A057944 -id:A057944 - cp's OEIS frontend

A145032 If t(n) is the maximal triangular number not exceeding n, then a(n) is the n-th prime for which a(n)-t(a(n)) is a triangular number.

2, 3, 7, 11, 13, 29, 31, 37, 61, 67, 79, 97, 101, 137, 139, 151, 163, 181, 191, 193, 211, 241, 263, 277, 331, 379, 409, 421, 463, 499, 571, 601, 631, 709, 739, 751, 769, 821, 823, 947, 967, 991, 1063, 1087, 1091, 1109, 1117, 1129, 1231, 1303, 1327, 1381, 1399

Offset: 1

Vladimir Shevelev, Sep 30 2008

nonn

Primes p for which p-A057944(p) is in A000217. [From R. J. Mathar, Oct 25 2010]

			E. g., t(181)=171 (see A000217) and 181-171=10 is triangular number. Therefore p=181 is in the sequence

A000217 A117112 A145016

Contribution from R. J. Mathar, Oct 25 2010: (Start)
A057944 := proc(n) for i from 0 do if i*(i+1)/2 > n then return (i-1)*i /2 ; end if; end do: end proc:
isA000217 := proc(n) issqr(8*n+1) ; end proc:
isA145032 := proc(p) if isprime(p) then tres := p-A057944(p) ; isA000217(tres) ; else false; end if; end proc:
for n from 1 to 400 do p := ithprime(n) ; if isA145032(p) then printf("%d,",p) ; end if; end do: (End)

More terms from R. J. Mathar, Oct 25 2010

A245235 Repeat 2^(n*(n+1)/2) n+1 times.

Original entry on oeis.org

1, 2, 2, 8, 8, 8, 64, 64, 64, 64, 1024, 1024, 1024, 1024, 1024, 32768, 32768, 32768, 32768, 32768, 32768, 2097152, 2097152, 2097152, 2097152, 2097152, 2097152, 2097152, 268435456, 268435456, 268435456, 268435456, 268435456, 268435456, 268435456, 268435456

Offset: 0

Paul Curtz, Jul 14 2014

For a(n), the successive exponents of 2 are 0, 1, 1, 3, 3, 3,... = A057944(n).

			n+1 times repeated 2^(n*(n+1)/2)= 1, 2, 8, 64, 1024,... = A139685(n).
By the formula: a(0)=1/1=1, a(1)=2/1=2, a(2)=4/2=2, a(3)=8/1=8, a(4)=16/2=8,...
As triangle:
   1,
   2,    2,
   8,    8,    8,
  64,   64,   64,   64,
1024, 1024, 1024, 1024, 1024,
etc.
Row sums: 1, 4, 24, 256,... = A095340.

Cf. A000079, A006125, A057944, A059268, A095340, A123903, A139685.

Table[2^(n*(n+1)/2), {n, 0, 7}, {n+1}] // Flatten (* Jean-François Alcover, Jul 15 2014 *)

from math import isqrt
def A245235(n): return 1<<((m:=isqrt(n+1<<3)-1>>1)*(m+1)>>1) # Chai Wah Wu, Dec 17 2024

a(n) = 2^n/A059268(n).

T(n, k) = 2^(n*(n+1)/2), 0 <= k <= n. - Michel Marcus, Jul 17 2014

A326923 a(n) is the number of iterations needed to reach 1 or 9 starting at n and using the map k -> (k/2 if k is even, otherwise k + (largest triangular number < k)). Set a(n) = -1 if the trajectory never reaches 1 or 9.

Original entry on oeis.org

0, 1, 3, 2, 4, 4, 9, 3, 0, 5, 4, 5, 8, 10, 10, 4, 6, 1, 8, 6, 3, 5, 7, 6, 9, 9, 8, 11, 15, 11, 17, 5, 19, 7, 19, 2, 27, 9, 41, 7, 33, 4, 39, 6, 47, 8, 10, 7, 12, 10, 9, 10, 14, 9, 12, 12, 14, 16, 16, 12, 18, 18, 18, 6, 14, 20, 26, 8, 32, 20, 24, 3, 26, 28, 40, 10, 32

Offset: 1

Ali Sada, Oct 21 2019

It is conjectured that this algorithm will always terminate at 1 or 9.
Jim Nastos verified the conjecture for n <= 64*10^5.
Jim Nastos verified the conjecture for n <= 45248000.

