A358236 Number of factorizations of n where the sum of the factors is carryfree when the addition is done in the primorial base.

1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 5, 1, 2, 1, 4, 1, 3, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 1, 5, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 9, 1, 2, 1, 4, 1, 4, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 5, 1, 1, 1, 1, 1, 8, 1, 3, 1, 4, 1, 3, 1, 2, 1

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to primorial base

Formula

For all n >= 1, a(2n-1) = 1, a(4n-2) = A358233(4n-2).

For all n >= 1, A358233(n) <= a(n) <= A001055(n).

Example

36 has in total 9 = A001055(36) factorizations:
  factors in decimal  in primorial base   Do they generate carries when summed?
  [3, 3, 2, 2]        [11, 11, 10, 10]    Yes, as A049345(3+3+2+2) = "120".
  [4, 3, 3]           [20, 11, 11]        Yes, in the least significant place.
  [6, 3, 2]           [100, 11, 10]       No, 6+3+2 = 11 = "121".
  [6, 6]              [100, 100]          No, 6+6 = 12 = "200".
  [9, 2, 2]           [111, 10, 10]       Yes, in the second place from right.
  [9, 4]              [111, 20]           Ditto.
  [12, 3]             [200, 11]           No, 12+3 = 15 = "211".
  [18, 2]             [300, 10]           No, 18+2 = 20 = "310".
  [36]                [1100]              No, as a single factor never does.
Thus only five of the sums are carryfree, and a(36) = 5.

Prog

(PARI)
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k, 2] = (f[k, 2]>=f[k, 1])); factorback(f); };
A358236(n, m=n, facs=List([])) = if(1==n, 1==A327936(factorback(apply(A276086, Vec(facs)))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A358236(n/d, d, newfacs))); (s));

Crossrefs

Cf. A001055, A049345, A276086, A327936, A358233.

Cf. also A317836.

Sequence in context: A111628 A161974 A316190 * A330738 A025921 A300978

Adjacent sequences: A358233 A358234 A358235 * A358237 A358238 A358239

Keyword

nonn,base

Author

Antti Karttunen, Nov 29 2022

Status

approved