A091089 -id:A091089 - cp's OEIS frontend

A365812 The smallest prime that is formed from the results in A091089.

16557103, 16718111, 26378113, 35921107, 46524113, 46867111, 50018107, 55187101, 58374101, 58452109, 60850129, 63714103, 68771149, 71299157, 78035107, 78269101, 81661121, 84213167, 89052101, 90157127, 95490103, 97080107, 102892117

Offset: 1

Robert Price, Sep 19 2023

Robert Price, Table of n, a(n) for n = 1..23419

Cf. A091089, A365813.

A068695 Smallest number (not beginning with 0) that yields a prime when placed on the right of n.

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 1, 3, 7, 1, 3, 7, 1, 9, 1, 3, 3, 1, 1, 11, 1, 3, 3, 1, 1, 3, 1, 1, 3, 7, 1, 17, 1, 7, 3, 7, 3, 3, 7, 1, 9, 1, 1, 3, 7, 1, 9, 7, 1, 3, 13, 1, 23, 1, 7, 3, 1, 7, 3, 1, 3, 11, 1, 1, 3, 1, 3, 3, 1, 1, 9, 7, 3, 3, 1, 1, 3, 7, 7, 9, 1, 1, 9, 19, 3, 3, 7, 1, 23, 7, 1, 9, 7, 1, 3, 7, 1, 3, 1, 9, 3, 1

Offset: 1

Amarnath Murthy, Mar 03 2002

Max Alekseyev (see link) shows that a(n) always exists. Note that although his argument makes use of some potentially large constants (see the comments in A060199), the proof shows that a(n) exists for all n. - N. J. A. Sloane, Nov 13 2020

Many numbers become prime by appending a one-digit odd number. Some numbers (such as 20, 32, 51, etc.) require a 2-digit odd number (A032352 has these). In the first 100000 values of n there are only 22 that require a 3-digit odd number (A091089). There probably are some values that require odd numbers of 4 or more digits, but these are likely to be very large. - Chuck Seggelin, Dec 18 2003

			a(20)=11 because 11 is the minimum odd number which when appended to 20 forms a prime (201, 203, 205, 207, 209 are all nonprime, 2011 is prime).

David W. Wilson, Table of n, a(n) for n=1..10000
Max Alekseyev, Given n, there is a k such that the concatenation n||k is a prime, Nov 09 2020
Index entries for primes involving decimal expansion of n

Cf. A032352 (a(n) requires at least a 2 digit odd number), A091089 (a(n) requires at least a 3 digit odd number).

Cf. also A060199, A228325, A336893.

d[n_]:=IntegerDigits[n]; t={}; Do[k=1; While[!PrimeQ[FromDigits[Join[d[n],d[k]]]],k++]; AppendTo[t,k],{n,102}]; t (* Jayanta Basu, May 21 2013 *)
mon[n_]:=Module[{k=1},While[!PrimeQ[n*10^IntegerLength[k]+k],k+=2];k]; Array[mon,110] (* Harvey P. Dale, Aug 13 2018 *)

A068695=n->for(i=1,oo,ispseudoprime(eval(Str(n,i)))&&return(i)) \\ M. F. Hasler, Oct 29 2013

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(1) if isprime(int(str(n)+str(k))))
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Oct 18 2022

More terms from Chuck Seggelin, Dec 18 2003
Entry revised by N. J. A. Sloane, Feb 20 2006
More terms from David Wasserman, Feb 14 2006

A365813 Numbers which form a prime by appending a 3-digit number and form no primes by appending 1 digit or 2 digits.

Original entry on oeis.org

16718, 26378, 46524, 55187, 58374, 58452, 60850, 71299, 78035, 78269, 84213, 89052, 95490, 97080, 115558, 121097, 128192, 131194, 133547, 140018, 140872, 145319, 147234, 149897, 149967, 151169, 155849, 158007, 158561, 159134, 167715, 169884, 170760, 176101, 177666, 186876, 189383, 198170, 198476

Offset: 1

Robert Price, Sep 19 2023

Similar to A091089 but without the restriction of eliminating appending of a number beginning with zero.

Robert Price, Table of n, a(n) for n = 1..13412

Cf. A091089. A365812.

A091088 a(n) is the minimum odd number that must be appended to n to form a prime.

