A104512 -id:A104512 - cp's OEIS frontend

A104513 The number of consecutive integers > 1 beginning with A104512(n), the sum of which equals n, or 0 if impossible.

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

Offset: 1

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 23 2005

			a(18) = 4 because 3+4+5+6 = 18.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 67.

Antti Karttunen, Table of n, a(n) for n = 1..16416

Cf. A104512, A104514, A104515, A104516.

f[n_] := Block[{r = Ceiling[n/2]}, If[ IntegerQ[ Log[2, n]], 0, m = Range[r]; lst = Flatten[ Table[ m[[k]], {i, r}, {j, i + 1, r}, {k, i, j}], 1]; Length[ lst[[ Position[ Plus @@@ lst, n, 1, 1][[1, 1]]] ]]]]; Table[ f[n], {n, 100}]

A104513(n) = if(!bitand(n,n-1), 0, my(b,d,u=1+sqrtint(2*n)); for(k=0,n-2,b = binomial(k+1,2); forstep(j=min(n,k+u),k+2,-1, d = binomial(j+1,2) - b; if(d==n,return(j-k),if(dAntti Karttunen, Mar 30 2021

a(n)=0 iff n=2^k.

A118235 Smallest positive number starting an interval of consecutive integers with element sum n.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 3, 8, 2, 1, 5, 3, 6, 2, 1, 16, 8, 3, 9, 2, 1, 4, 11, 7, 3, 5, 2, 1, 14, 4, 15, 32, 3, 7, 2, 1, 18, 8, 4, 6, 20, 3, 21, 2, 1, 10, 23, 15, 4, 8, 6, 3, 26, 2, 1, 5, 7, 13, 29, 4, 30, 14, 3, 64, 2, 1, 33, 5, 9, 7, 35, 4, 36, 17, 3, 6, 2, 1, 39, 14, 5, 19, 41, 7, 4, 20, 12, 3, 44, 2, 1, 8

Offset: 1

Reinhard Zumkeller, Apr 18 2006

Right border of A299765. - Omar E. Pol, Jul 24 2018

In other words: a(n) is smallest part of the partitions of n into consecutive parts. - Omar E. Pol, Mar 12 2019

			a(3)=1 since 3 = 1+2; a(5)=2 since 5 = 2+3; a(6)=1 since 6 = 1+2+3; etc.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Paul D. Hanna)

Cf. A000079, A000217, A001227, A002817, A104512, A109814, A212652, A299765.

a:= proc(n) local j, k, s; j, k, s:= 1$3;
      while s<>n do
         if sAlois P. Heinz, Aug 05 2018

a[n_] := Module[{j = 1, k = 1, s = 1}, While[True, If[s == n, Break[]]; If[s < n, k = k+1; s = s+k, s = s-j; j = j+1]]; j];
Array[a, 100] (* Jean-François Alcover, Mar 12 2019, after Alois P. Heinz *)

{a(n)=local(A=n);for(j=1,n,for(k=j,n+1,if(n==k*(k-1)/2-j*(j-1)/2,A=j;k=j=2*n+1)));A} /* Paul D. Hanna, Oct 28 2011 */

A109814(n) * (A109814(n) + 2*a(n) - 1) / 2 = n.
a(m) = n iff m = 2^k: a(A000079(n)) = A000079(n);
a(m) = 1 iff m = k*(k+1)/2: a(A000217(n)) = 1.
a(A002817(n-1)+1) = n; i.e., a(m) = n if m = k*(k-1)/2 + 1 and k = n*(n-1)/2 + 1. - Paul D. Hanna, Oct 28 2011
a(m) = 2 iff m = k*(k+3)/2: a(A000096(n)) = 2. - Bernard Schott, Mar 12 2019

A104514 a(n) = least number k > 1 of consecutive integers which sum to 2*n; or a(n) = 0 if n is a power of 2.

Original entry on oeis.org

0, 0, 3, 0, 4, 3, 4, 0, 3, 5, 4, 3, 4, 7, 3, 0, 4, 3, 4, 5, 3, 8, 4, 3, 4, 8, 3, 7, 4, 3, 4, 0, 3, 8, 4, 3, 4, 8, 3, 5, 4, 3, 4, 11, 3, 8, 4, 3, 4, 5, 3, 13, 4, 3, 4, 7, 3, 8, 4, 3, 4, 8, 3, 0, 4, 3, 4, 16, 3, 5, 4, 3, 4, 8, 3, 16, 4, 3, 4, 5, 3, 8, 4, 3, 4, 8, 3, 11, 4, 3, 4, 16, 3, 8, 4, 3, 4, 7, 3, 5, 4, 3, 4

Offset: 1

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 23 2005

nonn

a(2^k) = 0 and a(3*n) = 3.