			For n = 11: 11+10=21; 21+15=36; 36/2=18; 18/2=9; taking 4 steps to reach 9, so a(11)=4.

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

Cf. A006577, A057944, A326823, A326825.

LT:= proc(k) local n;
   n:= ceil((sqrt(1+8*k)-1)/2);
   n*(n-1)/2
end proc:
f:= proc(k) option remember; if k::even then 1+procname(k/2) else 1+procname(k+LT(k))fi
end proc:
f(1):=0: f(9):= 0:
map(f, [$1..100]); # Robert Israel, Oct 23 2019

LT[k_] := Module[{n}, n = Ceiling[(Sqrt[1+8k]-1)/2]; n(n-1)/2]; f[k_] := f[k] = If[EvenQ[k], 1+f[k/2], 1+f[k+LT[k]]]; f[1] = 0; f[9] = 0;
Array[f, 100] (* Jean-François Alcover, Aug 27 2022, after Robert Israel *)

A354762 Irregular triangle read by rows in which the row n lists the partition of n into the minimum number of triangular parts.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 3, 1, 1, 6, 6, 1, 6, 1, 1, 6, 3, 10, 10, 1, 10, 1, 1, 10, 3, 10, 3, 1, 15, 15, 1, 15, 1, 1, 15, 3, 15, 3, 1, 15, 3, 1, 1, 21, 21, 1, 21, 1, 1, 21, 3, 21, 3, 1, 21, 3, 1, 1, 21, 6, 28, 28, 1, 28, 1, 1, 28, 3, 28, 3, 1, 28, 3, 1, 1, 28, 6, 28, 6, 1

Offset: 0

Stefano Spezia, Jun 06 2022

The representation of the partitions (for fixed n) is as (weakly) decreasing list of the parts.

			The irregular triangle begins:
     0;
     1;
     1, 1;
     3;
     3, 1;
     3, 1, 1;
     6;
     6, 1;
     6, 1, 1;
     6, 3;
    10;
    10, 1;
    10, 1, 1;
    10, 3;
    10, 3, 1;
    15;
    15, 1;
    15, 1, 1;
    15, 3;
    15, 3, 1;
    15, 3, 1, 1;
    ...

Cf. A000041, A000217, A007294.
Cf. A001477 (row sums), A057944 (1st column), A057945 (row lengths).
Cf. A354763.

Flatten[Join[{0}, Table[First[IntegerPartitions[n, All, Table[k(k+1)/2, {k, (Sqrt[1+8n]-1)/2}]]], {n, 35}]]]

A094261 a(n) = n(n-1)(n-3)(n-6)...(n-t), where t is the largest triangular number less than n; number of factors in the product is ceiling((sqrt(1+8*n)-1)/2).

Original entry on oeis.org

1, 2, 6, 12, 40, 90, 168, 560, 1296, 2520, 4400, 14256, 32760, 64064, 113400, 187200, 586432, 1321920, 2560896, 4522000, 7484400, 11797632, 35784320, 78871968, 150480000, 263120000, 433060992, 681080400, 1033305728, 3044304000

Offset: 1

Amarnath Murthy, Apr 26 2004

nonn

			a(8) = 8*(8-1)*(8-3)*(8-6) = 8*7*5*2 = 560.

Cf. A057944, A094262.

a:=n->product(n-k*(k+1)/2,k=0..ceil((sqrt(1+8*n)-1)/2)-1): seq(a(n),n=1..35); # Emeric Deutsch, Feb 03 2006

Corrected and extended by Emeric Deutsch, Feb 03 2006

A274011 a(n) is the greatest number of elements in a partition of n into distinct parts such that no two elements add to another.

Original entry on oeis.org

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

Offset: 1

Gordon Hamilton, Jun 06 2016

nonn

A lower bound for a(n^2) is n (use the n^2 partition of the first n consecutive odd numbers.)

