 0, 1, 2, 3, 0, 3, 0, 2, 4, 1, 0, 4, 4, 4, 2, 2, 0, 5, 3, 3, 2, 1, 0, 4, 0, 1, 4, 5, 3, 3, 0, 2, 2, 4, 0, 5, 0, 1, 5, 2, 4, 3, 0, 4, 6, 1, 0, 4, 3, 1, 2, 3, 0, 5, 4, 2, 2, 1, 3, 5, 0, 5, 4, 5, 0, 3, 0, 3, 5

Offset: 1

Antti Karttunen, May 11 2020

nonn

a(n) is the sum of all other parts of the partition having Heinz number n except one instance of the largest part.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A052126, A056239, A061395, A122111, A318995.
Sum of A339895 and A339896.
Differs from A323077 for the first time at n=169, where a(169) = 6, while A323077(169) = 5.
Cf. also A334107.

Array[Total[# /. {p_, c_} /; p > 0 :> PrimePi[p] c] - PrimePi@ #[[-1, 1]] &@ FactorInteger[#] &, 105] (* Michael De Vlieger, May 14 2020 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A334201(n) = if(1==n,0,(bigomega(n)-1)+A334201(A064989(n)));

a(n) = A056239(n) - A061395(n) = A056239(A052126(n)).
a(n) = A318995(A122111(n)).
a(n) = a(A064989(n)) + A001222(n) - 1.
a(n) = A339895(n) + A339896(n). - Antti Karttunen, Dec 31 2020

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).

A301855 Number of divisors d|n such that no other divisor of n has the same Heinz weight A056239(d).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

			The a(24) = 4 special divisors are 1, 2, 12, 24.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000712, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301830, A301854, A301855, A301856.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
Table[Length[uqsubs[primeMS[n]]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A301855(n) = if(1==n,n,my(m=Map(),w,s); fordiv(n,d,w = A056239(d); if(!mapisdefined(m, w, &s), mapput(m,w,Set([d])), mapput(m,w,setunion(Set([d]),s)))); sumdiv(n,d,(1==length(mapget(m,A056239(d)))))); \\ Antti Karttunen, Jul 01 2018

More terms from Antti Karttunen, Jul 01 2018

A301900 Heinz numbers of strict non-knapsack partitions. Squarefree numbers such that more than one divisor has the same Heinz weight A056239(d).

Original entry on oeis.org

30, 70, 154, 165, 210, 273, 286, 330, 390, 442, 462, 510, 546, 561, 570, 595, 646, 690, 714, 741, 770, 858, 870, 874, 910, 930, 1045, 1110, 1122, 1155, 1173, 1190, 1230, 1254, 1290, 1326, 1330, 1334, 1365, 1410, 1430, 1482, 1495, 1590, 1610, 1653, 1770

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of strict non-knapsack partitions begins: (321), (431), (541), (532), (4321), (642), (651), (5321), (6321), (761), (5421), (7321), (6421), (752), (8321), (743), (871), (9321), (7421), (862), (5431), (6521).

Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301899.

wt[n_]:=If[n===1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
Select[Range[1000],SquareFreeQ[#]&&!UnsameQ@@wt/@Divisors[#]&]

Complement of A005117 in A299702.

A088880 Number of different values of A000005(m) when A056239(m) is equal to n.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 8, 6, 12, 10, 16, 13, 25, 18, 28, 25, 40, 32, 51, 40, 62, 51, 76, 62, 99, 77, 112, 92, 138, 109, 165, 130, 189, 153, 220, 178, 267, 208, 292, 240, 347, 274, 397, 315, 445, 361, 512, 407, 591, 464, 647, 524, 746, 588, 830, 664, 928, 746, 1034

Offset: 0

Naohiro Nomoto, Nov 28 2003

Number of distinct values of Product_{k=1..n} (m(k,P)+1) where m(k,P) is multiplicity of part k in partition P, as P ranges over all partitions of n. - Vladeta Jovovic, May 24 2008

Alois P. Heinz, Table of n, a(n) for n = 0..222

Cf. A088314, A215366.

multipl := proc(P,p)
        local a;
        a := 0 ;
        for el in P do
                if el = p then
                        a := a+1 ;
                end if;
        end do;
        a ;
end proc:
A088880 := proc(n)
        local pro,pa,m,p;
        pro := {} ;
        for pa in combinat[partition](n) do
                m := 1 ;
                for p from 1 to n do
                        m := m*(1+multipl(pa,p)) ;
                end do:
                pro := pro union {m} ;
        end do:
        nops(pro) ;
end proc: # R. J. Mathar, Sep 27 2011
# second Maple program
b:= proc(n, i) option remember; `if`(n=0 or i<2, {n+1},
       {seq(map(p->p*(j+1), b(n-i*j, i-1))[], j=0..n/i)})
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 09 2012

b[n_, i_] := b[n, i] = If[n==0 || i<2, {n+1}, Table[b[n-i*j, i-1]*(j+1), {j, 0, n/i}] // Flatten // Union]; a[n_] := Length[b[n, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

Original entry on oeis.org

1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is 5-smooth if its prime divisors are all <= 5.

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 sequence of terms together with their prime indices begins:
       1: {}            125: {3,3,3}
       3: {2}           135: {2,2,2,3}
       5: {3}           150: {1,2,3,3}
       9: {2,2}         225: {2,2,3,3}
      10: {1,3}         243: {2,2,2,2,2}
      15: {2,3}         250: {1,3,3,3}
      25: {3,3}         270: {1,2,2,2,3}
      27: {2,2,2}       300: {1,1,2,3,3}
      30: {1,2,3}       375: {2,3,3,3}
      45: {2,2,3}       405: {2,2,2,2,3}
      50: {1,3,3}       450: {1,2,2,3,3}
      75: {2,3,3}       500: {1,1,3,3,3}
      81: {2,2,2,2}     625: {3,3,3,3}
      90: {1,2,2,3}     675: {2,2,2,3,3}
     100: {1,1,3,3}     729: {2,2,2,2,2,2}

Allowing any number of parts and sum gives A051037, counted by A001399.
These are Heinz numbers of the partitions counted by A266755.
Allowing parts > 5 gives A344291, counted by A110618.
The non-3-smooth case is A344294, counted by A325691.
Requiring the sum of prime indices to be even gives A344295.
A000070 counts non-multigraphical partitions, ranked by A344292.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A300061 ranks partitions of even numbers, with 5-smooth case A344297.
Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]

Intersection of A051037 and A344291.

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

A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A365830 Heinz numbers of incomplete integer partitions, meaning not every number from 0 to A056239(n) is the sum of some submultiset.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A334201 a(n) = A056239(n) - A061395(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A301855 Number of divisors d|n such that no other divisor of n has the same Heinz weight A056239(d).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A301900 Heinz numbers of strict non-knapsack partitions. Squarefree numbers such that more than one divisor has the same Heinz weight A056239(d).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A088880 Number of different values of A000005(m) when A056239(m) is equal to n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs