A111055 -id:A111055 - cp's OEIS frontend

A111056 Minimal set of prime-strings in base 10 for primes of the form 4n+3 in the sense of A071062.

3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899

Offset: 1

Walter Kehowski, Oct 06 2005

The basic rule is: if no substring of p matches any smaller prime of the form 4n+3, add p to the list. The basic theorem of minimal sets says that the minimal set is always finite.

The sequence b-file is complete except for the number (2*10^19153 + 691)/9, i.e., the decimal number consisting of 19151 "2"s followed by two "9"s. - Curtis Bright, Jan 23 2015

			From _Danny Rorabaugh_, Mar 26 2015: (Start)
a(5) is not 23, even though 23 is the fifth prime of the form 4n+3, since 23 contains a(1)=3 as a substring. Similarly: 31 and 43 contain 3 and 47 contains a(2)=7. Thus a(5)=59.
This sequence contains 2099 since 2, 0, 9, 20, 09, 99, 209, 299, and 099 are not primes of the form 4n+3.
(End)

Walter A. Kehowski and Curtis Bright, Table of n, a(n) for n = 1..112 (first 103 terms from Walter A. Kehowski)
Walter A. Kehowski, Full list of terms
F. Morain, Primality certificate for the largest number of A111056, May 4 2015.
Carlos Rivera, Puzzle 178. Shallit Minimal Primes Set, The Prime Puzzles & Problems Connection.
Jeffrey Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Cf. A071062, A071070, A110600, A110615.

with(StringTools); wc := proc(s) cat("*",Join(convert(s,list),"*"),"*") end; M3:=[]: wcM3:=[]: p:=1: for z from 1 to 1 do for k while p<10^11 do p:=nextprime(p); if k mod 100000 = 0 then print(k,p,evalf((time()-st)/60,4)) fi; if p mod 4 = 3 then sp:=convert(p,string); if andmap(proc(w) not(WildcardMatch(w,sp)) end, wcM3) then M3:=[op(M3),p]; wcM3:=[op(wcM3),wc(sp)]; print(p) fi fi od od; # Let it run for a couple of days.

A174011 4*prime(n)-+3 are both prime.

Original entry on oeis.org

1, 3, 5, 8, 17, 24, 29, 35, 36, 67, 72, 77, 79, 85, 95, 98, 105, 109, 145, 160, 171, 175, 189, 204, 207, 215, 221, 230, 263, 271, 286, 321, 326, 327, 335, 364, 410, 444, 458, 487, 495, 501, 511, 541, 551, 580, 585, 633, 638, 651, 654, 681, 691, 708, 729, 735

Offset: 1

Juri-Stepan Gerasimov, Mar 05 2010

nonn

			a(1)=1 because 4*prime(1)-3=5=prime and 4*prime(1)+3=11=prime; a(2)=3 because 4*prime(3)-3=17=prime and 4*prime(3)+3=23=prime.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A000040, A002145, A111055.

Select[Range[800],And@@PrimeQ[4 Prime[#]+{3,-3}]&] (* Harvey P. Dale, Dec 09 2012 *)

Corrected (24 inserted, 104 replaced by 105, 114 removed etc.) by R. J. Mathar, Apr 28 2010

A330048 Cardinalities of the sets of minimal base-n representations of the primes.

Original entry on oeis.org

2, 3, 3, 8, 7, 9, 15, 12, 26, 152, 17, 228, 240, 100, 483

Offset: 2

Hugo Pfoertner, Nov 29 2019

For a continuation of the sequence see Figure 2 of the Bright, Devillers, Shallit article and Curtis Bright's GitHub repository.
a(17) >= 1279, a(18) = 50, a(19) >= 3462, a(20) = 651, a(21) >= 2599, a(22) = 1242, a(23) = 6021, a(24) = 306, a(25) >= 17597.
a(30) = 220, a(42) = 4551; private communication from Raymond Devillers. - Hugo Pfoertner, Jan 25 2021

			a(10) = 26 because the minimal set of prime-strings in base 10 consist of the 26 terms of A071062.

Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, June 11, 2015.
Curtis Bright, Computed data for the MEPN project, GitHub, 2017.

Cf. A071062, A111055, A111056, A327282, A330049.

A173487 Integers n such that 4*prime(n)-+3 are nonconsecutive primes.

Original entry on oeis.org

1, 3, 5, 95, 98, 175, 189, 458, 487, 501, 541, 638, 735, 969, 1015, 1027, 1526, 2133, 2256, 2290, 2329, 2506, 2600, 2699, 3106, 3939, 3999, 4552, 5548, 5585, 5611, 5803, 5838, 6355, 6756, 6856, 6982, 6983, 7070, 7107, 7167, 7383, 8421, 8868, 9157, 9249, 9405

Offset: 1

Juri-Stepan Gerasimov, Mar 05 2010

nonn

Subsequence of A174011: 4*prime(n)-+3 are both prime. - Michel Marcus, Apr 07 2014

			a(1)=1 because 4*prime(1)-3=5=3rd prime and 4*prime(1)+3=11=5th prime;
a(2)=3 because 4*prime(3)-3=17=7th prime and 4*prime(3)+3=23=9th prime;
a(3)=5 because 4*prime(5)-3=41=13th prime and 4*prime(5)+3=47=15th prime;
a(4)=95 because 4*prime(95)-3=1993=301st prime and 4*prime(95)=3=1999=303rd prime.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A000040, A002145, A111055.

