A260797 -id:A260797 - cp's OEIS frontend

A098743 Number of partitions of n into aliquant parts (i.e., parts that do not divide n).

1, 0, 0, 0, 0, 1, 0, 3, 1, 3, 3, 13, 1, 23, 10, 11, 9, 65, 8, 104, 14, 56, 66, 252, 10, 245, 147, 206, 77, 846, 35, 1237, 166, 649, 634, 1078, 60, 3659, 1244, 1850, 236, 7244, 299, 10086, 1228, 1858, 4421, 19195, 243, 17660, 3244, 12268, 4039, 48341, 1819, 27675

Offset: 0

Reinhard Zumkeller, Oct 01 2004

nonn

It seems very plausible that the low and high water marks occur when n is a factorial number or a prime: see A260797, A260798.

a(A000040(n)) = A002865(n) - 1.

			7 = 2 + 2 + 3 = 2 + 5 = 3 + 4, so a(7) = 3.
a(10) = #{7+3,6+4,4+3+3} = 3, all other partitions of 10 contain at least one divisor (10, 5, 2, or 1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 300 terms from N. J. A. Sloane, next 200 terms from Reinhard Zumkeller)

Cf. A000040, A000041, A002865, A018818, A200745, A260797, A260798.

See also A057562 (relatively prime parts).

a098743 n = p [nd | nd <- [1..n], mod n nd /= 0] n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m | m < k = 0 | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 22 2011

-- with memoization
import Data.MemoCombinators (memo3, integral)
a098743 n = a098743_list !! n
a098743_list = map (\x -> pMemo x 1 x) [0..] where
   pMemo = memo3 integral integral integral p
   p   0 = 1
   p x k m | m < k        = 0
           | mod x k == 0 = pMemo x (k + 1) m
           | otherwise    = pMemo x k (m - k) + pMemo x (k + 1) m
-- Reinhard Zumkeller, Aug 08 2015

a := [1,0,0,0,0]; M:=300; for n from 5 to M do t1:={seq(i,i=1..n)}; t3 := t1 minus divisors(n); t4 := mul(1/(1-x^i), i in t3); t5 := series(t4,x,n+2); a:=[op(a), coeff(t5,x,n)]; od: a; # N. J. A. Sloane, Aug 08 2015
# second Maple program:
a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)+
      `if`(irem(m, i)=0, 0, b(n-i, min(i, n-i))))) end; b(m$2)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 11 2018

a[m_] := a[m] = Module[{b}, b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 2, 0, b[n, i-1] + If[Mod[m, i] == 0, 0, b[n-i, Min[i, n-i]]]]]; b[m, m]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n)={polcoef(1/prod(k=1, n, if(n%k, 1 - x^k, 1) + O(x*x^n)), n)} \\ Andrew Howroyd, Aug 29 2018

a(0) added and offset changed by Reinhard Zumkeller, Nov 22 2011

New wording for definition suggested by Marc LeBrun, Aug 07 2015

A260798 Number of partitions of p=prime(n) into aliquant parts (i.e., parts that do not divide p, meaning any part except 1 and p).

Original entry on oeis.org

0, 0, 1, 3, 13, 23, 65, 104, 252, 846, 1237, 3659, 7244, 10086, 19195, 48341, 116599, 155037, 356168, 609236, 792905, 1716485, 2832213, 5887815, 15116625, 23911833, 29983570, 46873052, 58443395, 90374471, 394641602, 593224103, 1082063335, 1318608063, 3477935702, 4207389268, 7398721009, 12885091292, 18555597522, 31831360281, 54145147464, 64517020844

Offset: 1

Marc LeBrun and N. J. A. Sloane, Aug 07 2015

nonn

			For n=4, the fourth prime is 7, and we see the three partitions 7=2+5=2+2+3=3+4, so a(4)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 781 terms from Reinhard Zumkeller)

This is A098743(prime(n)). Cf. A260797.

Cf. A000040, A058698.

import Data.MemoCombinators (memo2, integral)
a260798 n = a260798_list !! (n-1)
a260798_list = map (subtract 1 . pMemo 2) a000040_list where
   pMemo = memo2 integral integral p
   p _ 0 = 1
   p k m | m < k     = 0
         | otherwise = pMemo k (m - k) + pMemo (k + 1) m
-- Reinhard Zumkeller, Aug 09 2015

b:= proc(n, i) option remember; `if`(n=0 or i=2, 1-irem(n, 2),
      `if`(i<2, 0, b(n, i-1)+b(n-i, min(i, n-i))))
    end:
a:= n-> (p-> b(p, p-1))(ithprime(n)):
seq(a(n), n=1..45);  # Alois P. Heinz, Mar 11 2018

b[n_, i_] := b[n, i] = If[n == 0 || i == 2, 1 - Mod[n, 2], If[i < 2, 0, b[n, i - 1] + b[n - i, Min[i, n - i]]]];
a[n_] := b[#, # - 1]&[Prime[n]];
Table[a[n], {n, 1, 45}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

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A098743 Number of partitions of n into aliquant parts (i.e., parts that do not divide n).

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A260798 Number of partitions of p=prime(n) into aliquant parts (i.e., parts that do not divide p, meaning any part except 1 and p).

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A261121 Number of partitions of n-th primorial into aliquant parts (i.e., parts that do not divide n#).

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