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

Author

Antti Karttunen, Nov 29 2022

Keywords

• nonn
• base

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to primorial base

Crossrefs

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

Cf. also A317836.

Programs

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));

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).