A085493 -id:A085493 - cp's OEIS frontend

A106431 Even elements of A085493.

944, 1574, 2204, 2834, 3464, 4094, 4724, 5354, 5774, 5984, 6434, 6614, 6824, 7244, 7424, 7874, 8084, 8414, 8504, 8924, 9134, 9554, 9764, 10394, 11024, 11654, 12284, 12704, 12914, 13544, 14174, 14804, 15014, 15434, 16064, 16694, 17324, 17954, 18584, 19214, 19304

Offset: 1

David W. Wilson, Feb 17 2006

nonn

Cf. A085493, A005231.

A085498 Primes p having at least one partition into distinct divisors of p + 1.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 179, 191, 197, 199, 223, 227, 233, 239, 251, 263, 269, 271, 293, 307, 311, 347, 349, 359, 367, 379, 383, 389, 419, 431, 439, 449, 461, 463, 467, 479, 499, 503

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2000

Subsequence of A085493.

Cf. A085496, A085497, A085499.

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[500], PrimeQ[#] && seqQ[#] &] (* Amiram Eldar, Jan 13 2020 *)
Select[Prime[Range[100]],MemberQ[Total/@Subsets[Divisors[#+1]],#]&] (* Harvey P. Dale, Oct 04 2020 *)

A085496(a(n)) > 0.

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

A085492 Numbers n having no partition into distinct divisors of n+1.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 13, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 37, 38, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96, 97

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

A085491(a(n)) = 0; complement of A085493;

A006093 is a subsequence (prime minus 1).

Cf. A085497.

A085494 Numbers k having exactly one partition into distinct divisors of k+1.

Original entry on oeis.org

1, 3, 5, 7, 15, 17, 19, 27, 31, 53, 63, 65, 69, 77, 87, 99, 103, 127, 161, 195, 255, 271, 303, 367, 413, 463, 485, 495, 499, 511, 579, 649, 725, 819, 859, 867, 967, 1013, 1023, 1035, 1147, 1183, 1311, 1315, 1351, 1371, 1375, 1457, 1483, 1503, 1695, 1887, 1951

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

Includes every 2^k-1 and is therefore infinite. - David W. Wilson, Feb 02 2006

Amiram Eldar, Table of n, a(n) for n = 1..330

Cf. A085491, A085499.

Subsequence of A085493.

q[k_] := Module[{d = Divisors[k+1], x}, CoefficientList[Product[1 + x^i, {i, d}], x][[1+k]] == 1]; Select[Range[2000], q] (* Amiram Eldar, Apr 16 2025 *)

A085491(a(n)) = 1.

More terms from David W. Wilson, Feb 02 2006

A343621 Numbers k such that the largest Dyck path of the symmetric representation of sigma(k) does not touch the largest Dyck path of the symmetric representation of sigma(k+1).

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 47, 53, 55, 59, 63, 65, 71, 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, 251, 255

Offset: 1

Omar E. Pol, Aug 04 2021

nonn

This property of a(n) is because the symmetric representation of sigma(a(n)+1) has only one part.
All terms are odd.
First differs from A085493 at a(22).

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..1390

Cf. A000203, A085493, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239931, A239932, A239933, A239934, A262626, A343621.

(* Functions a174973Q[ ] is defined in A279029 *)
a343621[n_] := Select[Range[n], a174973Q[#+1]&]
a343621[255] (* Hartmut F. W. Hoft, Feb 20 2025 *)

a(n) = A174973(n+1) - 1.

cp's OEIS Frontend

A106431 Even elements of A085493.

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A085498 Primes p having at least one partition into distinct divisors of p + 1.

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

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A085492 Numbers n having no partition into distinct divisors of n+1.

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A085494 Numbers k having exactly one partition into distinct divisors of k+1.

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Formula

Extensions

A343621 Numbers k such that the largest Dyck path of the symmetric representation of sigma(k) does not touch the largest Dyck path of the symmetric representation of sigma(k+1).

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