Original entry on oeis.org

3, 1, 3, 1, 1, 3, 1, 1, 3, 7, 1, 3, 7, 1, 9, 1, 3, 3, 1, 1, 11, 1, 3, 3, 1, 1, 3, 1, 1, 3, 7, 1, 17, 1, 7, 3, 7, 3, 3, 7, 1, 9, 1, 1, 3, 7, 1, 9, 7, 1, 3, 13, 1, 23, 1, 7, 3, 1, 7, 3, 1, 3, 11, 1, 1, 3, 1, 3, 3, 1, 1, 9, 7, 3, 3, 1, 1, 3, 7, 7, 9, 1, 1, 9, 19, 3, 3, 7, 1, 23, 7, 1, 9, 7, 1, 3, 7, 1, 3, 1, 9, 3

Offset: 0

Chuck Seggelin, Dec 18 2003

This is really a duplicate of A068695. See that entry for existence proof. - N. J. A. Sloane, Nov 07 2020
Note that of course a(n) is not allowed to begin with 0.
Many numbers become prime by appending a one-digit odd number. Some numbers (such as 20, 32, 51, etc.) require a 2 digit odd number (A032352 has these). In the first 100,000 values of n there are only 22 that require a 3 digit odd number (A091089). There probably are some values that require odd numbers of 4 or more digits, but these are likely to be very large.

			a(0)=3 because 3 is the minimum odd number which when appended to 0 forms a prime (03 = 3 = prime).
a(20)=11 because 11 is the minimum odd number which when appended to 20 forms a prime (201, 203, 205, 207, 209 are all nonprime, 2011 is prime).

Essentially the same as A068695, which is the main entry for this sequence.

Cf. A032352 (a(n) requires at least a 2 digit odd number), A091089 (a(n) requires at least a 3 digit odd number).

Table[Block[{k = 1}, While[! PrimeQ@ FromDigits[IntegerDigits[n] ~Join~ IntegerDigits[k]], k += 2]; k], {n, 0, 101}] (* Michael De Vlieger, Nov 24 2017 *)

a(n) = forstep(x=1, +oo, 2, if(isprime(eval(concat(Str(n), x))), return(x))) \\ Iain Fox, Nov 23 2017

A136295 Multiples of 100 which begin a span of 100 composite numbers.

Original entry on oeis.org

1671800, 2637800, 3117300, 3933600, 4640600, 4652400, 5178200, 5518700, 5837400, 5845200, 6012900, 6085000, 6333800, 6376200, 6789800, 6958700, 7129900, 7565200, 7803500, 7826900, 8027700, 8367400, 8421300, 8905200

Offset: 1

Julien Peter Benney (jpbenney(AT)gmail.com), Mar 21 2008

nonn

As numbers become large, the proportion of centuries with no primes increases according to the prime number theorem.

			a(2) = 2637800 because there is a prime gap of 112 from 2637799 to 2637911, which makes the century from 2637800 to 2637899 the second one consisting wholly of composite numbers.

J. W. L. Glaisher, Factor table for the sixth million : containing the least factor of every number not divisible by 2,3, or 5 between 5,000,000 and 6,000,000; Taylor and Francis, 1883
J. Murray, Report of the Annual Meeting, British Association for the Advancement of Science, Vol. 50 (1881), p. 308.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A091089, A002386.

100SequencePosition[PrimePi[100Range[90000]],{x_,x_}][[All,1]] (* Harvey P. Dale, Aug 19 2021 *)

a(n) ~ n. [Charles R Greathouse IV, Aug 21 2011]

Definition clarified by Don Reble, Sep 06 2008

A365853 The smallest prime that is formed from the results in A365813.

Original entry on oeis.org

16718003, 26378017, 46524017, 55187023, 58374053, 58452041, 60850043, 71299043, 78035011, 78269033, 84213047, 89052013, 95490049, 97080001, 115558063, 121097027, 128192011, 131194003, 133547119, 140018009, 140872009, 145319003, 147234001, 149897009

Offset: 1

Robert Price, Sep 20 2023

Robert Price, Table of n, a(n) for n = 1..13412

Cf. A091089, A365812, A365813.

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A365812 The smallest prime that is formed from the results in A091089.

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A068695 Smallest number (not beginning with 0) that yields a prime when placed on the right of n.

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A365813 Numbers which form a prime by appending a 3-digit number and form no primes by appending 1 digit or 2 digits.

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A091088 a(n) is the minimum odd number that must be appended to n to form a prime.

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A136295 Multiples of 100 which begin a span of 100 composite numbers.

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A365853 The smallest prime that is formed from the results in A365813.

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