A085492 -id:A085492 - cp's OEIS frontend

A085493 Numbers k having partitions into distinct divisors of k + 1.

1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 47, 53, 55, 59, 63, 65, 69, 71, 77, 79, 83, 87, 89, 95, 99, 103, 107, 111, 119, 125, 127, 131, 139, 143, 149, 155, 159, 161, 167, 175, 179, 191, 195, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

A085491(a(n)) > 0; complement of A085492.

			The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. Since 6 + 14 + 21 = 41, 41 is in the sequence.
The divisors of 43 are 1, 43. Since no selection of these divisors can possibly add up to 42, this means that 42 is not in the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Paul K. Stockmeyer, Of camels, inheritance, and unit fractions, Math Horizons, 21 (2013), 8-11.

Cf. A085498, A106431.

q:= proc(m) option remember; local b, l; b, l:=
      proc(n, i) option remember; n=0 or i>=1 and
        (l[i]<=n and b(n-l[i], i-1) or b(n, i-1))
      end, sort([numtheory[divisors](m+1)[]]);
      b(m, nops(l)-1)
    end:
select(q, [$1..300])[];  # Alois P. Heinz, Feb 04 2023

divNextableQ[n_] := TrueQ[Length[Select[Subsets[Divisors[n + 1]], Plus@@# == n &]] > 0]; Select[Range[100], divNextableQ] (* Alonso del Arte, Jan 07 2023 *)

def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
def divPartSums(n: Int): List[Int] = divisors(n).toSet.subsets.toList.map(_.sum)
(1 to 128).filter(n => divPartSums(n + 1).contains(n)) // Alonso del Arte, Jan 26 2023

{k > 0 : 0 < [x^k] Product_{d divides (k+1)} (1+x^d)}. - Alois P. Heinz, Feb 04 2023

A085491 Number of ways to write n as sum of distinct divisors of n+1.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 5, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 31, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 26, 0, 0, 0, 0, 0, 1, 0, 6, 0, 0, 0, 23, 0, 0, 0, 1, 0, 20, 0, 0, 0, 0, 0, 21, 0, 0, 0, 1

Offset: 0

Reinhard Zumkeller, Jul 03 2003

nonn

a(A085492(n)) = 0; a(A085493(n)) > 0; a(A085494(n)) = 1.

			n=11, divisors of 12=11+1 that are not greater 11: {1,2,3,4,6}, 11=6+5=6+4+1, therefore a(11)=2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A085496.

a:= proc(m) option remember; local b, l; b, l:=
      proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
        b(n, i-1)+`if`(l[i]>n, 0, b(n-l[i], i-1))))
      end, sort([numtheory[divisors](m+1)[]]);
      forget(b); b(m, nops(l)-1)
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 12 2019

a[n_] := Module[{dd}, dd = Select[Divisors[n+1], # <= n&]; Select[ IntegerPartitions[n, dd // Length, dd], Reverse[#] == Union[#]&] // Length]; Array[a, 100, 0] (* Jean-François Alcover, Mar 12 2019 *)

a(n) = [x^n] Product_{d divides (n+1)} (1 + x^d). - Alois P. Heinz, Feb 04 2023

a(0)=1 prepended by Alois P. Heinz, Mar 12 2019

A085497 Primes p having no partition into distinct divisors of p+1.

Original entry on oeis.org

2, 13, 37, 43, 61, 67, 73, 97, 101, 109, 113, 137, 151, 157, 163, 173, 181, 193, 211, 229, 241, 257, 277, 281, 283, 313, 317, 331, 337, 353, 373, 397, 401, 409, 421, 433, 443, 457, 487, 491, 523, 541, 547, 563, 577, 601, 613, 617, 631, 641, 653, 661, 673, 677

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

			p=13, divisors of p+1=13+1=14 that are not greater 13: {1,2,7} with sums of distinct summands 1,2,3=2+1,7,8=7+1,9=7+2 and 10=7+2+1, therefore 13 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..4000

Subsequence of A085492.

Cf. A085496, A085498.

seqQ[p_] := Module[{d = Most[Divisors[p+1]]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, p}], p] == 0]; Select[Range[700], PrimeQ[#] && seqQ[#] &] (* Amiram Eldar, Jan 13 2020 *)

A085496(a(n)) = 0.

More terms from Amiram Eldar, Jan 13 2020

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A085493 Numbers k having partitions into distinct divisors of k + 1.

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A085491 Number of ways to write n as sum of distinct divisors of n+1.

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Maple

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A085497 Primes p having no partition into distinct divisors of p+1.

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula

Extensions