A202259 -id:A202259 - cp's OEIS frontend

A276473 Number of terms of A202259 with precisely n digits.

6, 38, 320, 2819, 25668, 237586, 2224574, 21007948, 199725336, 1908845614, 18321586810, 176478166845

Offset: 1

Barry Carter, Sep 12 2016

When generating n random digits in order, number of ways to fail to generate a prime at any step.

a(n+1) >= 6*a(n), for n > 1, since any term of A202259 counted in a(n) may be extended with 0, 2, 4, 5, 6, or 8. - Michael S. Branicky, Nov 18 2024

Barry Carter, Mathematica work on probability of generating member of A202259 after generating n random numbers in order
Michael S. Branicky, Python program for OEIS A276473

Cf. A202259.

a(9)-a(11) from Michael S. Branicky, Nov 13 2024

a(12) from Michael S. Branicky, Nov 18 2024

A062115 Numbers with no prime substring in their decimal expansion.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 14, 16, 18, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 81, 84, 86, 88, 90, 91, 94, 96, 98, 99, 100, 104, 106, 108, 140, 144, 146, 148, 160, 164, 166, 168, 169, 180, 184, 186, 188, 400, 404, 406, 408, 440, 444, 446, 448, 460, 464, 466

Offset: 1

Erich Friedman, Jun 28 2001

This is a 10-automatic sequence, a consequence of the finitude of A071062. - Charles R Greathouse IV, Sep 27 2011

Subsequence of A202259 (right-truncatable nonprimes). Supersequence of A202262 (composite numbers in which all substrings are composite), A202265 (nonprime numbers in which all substrings and reversal substrings are nonprimes). - Jaroslav Krizek, Jan 28 2012

			25 is not included because 5 is prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jeffrey Shallit, Minimal primes, Journal of Recreational Mathematics 30:2 (1999-2000), pp. 113-117.
Index entries for 10-automatic sequences.

Subsequence of A084984. [Arkadiusz Wesolowski, Jul 05 2011]
Cf. A071062.
Cf. A163753 (complement).

a062115 n = a062115_list !! (n-1)
a062115_list = filter ((== 0) . a039997) a084984_list
-- Reinhard Zumkeller, Jan 31 2012

from sympy import isprime
def ok(n):
    s = str(n)
    ss = (int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))
    return not any(isprime(k) for k in ss)
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, May 02 2023

# faster for initial segment of sequence; uses ok, import above
from itertools import chain, count, islice, product
def agen(): # generator of terms
    yield from chain((0,), (int(t) for t in (f+"".join(r) for d in count(1) for f in "14689" for r in product("014689", repeat=d-1)) if ok(t)))
print(list(islice(agen(), 100))) # Michael S. Branicky, May 02 2023

A039997(a(n)) = 0. - Reinhard Zumkeller, Jul 16 2007
From Charles R Greathouse IV, Mar 23 2010: (Start)
a(n) = O(n^(log_4 10)) = O(n^1.661) because numbers containing only 0,4,6,8 are in this sequence.
a(n) = Omega(n^(log_13637 1000000)) = Omega(n^1.451) for similar reasons; this bound can be increased by considering longer sequences of digits. (End)

Offset corrected by Arkadiusz Wesolowski, Jul 27 2011

A202260 Right-truncatable composites: every decimal prefix is a composite number.

Original entry on oeis.org

4, 6, 8, 9, 40, 42, 44, 45, 46, 48, 49, 60, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 400, 402, 403, 404, 405, 406, 407, 408, 420, 422, 423, 424, 425, 426, 427, 428, 429, 440, 441, 442, 444, 445, 446, 447, 448

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A202259.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A012883 (right-truncatable noncomposites), A202259 (right-truncatable nonprimes), A024770 (right-truncatable primes).

Cf. A254750, A254752, A254754, A254755 (left-truncatable composites).

isComposite(n) = (n>2)&&(!isprime(n));
isRightTruncatableComposite(n,b=10) = {my(k=b);if(!isComposite(n),return(0););while(n\k>0,if(!isComposite(n\k),return(0););k*=b);return(1);} \\ Stanislav Sykora, Feb 15 2015

A202265 Nonprime numbers in which all substrings and reversal substrings are nonprimes.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 18, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 81, 84, 86, 88, 90, 94, 96, 99, 100, 108, 180, 184, 186, 400, 404, 406, 408, 440, 444, 446, 448, 460, 464, 466, 468, 469, 480, 481, 484, 486, 488, 490, 494, 496, 600, 604, 606, 608, 609

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A062115, A202259.

Supersequence of A202266.

			All substrings and reversal substrings of 186 are nonprimes: 1, 6, 8, 18, 68, 81, 86, 186, 681.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A068669 (noncomposite numbers in which all substrings and reversal substrings are noncomposites), A202266 (composite numbers in which all substrings and reversal substrings are composites).

