A236103 -id:A236103 - cp's OEIS frontend

A236108 Nonprimes whose proper divisors are partition numbers.

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 303, 505, 707, 1111, 10201, 35954, 53931, 89885, 125839, 197747, 1815677, 21239726, 31859589, 53099315, 74339041, 116818493, 323172529, 1072606163, 13241661778, 19862492667, 33104154445, 46345816223, 72829139779

Offset: 1

Omar E. Pol, Jan 22 2014

nonn

Known terms are squares of A049575 or products of 2 distinct terms of A049575. - Michel Marcus, Jan 25 2023

This conjecture holds for terms <= 10^16. - David A. Corneth, Jan 25 2023

			10 is in the sequence because 10 is a nonprime number and the proper divisors of 10 are 1, 2, 5, which are also partition numbers.

David A. Corneth, Table of n, a(n) for n = 1..81

Cf. A000041, A018252, A049575, A167392.

Cf. A236102, A236103, A236105, A236107, A236110.

isA000041 := proc(n)
    local k,P;
    for k from 1 do
        P := combinat[numbpart](k) ;
        if P > n then
            return false;
        elif P = n then
            return true ;
        end if;
    end do:
end proc:
isA236108 := proc(n)
    local pdvs,d ;
    if n =1 or isprime(n) then
        return false;
    end if;
    pdvs := numtheory[divisors](n) minus {n} ;
    for d in pdvs do
        if not isA000041(d) then
            return false;
        end if;
    end do:
    return true;
end proc:
for n from 1 to 300000 do
    if isA236108(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 29 2014

partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];
Select[Range[2, 10000],
 If[! PrimeQ[#],
ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* Julien Kluge, Dec 03 2016 *)

a(17)-a(26) from R. J. Mathar, Jan 29 2014
a(27)-a(32) from Jon E. Schoenfield, Feb 05 2014
a(33)-a(34) from Michel Marcus, Jan 24 2023
More terms from David A. Corneth, Jan 25 2023

A236102 Numbers whose divisors are partition numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 77, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939

Offset: 1

Omar E. Pol, Jan 21 2014

nonn

By definition all terms are partition numbers.
All members of A049575 are in this sequence.
Conjecture: the only composite numbers in this sequence are 15, 22, and 77. - Jon E. Schoenfield, Feb 05 2014

			15 is in the sequence because the divisors of 15 are 1, 3, 5, 15, which are also partition numbers.

Cf. A000041, A027750, A049575, A236103, A236105, A236107, A236108, A236110, A236111.

More terms from Jon E. Schoenfield, Feb 05 2014

A236105 Numbers whose proper divisors are partition numbers.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 202

Offset: 1

Omar E. Pol, Jan 21 2014

nonn

All prime numbers are in the sequence.

All terms of A236102 are in this sequence, except the first term.

			10 is in the sequence because the proper divisors of 10 are 1, 2, 5, which are also partition numbers.

Cf. A000040, A000041, A236102, A236103, A236107, A236108, A236110.

A236107 Nonprimes whose divisors are partition numbers.

Original entry on oeis.org

1, 15, 22, 77

Offset: 1

Omar E. Pol, Jan 22 2014

By definition all terms are partition numbers.

Conjecture: no terms exist beyond 77. - Jon E. Schoenfield, Feb 05 2014

			15 is in the sequence because 15 is a nonprime number and the divisors of 15 are 1, 3, 5, 15, which are also partition numbers.

Cf. A000041, A018252, A038753, A236102, A236103, A236105, A236108, A236110.

nmax = 1000;
pp = PartitionsP[Range[nmax]];
selQ[n_] := Module[{dd = Divisors[n]}, Intersection[pp, dd] == dd];
Select[Range[nmax], !PrimeQ[#] && selQ[#]&] (* Jean-François Alcover, Apr 09 2020 *)

A236110 Smallest number with the property that exactly n of its divisors are partition numbers.

Original entry on oeis.org

1, 2, 6, 15, 42, 30, 270, 210, 462, 1848, 3696, 11088, 2310, 9240, 18480, 55440, 83160, 166320, 498960, 2494800, 17463600, 331808400, 4418290800

Offset: 1

Omar E. Pol, Jan 22 2014

			a(3) = 6 because 6 is the smallest number with the property that exactly three of its divisors are partition numbers. The divisors of 6 are 1, 2, 3, 6, and 1, 2, 3 are also partition numbers.
a(5) = 42 because 42 is the smallest number with the property that exactly five of its divisors are partition numbers. The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42, and 1, 2, 3, 7, 42 are members of A000041.

