A046641 -id:A046641 - cp's OEIS frontend

A145771 Record values in A046641.

1, 2, 3, 11, 14, 21, 28, 54, 58, 66, 94, 129, 178, 340, 348, 372, 465, 960, 968, 995, 1453, 2037, 2088, 2309, 2620, 3978, 4873, 5307, 8149, 9441, 9590, 10669, 10919, 15687, 18908, 23136, 24023, 27058, 27564, 29623, 37756, 38010, 42187, 56154, 65761

Offset: 1

M. F. Hasler, Oct 18 2008

nonn

RECORDS transform of A046641: values A046641(k) larger than all A046641(m) for mA046641.

A145772 gives corresponding indices of A046641 (cf. formula).

a(n) = A046641( A145772(n))

A145772 Indices of record values in A046641.

Original entry on oeis.org

1, 2, 3, 4, 9, 12, 13, 17, 20, 32, 34, 39, 51, 54, 102, 126, 144, 163, 247, 258, 306, 326, 489, 563, 613, 657, 1112, 1123, 1334, 1409, 1913, 2494, 2518, 2703, 3548, 3672, 3798, 4481, 5905, 6014, 6186, 6581, 6759, 8543, 8893, 9453, 11892, 12708, 15346, 17183

Offset: 1

M. F. Hasler, Oct 18 2008

nonn

Indices k such that A046641(k) > A046641(m) for all m < k. Computed using the C program for A046641.

A145771(n) = A046641( a(n)) gives the corresponding record values.

i=1; c=1; m=0; while [ $c -le 50 ]; do if [ `./A046641 $i` -gt $m ]; then m=`./A046641 $i`; echo -n $i", "; c=`expr $c + 1`; fi; i=`expr $i + 1`; done

A091690 Smallest integer m such that the n-th prime A000040(n) divides partition number A000041(m).

Original entry on oeis.org

2, 3, 4, 5, 6, 28, 54, 20, 32, 26, 44, 86, 35, 27, 56, 52, 54, 115, 47, 43, 46, 123, 29, 25, 44, 13, 78, 96, 320, 63, 40, 73, 66, 60, 333, 203, 116, 960, 22, 79, 368, 310, 75, 41, 118, 59, 100, 64, 55, 115, 316, 353, 46, 23, 197, 236, 193, 871, 179, 37, 95, 183, 233

Offset: 1

Reinhard Zumkeller, Jan 29 2004

nonn

See A091689 for comment, links and example: A091689(n) = A000041(a(n)).

			a(10) = 26 as the 10th prime divides the 26th partition number, i.e., prime(10) = 29 | 2436 = A000041(26) and no smaller partition number is a multiple of 29. - _David A. Corneth_, Jan 05 2023

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2262 terms from M. F. Hasler)

forprime(p=2,400,n=2:while(numbpart(n)%p,n=n+1):print1(n","))

first(n) = { my(res = vector(n, i, oo), todo = [1..n], pr = Set(primes(n)), prmap = Map()); for(i = 1, n, mapput(prmap, pr[i], todo[i]) ); for(i = 1, oo, c = numbpart(i); j = 1; while(j <= #pr && pr[j] <= c, if(c % pr[j] == 0, res[mapget(prmap, pr[j])] = i; pr = setminus(pr, Set(pr[j])); ); j++ ); if(#pr == 0, return(res) ) ); res } \\ David A. Corneth, Jan 05 2023

i=2; c=1; while [ $i -le 20000 ]; do if [ `factor $i | wc -w` = 2 ]; then awk '/^'$i' / {print "'$c' " $2;exit}' b046641.txt; let c+=1; if [ $i = 2 ]; then i=1; fi; fi; let i+=2; done # M. F. Hasler, Oct 20 2008

a(n) = A046641(A000040(n)). - M. F. Hasler, Oct 20 2008

Corrected by Ralf Stephan, Jul 27 2004

A145523 Least integer k > 0 such that A000041(k) is divisible by 2^n.

