A370675 Number of unordered pairs of n-digit numbers k1, k2 such that their product has the same multiset of digits as in both k1 and k2 together.

0, 7, 156, 3399, 112025, 4505706, 213002162

Offset

1,2

Comments

Since multiplication and multiset union are commutative operations, we count unordered pairs, i.e. we can assume that k1 <= k2.

The sequence is nondecreasing, since for any x,y,p such that x*y=p, x0*y0=p00.

The numbers up to n=7 were verified by at least two independent implementations.

The property of possible residues mod 3 and mod 9 for A370676 also holds for this sequence.

Links

Table of n, a(n) for n=1..7.

Danila Potapov, Numbers that flip when multiplied (in Russian).

Example

For n=2 the a(2)=7 solutions are:
  15 * 93 = 1395
  21 * 60 = 1260
  21 * 87 = 1827
  27 * 81 = 2187
  30 * 51 = 1530
  35 * 41 = 1435
  80 * 86 = 6880

Prog

(PARI) a370675(n) = {my (np=0, n1=10^(n-1), n2=10*n1-1); for (k1=n1, n2, my(s1=digits(k1)); for (k2=k1, n2, my (s2=digits(k2)); my(sp=digits(k1*k2)); if (#s1+#s2==#sp && vecsort(concat(s1, s2)) == vecsort(sp), np++))); np} \\ Hugo Pfoertner, Feb 26 2024

Crossrefs

Cf. A114258, A370676 (number of such pairs with possibly unequal number of digits).

Cf. A370678, A370679, A370680.

Sequence in context: A220367 A220311 A218449 * A197595 A197979 A203584

Adjacent sequences: A370672 A370673 A370674 * A370676 A370677 A370678

Keyword

nonn,base,more

Author

Danila Potapov, Feb 26 2024

Status

approved