N:= 1000: # to get a(1) to a(N)
p:= 0:
A:= NULL:
count:= 0:
for n from 1 while count < N do
p:= nextprime(p);
if isprime(4*p-3) and isprime(4*p+3) and (isprime(4*p-1) or isprime(4*p+1))
    then count:= count+1; A:= A, n
fi
od:
A; # Robert Israel, Dec 20 2015

Select[Range@ 10000, And[PrimeQ[4 Prime[#] - 3], PrimeQ[4 Prime[#] + 3], NextPrime[4 Prime[#] - 3] != 4 Prime[#] + 3] &] (* Michael De Vlieger, Dec 20 2015 *)

isok(n) = isprime(pa=4*prime(n)-3) && isprime(pb=4*prime(n)+3) && (primepi(pb) - primepi(pa) != 1); \\ Michel Marcus, Apr 07 2014

Corrected (85 replaced by 95, 286 and 321 removed) and extended by R. J. Mathar, May 02 2010

More terms from Michel Marcus, Apr 07 2014

A330049 Maximum width of an element in the set of minimal base-n representations of the primes.

Original entry on oeis.org

2, 3, 2, 5, 5, 5, 9, 4, 8, 45, 8, 32021, 86, 107, 3545

Offset: 2

Hugo Pfoertner, Nov 29 2019

See A330048 for more information.
a(17) >= 111334, a(18) = 33, a(19) >= 110986, a(20) = 449, a(21) >= 47336, a(22) = 764, a(23) = 800874, a(24) = 100, a(25) >= 136967.
a(30) = 1024, a(42) = 487, a(60) = 1938; private communication from Raymond Devillers. - Hugo Pfoertner, Jan 25 2021

			a(10) = 8 because the largest member of the minimal set of prime-strings in base 10 is A071062(26) = 66600049 with 8 decimal digits.

Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, June 11, 2015.

Cf. A071062, A111055, A111056, A330048.

A347819 Minimal elements for the base-10 representations of the primes greater than 10.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501

Offset: 1

Eric Chen, Sep 16 2021

Sequence is finite with 77 terms, the largest being 5*10^30 + 27 (which can be written 5(0_28)27, where 0_28 means the string of 28 0's). See text file for proof (this file also has proofs for bases 2, 3, 4, 5, 6, 8, 12).
Minimal elements for the base b representations of the primes > b for other bases b: (see the text file for 9 <= b <= 16) (all written in base b)
b=2: {11}
b=3: {12, 21, 111}
b=4: {11, 13, 23, 31, 221}
b=5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 10^95 + 13}
b=6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}
b=7: {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} (conjectured, not proven)
b=8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, (10^220-1)/9*40 + 7}.
Equivalently: primes > 10 such that no proper substring (i.e., deleting any positive number of digits) is again a prime > 10. - M. F. Hasler, May 03 2022

			277 is in this sequence because none of 2, 7, 27, 77 is a prime > 10.
857 is in this sequence because none of 8, 5, 7, 85, 87, 57 is a prime > 10.
991 is in this sequence because none of 9, 1, 99, 91 is a prime > 10.
149 is not in this sequence because 19 is subsequence of 149 and 19 is a prime > 10.
389 is not in this sequence because 89 is subsequence of 389 and 89 is a prime > 10.
439 is not in this sequence because 43 is subsequence of 439 and 43 is a prime > 10.

Jinyuan Wang, Table of n, a(n) for n = 1..77
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics 25 (3) (2016), pp. 321-331. DOI:10.1080/10586458.2015.1064048
Eric Chen, Minimal elements for the base b representations of the primes which are > b
Eric Chen, Data for minimal elements for the base b representations of the primes which are > b for 2 <= b <= 16
Eric Chen, Proof for that the data for bases 2, 3, 4, 5, 6, 8, 10, 12 is complete
Eric Chen, Known values or lower bounds of the largest minimal element for the base b representations of the primes which are > b for 2 <= b <= 36
Eric Chen, Condensed table for the status of the minimal element for the base b representations of the primes which are > b for 2 <= b <= 16
Prime Glossary, Minimal prime
J. Shallit, Minimal primes, J. Recreational Math., 30:2 (2000) pp. 113-117.

Cf. A071062 (primes > 10 are not required).

Minimal sets for other sets: A071070 (for composites), A071071 (powers of 2), A071072 (multiples of 4), A071073 (multiples of 3), A111055 (primes of the form 4*n+1), A111056 (primes of the form 4*n+3), A114835 (palindromic primes), A130448 (minimal set of squares).

a(n, k, b)=v=[]; for(r=1, length(digits(n, b)), if(r+length(digits(k, 2))-length(digits(n, b))>0 && digits(k, 2)[r+length(digits(k, 2))-length(digits(n, b))]==1, v=concat(v, digits(n, b)[r]))); fromdigits(v, b)
iss(n, b)=for(k=1, 2^length(digits(n, b))-2, if(ispseudoprime(a(n, k, b)) && a(n, k, b)>b, return(0))); 1
is(n, b=10)=isprime(n) && n>b && iss(n, b) \\ Test whether n is a minimal element for the base b representations of the primes > b. Default value b = 10 for this sequence.
select( {is_A347819(n,b=10)=for(L=2, #n=digits(n,b), forvec(d=vector(L, i, [1,#n]), n[d[1]]&& isprime(fromdigits(vecextract(n,d),b))&& return(L==#n), 2))}, [1..8888]) \\ Better select among primes([1,N]). - M. F. Hasler, May 03 2022

Edited by M. F. Hasler, May 03 2022

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A111056 Minimal set of prime-strings in base 10 for primes of the form 4n+3 in the sense of A071062.

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A174011 4*prime(n)-+3 are both prime.

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A330048 Cardinalities of the sets of minimal base-n representations of the primes.

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A173487 Integers n such that 4*prime(n)-+3 are nonconsecutive primes.

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A330049 Maximum width of an element in the set of minimal base-n representations of the primes.

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A347819 Minimal elements for the base-10 representations of the primes greater than 10.

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