Original entry on oeis.org

1, 2, 11, 11, 15, 66, 66, 96, 96, 96, 96, 96, 3693, 15005, 18978, 18978, 18978, 43002, 55943, 972190, 1151214, 2799146, 15519397, 15519397, 15519397, 122101417, 210553237, 289585489, 473093534

Offset: 0

M. F. Hasler, Oct 12 2008

The requirement a(n) > 0 is somewhat arbitrary, chosen for agreement with A046641; a(n) >= 0 would have been possible, too, yielding a(0)=0.

a(29) > 10^9.

Cf. A000041, A046641, A000123, A046063.

i=1; c=1; while [ $c -le 21 ]; do echo -n `./A046641 $i`", "; c=`expr $c + 1`; i=`expr $i + $i`; done # M. F. Hasler, Oct 18 2008

a(n) = A046641(2^n).

More terms from M. F. Hasler, Oct 18 2008

a(22)-a(28) from Max Alekseyev, Dec 16 2011

A235704 a(n) is the smallest number k such that n*k is a partition number.

Original entry on oeis.org

1, 1, 1, 14, 1, 5, 1, 7, 15, 3, 1, 66, 286, 3, 1, 11, 22715, 44, 33, 35761, 2, 1, 363, 33, 63, 143, 5, 2, 84, 1, 2425, 72610, 7, 2725580, 11, 22, 926026, 3283, 123981330, 58088, 363, 1, 70, 4, 3, 176484, 11209, 85166, 10, 141790, 11209835405

Offset: 1

Omar E. Pol, Jan 14 2014

nonn

			For n = 4, a(4) = 14 because if 1 <= k <= 13 we have that 4*k is not a partition number, but if k = 14 then 4*14 = 56 and 56 is the number of partitions of 11.

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

Cf. A000041, A046641, A072871, A213179, A213365, A216258, A217725, A217726, A222175, A222178, A222179, A225317, A225361.

a[n_] := PartitionsP[NestWhile[(# + 1)&, 1, Mod[PartitionsP@ #, n] > 0 &]]/n; Array[a,51] (* Giovanni Resta, Jan 15 2014 *)

a(n) = 1 iff n is a partition number.

a(n) = A072871(n)/n.

A324060 Smallest positive index m such that A000009(m) is divisible by n.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 33, 9, 13, 10, 14, 11, 23, 33, 12, 16, 31, 13, 17, 31, 40, 14, 18, 27, 65, 23, 15, 33, 52, 32, 42, 16, 26, 31, 41, 38, 28, 17, 32, 37, 112, 41, 149, 79, 35, 18, 121, 27, 203, 65, 49, 23, 40, 19, 26, 33, 87, 52, 50, 41, 24, 42, 41, 20, 32, 79

Offset: 1

Robert G. Wilson v, Jul 31 2019

nonn

Analogous to A046641.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000009, A046641.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) local k; for k while irem(b(k), n)>0 do od; k end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 17 2019

t[] := 0; k = 1; f[n] := Block[{p = PartitionsQ@n, lst}, lst = Select[Range@100, Mod[p, #] == 0 &]; If[t[#] == 0, t[#] = n] & /@ lst]; While[k < 1001, f[k]; k++]; t@# & /@ Range@100

A086146 a(n) is the smallest k>=n such that the number of partitions of k is a multiple of n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 3, 11, 7, 9, 10, 11, 14, 19, 12, 21, 28, 19, 24, 66, 54, 21, 20, 58, 24, 25, 32, 70, 44, 28, 39, 55, 91, 97, 44, 66, 35, 94, 39, 80, 86, 47, 129, 66, 45, 75, 100, 58, 129, 75, 56, 70, 68, 74, 178, 62, 66, 340, 58, 75, 209, 97, 93, 124, 115, 101, 138, 66, 84, 75, 111, 94

Offset: 1

Jim Nastos, Aug 25 2003

nonn

I do not know if a(n) exists for all n. First term which is currently unknown is a(2219) (a(2219) > 11600, while a(2218) = 2602).

			a(4) is 11 because 11 is the smallest number for which P(11) is divisible by 4, where P() is the partition function.

