A219320 -id:A219320 - cp's OEIS frontend

A219311 Number T(n,k) of standard Young tableaux for partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 9, 0, 1, 14, 16, 0, 1, 34, 35, 0, 1, 55, 134, 0, 1, 125, 435, 0, 1, 209, 1213, 768, 0, 1, 461, 3454, 2310, 0, 1, 791, 10484, 11407, 0, 1, 1715, 28249, 44187, 0, 1, 3002, 80302, 200044, 0, 1, 6434, 231895, 680160, 292864

Offset: 0

Alois P. Heinz, Nov 17 2012

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=A003056(n). T(n,k) = 0 for k>A003056(n).

			A(4,2) = 3:
  +---------+  +---------+  +---------+
  | 1  2  3 |  | 1  2  4 |  | 1  3  4 |
  | 4 .-----+  | 3 .-----+  | 2 .-----+
  +---+        +---+        +---+
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1;
  0,  1,    2;
  0,  1,    3;
  0,  1,    9;
  0,  1,   14,    16;
  0,  1,   34,    35;
  0,  1,   55,   134;
  0,  1,  125,   435;
  0,  1,  209,  1213,   768;
  0,  1,  461,  3454,  2310;
  0,  1,  791, 10484, 11407;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Wikipedia, Young tableau

Columns k=0-10 give: A000007, A000012 (for n>0), A047171(n) = A037952(n)-1, A219316, A219317, A219318, A219319, A219320, A219321, A219322, A219323.
Row sums give: A218293.
Row lengths are 1 + A003056(n).
T(A000217(k),k) = A005118(k+1).
Cf. A219272, A219274.

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, k, l) `if`(n=0, h(l), `if`(n>k*(i-(k-1)/2), 0,
      g(n, i-1, min(k, i-1), l)+`if`(i>n, 0, g(n-i, i-1, k-1, [l[], i]))))
    end:
A:= proc(n, k) option remember; `if`(k<0, 0, g(n, n, k, [])) end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] ];
g[n_, i_, k_, l_] := If[n == 0, h[l], If[n > k*(i-(k-1)/2), 0, g[n, i-1, Min[k, i-1], l] + If[i > n, 0, g[n-i, i-1, k-1, Append[l, i]]]]];
a[n_, k_] := a[n, k] = If[k < 0, 0, g[n, n, k, {}]];
t[n_, k_] := a[n, k] - a[n, k-1];
Table[Table[t[n, k], {k, 0, Floor[(Sqrt[1+8*n]-1)/2]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

A210445 Least positive integer k with k*n practical.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 1, 2, 2, 6, 1, 6, 2, 2, 1, 12, 1, 12, 1, 2, 3, 12, 1, 4, 3, 2, 1, 12, 1, 16, 1, 2, 6, 4, 1, 18, 6, 2, 1, 20, 1, 20, 2, 2, 6, 24, 1, 4, 2, 4, 2, 24, 1, 4, 1, 4, 6, 24, 1, 24, 8, 2, 1, 4, 1, 30, 3, 4, 2, 30, 1, 30, 9, 2, 3, 4, 1, 36, 1, 2, 10, 36, 1, 4, 10, 4, 1, 36, 1, 4, 3, 6, 12, 4, 1, 42, 2, 2, 1

Offset: 1

Zhi-Wei Sun, Jan 20 2013

Conjecture: a(n) < n for all n>1, and a(n) < n/2 for all n>47.

Large values are obtained for prime n: The corresponding subsequence is a(p(n)) = (1, 2, 4, 4, 6, 6, 12, 12, 12, 12, 16, 18, 20, 20, 24, 24, 24, 24, ...), while for composite indices, a(c(n)) = (1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 1, 2, ...). - M. F. Hasler, Jan 21 2013

			a(10)=2 since 2*10=20 is practical but 1*10=10 is not.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Cf. A005153, A210444, A071558, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315, A219320.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
Do[Do[If[pr[k*n]==True,Print[n," ",k];Goto[aa]],{k,1,n}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]

A210445(n)={for(k=1,n,is_A005153(k*n)&&return(k))} \\ (Would return 0 if a(n)>n.) - M. F. Hasler, Jan 20 2013

a(n) = 1 iff n is in A005153, therefore a(n) > 1 for all odd n>1. - M. F. Hasler, Jan 21 2013

A210444 a(n) = |{0

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 1, 0, 2, 2, 0, 4, 0, 1, 4, 2, 0, 6, 1, 3, 2, 2, 0, 5, 2, 1, 3, 1, 2, 11, 0, 1, 4, 1, 2, 6, 0, 2, 4, 3, 1, 9, 2, 3, 4, 2, 0, 7, 1, 4, 4, 5, 0, 8, 4, 1, 3, 3, 0, 15, 0, 3, 4, 4, 4, 13, 2, 4, 2, 5, 2, 10, 0, 2, 11, 2, 3, 12, 0, 6, 6, 2, 2, 13, 3, 5, 7, 5, 1, 16, 4, 4, 6, 3, 2, 11, 0, 8, 6, 7

Offset: 1

Zhi-Wei Sun, Jan 20 2013

nonn

Conjecture: a(n)>0 for all n>911.

This implies that for each n=2,3,4,... there is a positive integer k

The conjecture has been verified for n up to 10^6.

			a(7) = 1 since 6*7 = 42 is practical, and 41 and 43 are twin primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A071558, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A219185, A219312, A219315, A219320.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[k*n-1]==True&&PrimeQ[k*n+1]==True&&pr[k*n]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A210452 Number of integers k

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 20 2013

nonn

Conjecture: a(n)>0 for all n>4.
This implies the twin prime conjecture since k*p is not practical for any prime p>sigma(k)+1.
Zhi-Wei Sun also made the following conjectures:
(1) For each integer n>197, there is a practical number k(2) For every n=9,10,... there is a practical number k(3) For any integer n>26863, the interval [1,n] contains five consecutive integers m-2, m-1, m, m+1, m+2 with m-1 and m+1 both prime, and m-2, m, m+2, m*n all practical.

			a(11)=1 since 5 and 7 are twin primes, and 6 and 6*11 are both practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.

Cf. A005153, A210444, A210445, A071558, A208243, A208244, A208246, A208249, A219185, A209253, A209254, A219312, A219315, A219320.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[k-1]==True&&PrimeQ[k+1]==True&&pr[k]==True&&pr[k*n]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

Showing 1-4 of 4 results.

cp's OEIS Frontend

A219311 Number T(n,k) of standard Young tableaux for partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A210445 Least positive integer k with k*n practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A210444 a(n) = |{0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A210452 Number of integers k

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica