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

Author

Danila Potapov, Feb 26 2024

Keywords

• nonn
• base
• more

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.

Examples

			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
		

Links

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

Crossrefs

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

Cf. A370678, A370679, A370680.

Programs

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