A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

111, 101, 87, 23, 21, 83, 21, 21, 27, 101, 87, 29, 23, 21, 33, 21, 83, 21, 39, 101, 87, 23, 21, 21, 21, 83, 101, 87, 59, 23, 21, 99, 57, 21, 27, 21, 101, 87, 29, 23, 21, 83, 69, 21, 71, 21, 101, 87, 33, 23, 21, 21, 83, 21, 101, 87, 23, 21, 27, 21, 39, 21, 83, 101, 87, 29, 23, 21, 21, 107, 21, 101

Offset: 1

Author

Bernard Schott, Feb 07 2022

Keywords

• nonn
• base

Comments

Except for a(1) = 111, which is unique, all terms appear infinitely many times and belong to this set of fifteen integers: {21, 23, 27, 29, 33, 39, 57, 59, 69, 71, 83, 87, 99, 101, 107}; see A329914.

The corresponding indices where these integers appear the first time are respectively: 5, 4, 9, 12, 15, 19, 33, 29, 43, 45, 6, 3, 32, 2, 70.

Examples

			A116436(1) = 1 and 111 * 1 = 1.1.1, hence a(1) = 111.
A116436(2) = 11 and 101 * 11 = 1.11.1, hence a(2) = 101.
A116436(32) = 112359550561797752809 and 99 * 112359550561797752809 = 1.112359550561797752809.1 hence a(32) = 99 (see Penguin reference).
		

References

• David Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.

Crossrefs

Cf. A116436, A329914, A329915.

M such that k*M=1M1 for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), A351238 (k=87), A351239 (k=101).

Programs

PARI

A116436(k) = {local(l, d, lb, ub); d=divisors(10^(k+1)+1); l=[]; lb=10^(k-1); ub=10*lb; for(i=1, #d, if(d[i]>=lb&&d[i]A116436
a(n) = {my(v6=[], i=1); while (#v6 < n, v6 = concat(v6, A116436(i)); i++); my(x= v6[n]); my(k=1); while (eval(Str(1, x, 1)) % x, k++); eval(Str(1, x, 1))/x;} \\ Michel Marcus, Feb 10 2022

Formula

a(n) * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.