Eduardo P. Feitosa - cp's OEIS frontend

A334537 a(n) is the largest n-digit number using each digit 0 to n-1 once, such that the numbers formed by its last k digits are divisible by k, (k = 1..n).

0, 10, 210, 3120, 43120, 543120, 6531420, 76351240, 876351240, 9876351240

Offset: 1

Eduardo P. Feitosa, May 05 2020

			a(4) = 43120 because 0, 20, 120, 3120 and 43120 are divisible by 1, 2, 3, 4 and 5, and it is the largest such number with distinct digits 0 to 4.

Cf. A147636, A158242, A220491, A331475.

A331475 a(n) is the smallest n-digit number using each digit 0 to n-1 once, such that the numbers formed by its last k digits are divisible by k, (k = 1..n).

Original entry on oeis.org

0, 10, 102, 3012, 13240, 123540, 3516240, 16453720, 123567480, 9123567480

Offset: 1

Eduardo P. Feitosa, May 03 2020

a(n) = A147636(n) for n=1, 2, 3, 9 and 10.

			a(3) = 3012 because 2, 12, 012, 3012 are divisible by 1, 2, 3, 4 and it is the least such number with distinct digits 0 to 3.

Cf. A147636, A158242, A220491.

ok[n_] := AllTrue[Range@ IntegerLength@ n, Mod[ Mod[n, 10^#], #] == 0 &]; a[n_] := SelectFirst[ FromDigits /@ Permutations[Range[0, n-1]], # >= 10^(n-1) - 1 && ok[#] &]; Array[a, 10] (* Giovanni Resta, May 04 2020 *)

A328022 Prime numbers p such that all 4 variables of the equation (p = i * q + r) are prime, with i being the index of p, q the quotient of p/i, and r the remainder of p/i.

Original entry on oeis.org

17, 41, 367, 514275529

Offset: 1

Eduardo P. Feitosa, Oct 01 2019

The other two variables in the equation result from the division of a prime p by its index i, giving quotient q and remainder r. All four of p, i, q, r are required to be prime.
For all remaining terms, q (which has become greater than 2) will be an odd prime, and q increases exponentially slowly. And when q is odd, exactly one of i and r will be odd. Consequently, a new term will only occur when r = 2 and both q and i are prime.
a(5) > 10^22, if it exists. - Giovanni Resta, Oct 02 2019

			Known values:
   n |  a(n) = p =        i *  q + r
  ===+==============================
   1 |        17 =        7 *  2 + 3
   2 |        41 =       13 *  3 + 2
   3 |       367 =       73 *  5 + 2
   4 | 514275529 = 27067133 * 19 + 2

Cf. A023144, A156152, A156153.

Select[Prime@ Range[10^5], AllTrue[Join[{#1, #2}, QuotientRemainder[#1, #2]], PrimeQ] & @@ {#, PrimePi@ #} &] (* Michael De Vlieger, Oct 01 2019 *)

lista(nn)={my(i=1); forprime(p=3, nn, i++; if(isprime(i), my(q=p\i); if(isprime(q)&&isprime(p-q*i), print1(p, ", ")) ))} \\ Andrew Howroyd, Oct 01 2019

a(4) from Andrew Howroyd, Oct 01 2019

cp's OEIS Frontend

User: Eduardo P. Feitosa

A334537 a(n) is the largest n-digit number using each digit 0 to n-1 once, such that the numbers formed by its last k digits are divisible by k, (k = 1..n).

Views

Author

Keywords

Examples

Crossrefs

A331475 a(n) is the smallest n-digit number using each digit 0 to n-1 once, such that the numbers formed by its last k digits are divisible by k, (k = 1..n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A328022 Prime numbers p such that all 4 variables of the equation (p = i * q + r) are prime, with i being the index of p, q the quotient of p/i, and r the remainder of p/i.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions