A285175 -id:A285175 - cp's OEIS frontend

A153452 a(1) = 1; if n > 1, then a(n) = Sum_{prime q |n} a(n*q' /q), where q' = prevprime(q) for q>2 and 2' = 1.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 3, 1, 4, 5, 1, 1, 5, 1, 6, 9, 5, 1, 4, 5, 6, 5, 10, 1, 16, 1, 1, 14, 7, 14, 9, 1, 8, 20, 10, 1, 35, 1, 15, 21, 9, 1, 5, 14, 21, 27, 21, 1, 14, 28, 20, 35, 10, 1, 35, 1, 11, 56, 1, 48, 64, 1, 28, 44, 70, 1, 14, 1, 12, 42, 36, 42

Offset: 1

Naohiro Nomoto, Dec 26 2008

Original name was: a(1)=1, for n>1, if 2*m = n or (m/p)*nextprime(p) = n, where p is a prime factor of m ( m runs from 1 to n-1 ), then a(n) = Sum_{m} a(m).
The number of standard tableaux of the integer partition with Heinz number n (for the definition of the Heinz number of a partition see the next comment). The proof follows from Lemma 2.8.2 of the Sagan reference. Examples: (i) a(6)=2; indeed 6 = 2*3 is the Heinz number of the partition [1,2] and, obviously, the Ferrers board admits 2 standard tableaux; (ii) a(60)=35; indeed, 60 = 2*2*3*5 is the Heinz number of the partition [1,1,2,3] and the hook-lengths of its Ferrer board are 6,3,1,4,1,2,1; then, the number of standard tableaux is 7!/(6*3*4*2) = 35. - Emeric Deutsch, May 24 2015
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition; for example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436). - Emeric Deutsch, May 24 2015

			For n=10; (m=5; 2*5 = 10), (m=6; (6/3)*nextprime(3) = 10), hence a(10) = a(5) + a(6) = 3.

B. E. Sagan, The Symmetric Group, Springer, 2001, New York.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A285175.

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 09 2012

a[n_] := a[n] = If[n == 1, 1, Sum[a[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 04 2016, after Alois P. Heinz *)

New name from Emeric Deutsch, May 14 2015

A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 189, 753, 3248, 14738, 70658, 354178, 1857703, 10121033, 57224955, 334321008, 2017234773, 12530668585, 80083779383, 525284893144, 3533663143981, 24336720018666, 171484380988738, 1234596183001927, 9075879776056533, 68052896425955296

Offset: 0

Gus Wiseman, Feb 26 2018

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(4) = 15 tableaux:
1 2 3 4
.
1 2 3   1 2 4   1 3 4   1 2 3   1 2 3
4       3       2       2       3
.
1 2   1 3   1 2
3 4   2 4   2 3
.
1 2   1 3   1 2   1 4   1 3
3     2     2     2     2
4     4     3     3     3
.
1
2
3
4

Alois P. Heinz, Table of n, a(n) for n = 0..50 (first 46 terms from Ludovic Schwob)
D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics, Vol. 34, No. 3 (1970), 709-727.
Wikipedia, Young tableau

Cf. A000085, A000898, A001221, A005117, A006958, A015128, A138178, A238121, A238690, A285175, A296561, A297388, A299926, A300120, A300122.

unddis[y_]:=DeleteCases[y-#,0]&/@Tuples[Table[If[y[[i]]>Append[y,0][[i+1]],{0,1},{0}],{i,Length[y]}]];
dos[y_]:=With[{sam=Rest[unddis[y]]},If[Length[sam]===0,If[Total[y]===0,{{}},{}],Join@@Table[Prepend[#,y]&/@dos[sam[[k]]],{k,1,Length[sam]}]]];
Table[Sum[Length[dos[y]],{y,IntegerPartitions[n]}],{n,1,8}]

a(n) = Sum_{k=0..n} 2^k * A238121(n,k). - Ludovic Schwob, Sep 23 2023

More terms from Ludovic Schwob, Sep 23 2023

A300002 Lexicographically earliest sequence of positive integers such that no k+2 points fall on any polynomial of degree k.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 16, 14, 20, 7, 15, 8, 12, 18, 31, 26, 27, 40, 30, 49, 38, 19, 10, 23, 53, 11, 32, 21, 25, 13, 47, 83

Offset: 1

Peter Kagey, Apr 17 2017

a(n) must avoid 2^(n-1)-1 polynomials: the polynomials defined by each nonempty subset of the first (n-1) terms of the sequence.
Conjecture: This sequence is a permutation of the natural numbers.
From David A. Corneth, May 10 2017: (Start)
Sequence is also "Lexicographically earliest sequence of positive integers such that any k+1 points fall on a polynomial of degree k."
Conjecture: a(27)-a(32) are 11, 32, 21, 25, 13, 47. If all previous data are correct, no polynomial of degree ceiling(n/2.5) - 1 goes through any set of points. (End)
Formerly A285175. - Peter Kagey, Mar 06 2018

			a(1) = 1.
a(2) != 1 or else (1, 1) and (2, 1) fall on y = 1. (Similarly all terms must be distinct.)
a(2) = 2.
a(3) != 1 or else (1, 1) and (3, 1) fall on y = 1.
a(3) != 2 or else (2, 2) and (3, 2) fall on y = 2.
a(3) != 3 or else (1, 1), (2, 2) and (3, 3) fall on y = x.
a(3) = 4.
a(4) != 1 or else (1, 1) and (4, 1) fall on y = 1.
a(4) != 2 or else (2, 2) and (4, 2) fall on y = 2.
a(4) = 3

Rok Cestnik, Graphical example
David A. Corneth, Seqfan post about this sequence, May 01 2017.

Cf. A231334, A236335, A301802.

A = {{1, 1}, {2, 2}};
n = 3;
While[n < 50,
c = Sort[Select[Select[InterpolatingPolynomial[#, n] & /@ Subsets[A, {1, n - 1}], # > 0 & ] , IntegerQ]];
B = Differences[c];
If[Max[B] == 1,
d = Max[c] + 1,
d = Part[c, First[Position[B, Select[B, # > 1 &][[1]]]][[1]]] + 1];
A = Append[A, {n, d}];
Print[{n, d}]
n++;
] (* Luca Petrone, Apr 18 2017 *)

a(21)-a(26) from Luca Petrone, Apr 19 2017
a(27) from Robert G. Wilson v, Jul 09 2017
a(28)-a(33) from Bert Dobbelaere, Apr 12 2024

cp's OEIS Frontend

A153452 a(1) = 1; if n > 1, then a(n) = Sum_{prime q |n} a(n*q' /q), where q' = prevprime(q) for q>2 and 2' = 1.

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A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

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Examples

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Crossrefs

Programs

Mathematica

Formula

Extensions

A300002 Lexicographically earliest sequence of positive integers such that no k+2 points fall on any polynomial of degree k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions