A331045 -id:A331045 - cp's OEIS frontend

A331044 a(n) is the greatest prime number of the form floor(n/10^k) for some k >= 0, or 0 if no such prime number exists.

0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 2, 2, 2, 23, 2, 2, 2, 2, 2, 29, 3, 31, 3, 3, 3, 3, 3, 37, 3, 3, 0, 41, 0, 43, 0, 0, 0, 47, 0, 0, 5, 5, 5, 53, 5, 5, 5, 5, 5, 59, 0, 61, 0, 0, 0, 0, 0, 67, 0, 0, 7, 71, 7, 73, 7, 7, 7, 7, 7, 79, 0

Offset: 0

Rémy Sigrist, Jan 08 2020

In other words, a(n) is the greatest prime prefix of n, or 0 if every prefix of n is nonprime.

This sequence is a decimal variant of A039634.

			For n = 42:
- neither 42 nor 4 is a prime number,
- hence a(42) = 0.
For n = 290:
- 290 is not a prime number,
- 29 is a prime number,
- hence a(290) = 29.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A331045 for a similar sequence.

Cf. A039634, A202259, A320584.

A331044[n_] := NestWhile[Quotient[#, 10] &, n, # > 0 && !PrimeQ[#] &];
Array[A331044, 100, 0] (* Paolo Xausa, Nov 22 2024 *)

a(n, base=10) = while (n, if (isprime(n), return (n), n\=base)); 0

a(n) <= n with equality iff n = 0 or n is a prime number.

a(n) >= 0 with equality iff n belongs to A202259.

A331046 Numbers k such that floor(k/10^m) is a prime number for some m >= 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 83, 89, 97, 101, 103, 107, 109, 110, 111, 112, 113

Offset: 1

Rémy Sigrist, Jan 08 2020

In other words, these are the numbers with a prime prefix.
For any m > 0:
- let f(m) be the proportion of positive numbers <= 10^m belonging to this sequence,
- we have f(m) = Sum_{p < 10^m in A069090} 1/10^A055642(p),
- also f(m) <= f(m+1) <= 1,
- so {f(m)} has a limit, say F, as m tends to infinity,
- what is the value of F?

			The number 2 is prime, so every number in A217394 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A055642, A069090, A202259 (complement), A217394, A331044, A331045.

is(n,base=10) = while (n, if (isprime(n), return (1), n\=base)); return (0)

cp's OEIS Frontend

A331044 a(n) is the greatest prime number of the form floor(n/10^k) for some k >= 0, or 0 if no such prime number exists.

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