Least proper divisor d of 4*n (if any) such that d or 4*n/d is odd. - Robert Israel, May 06 2015

			a(9) = 3 because 3+4+5+6 = 5+6+7 = 2*9 = 18.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 67.

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

Cf. A104512, A104513, A104515, A104516, A163169.

a:= proc(n) local divs,r;
   divs:= select(t -> t::odd or (4*n/t)::odd, numtheory:-divisors(4*n) minus {1,4*n});
   if nops(divs)=0 then 0 else min(divs) fi
end proc:
seq(a(n), n=1..200); # Robert Israel, May 06 2015

f[n_] := Block[{r = Ceiling[n/2]}, If[IntegerQ[Log[2, n]], 0, m = Range[r]; lst = Flatten[Table[m[[k]], {i, r}, {j, i + 1, r}, {k, i, j}], 1]; Min[Length /@ lst[[Flatten[Position[Plus @@@ lst, n]]]]]]]; Table[f[2n], {n, 103}]

a(n) = A163169(2*n). Robert Israel, May 06 2015

A104515 Difference between the maximum number of consecutive integers and the least number >1 of consecutive integers, the sum of which equals 2n.

Original entry on oeis.org

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

Offset: 1

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 23 2005

nonn

a(n)=0 iff n=2^k.

			a(18) = 1 because 3+4+5+6 = 5+6+7 = 18.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 67.

Cf. A104512, A104513, A104514, A104516.

f[n_] := Block[{r = Ceiling[n/2]}, If[ IntegerQ[ Log[2, n]], 0, m = Range[r]; lst = Flatten[ Table[ m[[k]], {i, r}, {j, i + 1, r}, {k, i, j}], 1]; l = Length /@ lst[[ Flatten[ Position[ Plus @@@ lst, n]]]]; Max[l] - Min[l]]]; Table[ f[2n], {n, 105}]

A104516 a(n) is the first occurrence of k in A104515, the difference between the maximum number of consecutive integers and the minimum number >1 of consecutive integers, the sum of which equals n.

Original entry on oeis.org

1, 9, 30, 15, 21, 35, 54, 45, 55, 77, 156, 91, 105, 135, 204, 153, 171, 209, 252, 231, 253, 299, 450, 325, 351, 405, 522, 435, 465, 527, 594, 561, 595, 665, 888, 703, 741, 819, 984, 861, 903, 989

Offset: 0

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 23 2005

nonn

a(n)=0 iff n=2^k.

Where a(n)=k & a(n+2)=k+1 for k=54,252,594,...

			a(2)=30 because 4+5+6+7+8 = 9+10+11 = 30.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 67.

Cf. A104512, A104513, A104514, A104515.

f[n_] := Block[{r = Ceiling[n/2]}, If[ IntegerQ[ Log[2, n]], 0, m = Range[r]; lst = Flatten[ Table[ m[[k]], {i, r}, {j, i + 1, r}, {k, i, j}], 1]; l = Length /@ lst[[ Flatten[ Position[ Plus @@@ lst, n]]]]; Max[l] - Min[l]]]; t = Table[0, {50}]; Do[ c = f[n]; If[ t[[c + 1]] == 0, t[[c + 1]] = n; Print[{n, c}]], {n, 10^4}]; t

cp's OEIS Frontend

A104513 The number of consecutive integers > 1 beginning with A104512(n), the sum of which equals n, or 0 if impossible.

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A118235 Smallest positive number starting an interval of consecutive integers with element sum n.

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A104514 a(n) = least number k > 1 of consecutive integers which sum to 2*n; or a(n) = 0 if n is a power of 2.

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A104515 Difference between the maximum number of consecutive integers and the least number >1 of consecutive integers, the sum of which equals 2n.

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A104516 a(n) is the first occurrence of k in A104515, the difference between the maximum number of consecutive integers and the minimum number >1 of consecutive integers, the sum of which equals n.

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