Select[Range[0,1000],NoneTrue[Union[Flatten[{#,IntegerReverse[#]}&/@Flatten[Table[ FromDigits/@Partition[IntegerDigits[#],d,1],{d,IntegerLength[#]}]]]],PrimeQ]&] (* Harvey P. Dale, Jul 30 2024 *)

Corrected (498 deleted) by Harvey P. Dale, Jul 30 2024

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.

Original entry on oeis.org

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.

A329428 Starting from n: as long as the decimal representation starts with a prime number, replace the largest such prefix with the index of the corresponding prime number; a(n) corresponds to the final value.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 6, 4, 8, 9, 10, 1, 12, 6, 14, 15, 16, 4, 18, 8, 10, 1, 12, 9, 14, 15, 16, 4, 18, 10, 10, 1, 12, 9, 14, 15, 16, 12, 18, 10, 40, 6, 42, 14, 44, 45, 46, 15, 48, 49, 10, 1, 12, 16, 14, 15, 16, 12, 18, 4, 60, 18, 62, 63, 64, 65, 66, 8, 68, 69, 40

Offset: 0

Rémy Sigrist, Nov 30 2019

As long as we have a number whose decimal representation is the concatenation of a prime number, say the k-th prime number, and a minimal string possibly empty or with leading zeros, say v, we replace this number with the concatenation of k and v; eventually none of the prefixes will be a prime number.

			For n = 991:
- let pi denote A000720,
- 991 gives pi(991) = 167,
- 167 gives pi(167) = 39,
- 39 gives pi(3) followed by 9 = 29,
- 29 gives pi(29) = 10,
- neither 1 nor 10 is a prime number, so a(991) = 10.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

See A327539 for similar sequences.

Cf. A007097, A202259.

Array[Which[# == 0, 0, # == 1, 1, True, FixedPoint[If[! IntegerQ@ #1, FromDigits[#2], FromDigits[Join @@ {IntegerDigits@ PrimePi@ #1, Drop[#2, IntegerLength@ #1]}]] & @@ {SelectFirst[Table[FromDigits[#[[1 ;; i]]], {i, Length@ #, 1, -1}], PrimeQ], #} &@ IntegerDigits[#] &, #]] &, 71, 0] (* Michael De Vlieger, Dec 01 2019 *)
f[n_]:=Block[{p,r,d = IntegerDigits@ n,v=n}, Do[{p,r}= FromDigits/@ TakeDrop[d,k]; If[PrimeQ@ p, v=PrimePi[p] 10^(Length[d]-k)+r; Break[]],{k, Length@d, 1, -1}]; v]; a[n_]:= FixedPoint[f, n]; Array[a,71,0] (* Giovanni Resta, Dec 02 2019 *)

t(n) = if (n==0, 0, isprime(n), primepi(n), 10*t(n\10)+(n%10))
a(n) = while (n!=n=t(n),); n

a(n) <= n with equality iff n belongs to A202259.
a(prime(k)) = a(k) for any k > 0 where prime(k) denotes the k-th prime number.
a(10*k + v) = 10*a(k) + v for any k > 0 and v in {0, 2, 4, 5, 6, 8}.
a(A007097(k)) = 1 for any k >= 0.

A331045 a(n) is the least prime number of the form floor(n/10^k) for some k >= 0, or 0 if no such prime number exists.

Original entry on oeis.org

0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 41, 0, 43, 0, 0, 0, 47, 0, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 61, 0, 0, 0, 0, 0, 67, 0, 0, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 0, 0, 0, 83

Offset: 0

Rémy Sigrist, Jan 08 2020

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

This sequence is a variant of A331044.

			For n = 23:
- 2 is a prime number,
- hence a(23) = 2.

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

Cf. A069090, A202259, A331044.

a(n, base=10) = my (d=digits(n, base), p=0); for (k=1, #d, if (isprime(p=base*p+d[k]), return (p))); return (0)

a(n) <= n with equality iff n = 0 or n belongs to A069090.
a(n) >= 0 with equality iff n belongs to A202259.
a(n) <= A331044(n).

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

A276473 Number of terms of A202259 with precisely n digits.

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A062115 Numbers with no prime substring in their decimal expansion.

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A202260 Right-truncatable composites: every decimal prefix is a composite number.

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A202265 Nonprime numbers in which all substrings and reversal substrings are nonprimes.

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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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A329428 Starting from n: as long as the decimal representation starts with a prime number, replace the largest such prefix with the index of the corresponding prime number; a(n) corresponds to the final value.

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A331045 a(n) is the least 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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A331046 Numbers k such that floor(k/10^m) is a prime number for some m >= 0.

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