A085498 -id:A085498 - 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

A085496 Number of ways to write prime(n) as sum of distinct divisors of prime(n)+1.

Original entry on oeis.org

0, 1, 1, 1, 2, 0, 1, 1, 5, 3, 1, 0, 2, 0, 10, 1, 31, 0, 0, 26, 0, 6, 23, 20, 0, 0, 1, 13, 0, 0, 1, 15, 0, 14, 9, 0, 0, 0, 190, 0, 713, 0, 42, 0, 7, 9, 0, 9, 6, 0, 6, 2148, 0, 509, 0, 120, 109, 1, 0, 0, 0, 4, 6, 100, 0, 0, 0, 0, 2, 4, 0, 21897, 1, 0, 3, 85, 79, 0, 0, 0, 19172, 0, 1130

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

a(n) = A085491(A000040(n));

a(A085498(n)) > 0.

			n=5, divisors of A000040(5)+1=11+1=12 that are not greater 11: {1,2,3,4,6}, 11=6+4+1=6+3+2, therefore a(5)=2.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

b:= proc(n, i) option remember; global l;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      `if`(l[i]>n, 0, b(n-l[i], i-1))))
    end:
a:= proc(n) global l; local p;
      forget(b);
      p:= ithprime(n);
      l:= sort([numtheory[divisors](p+1)[]]);
      b(p, nops(l)-1)
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 01 2012

Count[Total/@Subsets[Most[Divisors[Prime[#]+1]]],Prime[#]]&/@Range[90] (* Harvey P. Dale, Jan 31 2016 *)

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

A225223 Primes of the form p - 1, where p is a practical number (A005153).

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, 359, 367, 379, 383, 389, 419, 431, 439, 449, 461, 463, 467, 479, 499, 503, 509

Offset: 1

Frank M Jackson, May 02 2013

nonn

			a(5)=17 as 18 is a practical number, 18-1=17 and it is the 5th such prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005153, A085498.

PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
Select[Table[Prime[n]+1, {n, 1, 200}], PracticalQ]-1 (* using T. D. Noe's program A005153 *)

isPractical(n)={
    if(n%2,return(n==1));
    my(f=factor(n),P=1);
    for(i=1,#f[,1]-1,
        P*=sigma(f[i,1]^f[i,2]);
        if(f[i+1,1]>P+1,return(0))
    );
    n>0
};
select(p->isPractical(p+1),primes(300)) \\ Charles R Greathouse IV, May 03 2013

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

A164319 Primes p such that the sum of divisors of p+1 is larger than 2*p.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 167, 173, 179, 191, 197, 199, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 347, 349, 353, 359, 367, 379, 383, 389, 401, 419

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

For a subset of these, namely p=179, 239, 359, 419, etc, sigma(p+1) is even larger than 3*p.

			For p=3, the sum of divisors of p+1 is A000203(4)=7 > 2*3, so p=3 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A008333, A085498, A164318.

f[n_]:=Plus@@Divisors[n]; lst={};Do[p=Prime[n];If[f[p+1]>2*p,AppendTo[lst, p]],{n,6!}];lst
Select[Prime[Range[100]], DivisorSigma[1, # + 1] > 2 # &] (* G. C. Greubel, Sep 13 2017 *)

lista(nn) = forprime(p=2, nn, if (sigma(p+1) > 2*p, print1(p, ", "))); \\ Michel Marcus, Sep 13 2017

Edited by R. J. Mathar, Aug 21 2009

A346794 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) does not touch the largest Dyck path of the symmetric representation of sigma(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, 359, 367, 379, 383, 389, 419, 431, 439, 449, 461, 463, 467, 479, 499, 503, 509, 521

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.

First differs from both A085498 and A225223 at a(40).

Primes in A343621.

Cf. A000040, A000203, A085498, A174973, A196020, A225223, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239931, A239932, A239933, A239934, A262626.

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

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A085496 Number of ways to write prime(n) as sum of distinct divisors of prime(n)+1.

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

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A225223 Primes of the form p - 1, where p is a practical number (A005153).

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

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Mathematica

Formula

Extensions

A164319 Primes p such that the sum of divisors of p+1 is larger than 2*p.

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Mathematica

PARI

Extensions

A346794 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) does not touch the largest Dyck path of the symmetric representation of sigma(p+1).

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