An upper bound u for a(n) is found by a partition of A057944(n); [1,..,u]. This gives u = floor((floor(sqrt(8*n+7))-1)/2). - David A. Corneth, Jun 06 2016

			a(24) = 5 because {1,2,4,7,10} is a partition of 24 and there are no sum-free partitions with more parts.
Candidates for such a partition of size 5 of 24 are found by adding [0,1,2,3,4] to partitions of 5 of 24 - (0+1+2+3+4). - _David A. Corneth_, Jun 06 2016
a(25) = 5 because {1,3,5,7,9} is a partition of 25. {1,2,4,7,11} does not show that a(25) >= 5 because 4,7, and 11 are all elements of the set and 4+7=11.

dif[w_] := Length[w] <= 2 || {} == Intersection[w, Reap[ Do[ Sow[w[[i]] + w[[j]]], {i, Length@ w}, {j, i-1}]][[2, 1]]]; p[tg_, w_] := If[tg == 0, bst = Max[bst, Length@ w], Block[{v=If[w == {}, 0, Last@w], u}, Do[u = Append[w, k]; If[dif@ u, p[tg-k, u]], {k, v+1, tg}]]]; a[n_] := (bst = 0; p[n, {}]; bst); Array[a, 50] (* Giovanni Resta, Jun 06 2016 *)

a(25)-a(86) from Giovanni Resta, Jun 06 2016

A307392 Number of partitions of n with at most one part in the interval [i(i+1)/2, i+(i(i+1)/2)] for all nonnegative integers i.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 6, 9, 11, 12, 12, 12, 13, 15, 18, 22, 27, 34, 42, 50, 56, 60, 63, 66, 70, 76, 84, 94, 106, 120, 136, 154, 177, 206, 241, 279, 317, 352, 381, 404, 423, 442, 464, 492, 528, 574, 630, 694, 764, 839, 920, 1008, 1104, 1213, 1341, 1494, 1674, 1878

Offset: 0

Igor Haladjian, Apr 06 2019

nonn

The intervals are: [1,2], [3,5], [6,9], [10,14], [15,20], [21,27], [28,35], [36,44], [45,54], [55,65], ... .

			a(0)=1 by definition of the empty partition.
a(10)=6 because 10=9+1=8+2=7+3=6+4=6+3+1 (for example, you cannot take 5+5 or 7+2+1 because of the definition of a(n)).

Cf. A000217, A057944.

f:= n-> 1+add(x^j, j=n*(n+1)/2..n*(n+3)/2):
a:= n-> coeff(mul(f(k), k=1..ceil((sqrt(9+8*n)-3)/2)), x, n):
seq(a(n), n=0..61);

f(n, x) = (1+sum(j=n*(n+1)/2, n*(n+3)/2, x^j));
a(n) = polcoef(prod(k=1, ceil((sqrt(9+8*n)-3)/2), f(k, x)), n, x); \\ version 2.11.0 or newer; Michel Marcus, Apr 08 2019

first(n) = v = Vecrev(Vec(a(n))); vector(n, i, v[i]) \\ using a(n) from above \\ David A. Corneth, Apr 08 2019

G.f.: Product_{n>=0} (1 + Sum_{k=(n*(n+1)/2)..(n*(n+3)/2)} x^k).

cp's OEIS Frontend

A145032 If t(n) is the maximal triangular number not exceeding n, then a(n) is the n-th prime for which a(n)-t(a(n)) is a triangular number.

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A245235 Repeat 2^(n*(n+1)/2) n+1 times.

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A326923 a(n) is the number of iterations needed to reach 1 or 9 starting at n and using the map k -> (k/2 if k is even, otherwise k + (largest triangular number < k)). Set a(n) = -1 if the trajectory never reaches 1 or 9.

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A354762 Irregular triangle read by rows in which the row n lists the partition of n into the minimum number of triangular parts.

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A094261 a(n) = n(n-1)(n-3)(n-6)...(n-t), where t is the largest triangular number less than n; number of factors in the product is ceiling((sqrt(1+8*n)-1)/2).

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A274011 a(n) is the greatest number of elements in a partition of n into distinct parts such that no two elements add to another.

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Mathematica

Extensions

A307392 Number of partitions of n with at most one part in the interval [i*(i+1)/2, i+(i*(i+1)/2)] for all nonnegative integers i.

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A307392 Number of partitions of n with at most one part in the interval [i(i+1)/2, i+(i(i+1)/2)] for all nonnegative integers i.