Cf. A046641.

for i from 2 while i < 30000 do for j from i while j < 1000000000 do c := numbpart(j); if (c mod i = 0) then print(i,j); break; end if; end do; end do;
# alternative
A086146 := proc(n)
    local k ;
    for k from n do
        if combinat[numbpart](k) mod n =0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Apr 22 2013

kmax = 10^9;
a[n_] := Module[{k}, For[k = n, k <= kmax, k++, If[Divisible[ PartitionsP[k], n], Return[k]]]] /. Null -> -1;
Table[a[n], {n, 1, 68}] (* Jean-François Alcover, Jul 08 2024 *)

More terms from Ray Chandler, Sep 16 2003

A145524 Least integer k>0 such that A000041(k) is divisible by 10^n.

Original entry on oeis.org

1, 9, 74, 449, 599, 11224, 55374, 3099324, 3099324

Offset: 0

M. F. Hasler, Oct 12 2008

The requirement a(n)>0 is somewhat arbitrary, chosen for agreement with A046641 ; a(n)>=0 would have been possible, too, yielding a(0)=0.

a(9) > 10^7. [From Max Alekseyev, Oct 18 2008]

Cf. A000041, A046641, A145523

a(n) = A046641(10^n)

a(6)..a(8) from Max Alekseyev, Oct 18 2008

A307547 a(n) is the smallest divisor of the partition number P(n) = A000041(n) not already in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 6, 14, 4, 77, 101, 9, 8, 21, 27, 35, 10, 19, 12, 167, 251, 25, 89, 28, 43, 13, 55, 467, 311, 23, 49, 1231, 33, 17977, 281, 121, 45, 42, 193, 2417, 71, 31, 41, 73, 38, 7013, 275, 9283, 363, 53, 63, 17, 142, 47, 102359, 20, 44, 139

Offset: 1

Rémy Sigrist, Jul 27 2019

nonn

Provided A046641(m) is defined for any number m > 0, this sequence is a permutation of the natural numbers.

			The first terms, alongside the divisors of P(n), are:
  n   a(n)  div(P(n))
  --  ----  --------------------
   1     1  (1)
   2     2  (1, 2)
   3     3  (1, 3)
   4     5  (1, 5)
   5     7  (1, 7)
   6    11  (1, 11)
   7    15  (1, 3, 5, 15)
   8    22  (1, 2, 11, 22)
   9     6  (1, 2, 3, 5, 6, 10, 15, 30)
  10    14  (1, 2, 3, 6, 7, 14, 21, 42)

Rémy Sigrist, Table of n, a(n) for n = 1..5000
Rémy Sigrist, PARI program for A307547

Cf. A000041, A046641, A049575, A309200.

PARI
```
See Links section.
```

a(A049575(k)) = A049575(k).

cp's OEIS Frontend

A145771 Record values in A046641.

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A145772 Indices of record values in A046641.

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sh

A091690 Smallest integer m such that the n-th prime A000040(n) divides partition number A000041(m).

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PARI

PARI

sh

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A145523 Least integer k > 0 such that A000041(k) is divisible by 2^n.

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sh

Formula

Extensions

A235704 a(n) is the smallest number k such that n*k is a partition number.

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Mathematica

Formula

A324060 Smallest positive index m such that A000009(m) is divisible by n.

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Maple

Mathematica

A086146 a(n) is the smallest k>=n such that the number of partitions of k is a multiple of n, or -1 if no such k exists.

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Examples

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Programs

Maple

Mathematica

Extensions

A145524 Least integer k>0 such that A000041(k) is divisible by 10^n.

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Author

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Comments

Crossrefs

Formula

Extensions

A307547 a(n) is the smallest divisor of the partition number P(n) = A000041(n) not already in the sequence.

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