A085496 -id:A085496 - cp's OEIS frontend

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

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

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

A085499 Primes p having exactly one partition into distinct divisors of p+1.

Original entry on oeis.org

3, 5, 7, 17, 19, 31, 53, 103, 127, 271, 367, 463, 499, 859, 967, 1013, 1483, 1951, 3229, 3533, 3769, 3833, 4373, 5477, 6101, 7069, 7457, 8191, 8501, 9041, 9521, 11527, 11621, 11777, 13121, 14551, 17791, 20071, 21943, 23167, 25471, 29311, 33619, 36979, 44491, 45667

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..68

Subsequence of A085498 and of A085494.

Cf. A085496.

seqQ[p_] := Module[{d = Most[Divisors[p+1]]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, p}], p]  == 1]; Select[Range[1000], PrimeQ[#] && seqQ[#] &] (* Amiram Eldar, Jan 13 2020 *)

A085496(a(n)) = 1.

a(14)-a(38) from Alois P. Heinz, Apr 30 2012

a(39)-a(46) from Amiram Eldar, Jan 13 2020

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

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A085499 Primes p having exactly one partition into distinct divisors of p+1.

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