Cf. A000041, A236102, A236103, A236105, A236107, A236108, A236111.

a(12) and a(15)-a(18) from Alois P. Heinz, Jan 22 2014
a(19)-a(22) from Giovanni Resta, Feb 06 2014
a(23) from Amiram Eldar, Jun 23 2023

A239930 Number of distinct quarter-squares dividing n.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 1, 4, 1, 4, 1, 4, 1, 2, 1, 5, 2, 2, 2, 3, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 6, 2, 3, 1, 3, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 2, 5, 1, 3, 1, 3, 1, 2, 1, 8, 1, 2, 2, 3, 1, 3, 1, 5, 3, 2, 1, 6, 1, 2, 1, 3, 1, 6, 1, 3, 1, 2, 1, 6, 1, 3, 2, 6, 1, 3, 1, 3, 1, 2, 1, 7, 1, 3

Offset: 1

Omar E. Pol, Jun 19 2014

nonn

For more information about the quarter-squares see A002620.

			For n = 12 the quarter-squares <= 12 are [0, 0, 1, 2, 4, 6, 9, 12]. There are five quarter-squares that divide 12; they are [1, 2, 4, 6, 12], so a(12) = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Table of divisors.

Cf. A000005, A001221, A001511, A002620, A005086, A006519, A007862, A013661, A027750, A046951, A147645, A236103.

Cf. A240025.

a239930 = sum . map a240025 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

isA002620 := proc(n)
    local k,qsq ;
    for k from 0 do
        qsq := floor(k^2/4) ;
        if n = qsq then
            return true;
        elif qsq > n then
            return false;
        end if;
    end do:
end proc:
A239930 := proc(n)
    local a,d ;
    a :=0 ;
    for d in numtheory[divisors](n) do
        if isA002620(d) then
            a:= a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 03 2014

qsQ[n_] := AnyTrue[Range[Ceiling[2 Sqrt[n]]], n == Floor[#^2/4]&]; a[n_] := DivisorSum[n, Boole[qsQ[#]]&]; Array[a, 110] (* Jean-François Alcover, Feb 12 2018 *)

a(n) = sumdiv(n, d, issquare(d) + issquare(4*d + 1)); \\ Amiram Eldar, Dec 31 2023

a(n) = Sum_{k=1..A000005(n)} A240025(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) + 1 = A013661 + 1 = 2.644934... . - Amiram Eldar, Dec 31 2023

A236111 Numbers that set a record for the number of divisors that are partition numbers.

Original entry on oeis.org

1, 2, 6, 15, 30, 210, 462, 1848, 2310, 9240, 18480, 55440, 83160, 166320, 498960, 2494800, 17463600, 331808400, 4418290800

Offset: 1

Omar E. Pol, Jan 22 2014

Where records occur in A236103.

Cf. A000041, A236102, A236103, A236105, A236107, A236108, A236110.

More terms from Jon E. Schoenfield, Feb 08 2014

a(18)-a(19) from Amiram Eldar, Jun 23 2023

A244964 Number of distinct generalized pentagonal numbers dividing n.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 4, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 2, 2, 2, 3, 1, 2, 1, 6, 1, 3, 1, 2, 3, 2, 3, 3, 1, 4, 1, 2, 1, 4, 2, 2, 1, 3, 1, 4, 2, 3, 1, 2, 2, 3, 1, 3, 1, 4, 1, 3, 1, 3, 5

Offset: 1

Omar E. Pol, Jul 10 2014

nonn

For more information about the generalized pentagonal numbers see A001318.

			For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.

a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ Amiram Eldar, Dec 31 2023

From Amiram Eldar, Dec 31 2023: (Start)
a(n) = Sum_{d|n} A080995(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)

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A236108 Nonprimes whose proper divisors are partition numbers.

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A236102 Numbers whose divisors are partition numbers.

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A236105 Numbers whose proper divisors are partition numbers.

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A236107 Nonprimes whose divisors are partition numbers.

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A236110 Smallest number with the property that exactly n of its divisors are partition numbers.

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A239930 Number of distinct quarter-squares dividing n.

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A236111 Numbers that set a record for the number of divisors that are partition numbers.

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A244964 Number of distinct generalized pentagonal numbers dividing n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula