A071062 -id:A071062 - cp's OEIS frontend

A071073 Minimal "multiples of 3" set in base 10.

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 42, 45, 48, 51, 54, 57, 72, 75, 78, 81, 84, 87, 111, 114, 117, 141, 144, 147, 171, 174, 177, 222, 225, 228, 252, 255, 258, 282, 285, 288, 411, 414, 417, 441, 444, 447, 471, 474, 477, 522, 525, 528, 552, 555, 558, 582, 585, 588

Offset: 1

Benoit Cloitre, May 26 2002

Any multiple of 3 contains in its digits at least one of the terms of this sequence. There are 76 terms in the sequence; Delahaye gives all 76 terms and proves that there are no further terms (his statement that there are 280 terms seems to be a typo). There is no smaller set.

Nathaniel Johnston, Table of n, a(n) for n = 1..76 (full sequence)
J.-P. Delahaye, Nombres premiers inévitables et pyramidaux, Pour la Science 296 (2002), p 102.

Cf. A071062, A071071, A071072, A071070.

A071072 Minimal "multiples of 4" set in base 10.

Original entry on oeis.org

0, 4, 8, 12, 16, 32, 36, 52, 56, 72, 76, 92, 96

Offset: 1

Benoit Cloitre, May 26 2002

Any multiple of 4 contains in its digits at least one of the terms of this sequence, and there is no smaller set.

J.-P. Delahaye, "Pour la science" (French edition of Scientific American), Juin 2002, p. 98

Cf. A071062, A071071, A071070, A071073.

A163753 At least one prime occurs as a substring of the digits of n.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 87, 89, 92, 93, 95, 97, 101, 102

Offset: 1

Gil Broussard, Aug 03 2009

A039997(a(n)) > 0. - Reinhard Zumkeller, Jan 31 2012

This sequence (written in decimal) is automatic in the terminology of Allouche & Shallit since A071062 is finite. - Charles R Greathouse IV, Jan 31 2012

			a(6) = 12 because "2" is a prime substring of "12".

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A062115 (complement), A205667 (subsequence), A071062.

a163753 n = a163753_list !! (n-1)
a163753_list = filter ((> 0) . a039997) [0..]
-- Reinhard Zumkeller, Jan 31 2012

A110615 Minimal set of composite-strings in base 12 in the sense of A071070.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 24, 25, 26, 27, 35, 36, 38, 39, 60, 62, 63, 65, 84, 85, 86, 87, 91, 95, 132, 133, 134, 135, 143, 161, 205, 215, 355, 377, 445, 451, 455, 493, 1651, 1673, 1885, 1891, 1895, 8797

Offset: 1

Walter Kehowski, Sep 14 2005; name corrected Sep 18 2005

nonn

Maple worksheet available upon request. Here is the sequence of minimal composites in base 12, where X is 10 and E is 11. 4, 6, 8, 9, X, 10, 12, 13, 20, 21, 22, 23, 2E, 30, 32, 33, 50, 52, 53, 55, 70, 71, 72, 73, 77, 7E, E0, E1, E2, E3, EE, 115, 151, 15E, 257, 275, 311, 317, 31E, 351, E57, E75, 1111, 1117, 111E, 5111.

			a(13)=35=2E since no earlier composite is of the form "*2*E*". The list of minimal composites can be constructed using a sieve-like process: subsequently eliminate from the list of composites all composites of the form "*2*E*". Assuming all previous terms have been similarly determined, then the next remaining composite should be 30.

Cf. A071062, A071070.

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.

A131480 Minimal set of prime-strings in Roman numerals.

Original entry on oeis.org

2, 5, 11, 59, 101, 509, 1009, 1051, 3001

Offset: 1

Robert Happelberg (roberthappelberg(AT)yahoo.com), Jul 27 2007

Written in Roman numerals this sequence begins II, V, XI, LIX, CI, DIX, MIX, MLI, MMMI.

For a while I actually considered putting in 97 (XCVII) instead of 101 (CI), since XCVII contains CI as a substring. But it also contains II, V and XI as substrings, so going in order strictly from smallest to greatest, 97 must be excluded.

			a(4) = 59 because it is a prime number and in Roman numerals it's written LIX (all the substrings are composite or not prime: L, LI, LX, I, IX, X).

Cf. A071062, minimal set of prime-strings in base 10.

A225673 Number of integers whose sum of substrings = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 7, 7, 5, 5, 3, 3, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 3, 4, 3, 4, 3, 4, 3, 3, 2, 3, 3, 4, 4, 5, 4, 5, 4, 5, 3, 4, 3, 4, 4, 5, 5, 6, 5, 6, 5, 5, 4, 5, 4, 5, 5, 6, 6, 7, 6, 7, 5, 6

Offset: 1

Christian N. K. Anderson and Kevin L. Schwartz, May 11 2013

20 is the first number that is not the sum of substrings of any positive integer. There are 203 such numbers < 10000, and they disproportionately begin with 2 and 3 -- 123 of them and 70 of them, respectively.

			For a(59)=5, the five solutions are:
136 (because 13+36+1+3+6=59),
140 (because 14+40+1+4+0=59),
317 (because 31+17+3+1+7=59),
321 (because 32+21+3+2+1=59), and
502 (because 50+02+5+0+2=59).

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), and all integer solutions.
Christian N. K. Anderson, Ulam Spiral of n = 1..10000, color-coded by their value in this sequence.

Cf. A071062, A007953.

table(factor(sapply(1:900,function(n) { tot=0; s=as.character(n); len=nchar(s); for(i in 1:len) for(j in i:len) tot=tot+as.numeric(substr(s,i,j)); tot-n } ),levels=1:100))

A320725 Prime numbers such that all other numbers obtained from all permutations of all subsets of the digits are nonprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 6469

Offset: 1

Daniel Lignon, Oct 19 2018

Sequence is finite since it is a subsequence of a finite sequence (A071062).

This is complete: there are only 14 terms in the sequence.

			449 is in this sequence because it's prime and none of the numbers 4, 9, 44, 49, 94, 494 and 944 is prime.

Subsequence of A071062.

Cf. A320726 (the same for composite numbers).

Select[Prime@ Range[10^3], NoneTrue[DeleteCases[FromDigits /@ Rest@ Union@ Apply[Join, Permutations /@ Subsets@ IntegerDigits@ #], #], PrimeQ] &] (* Michael De Vlieger, Oct 22 2018 *)

A326609 Largest minimal prime in base n (written in base 10).

Original entry on oeis.org

3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921

Offset: 2

Richard N. Smith, Jul 13 2019

a(13) is (probably) 13^32020*8+183, it has 35670 digits, a(14) = 14^85*4+65, it has 99 digits, a(15) = (15^106*66-619)/7, it has 126 digits, a(16) = 16^3544*9+145, it has 4269 digits.
a(17) is the smallest prime of the form (4105*17^k-9)/16 if it exists, otherwise (probably) (73*17^111333-9)/16 (136991 digits), a(18) = 18^31*304+1 (42 digits).
Other known terms: a(20) = (20^449*16-2809)/19 (585 digits), a(22) = 22^763*20+7041 (1026 digits), a(23) is (probably) (23^800873*106-7)/11 (1090573 digits), a(24) = (24^99*512-121)/23 (138 digits), a(30) = 30^1023*12+1 (1513 digits), a(42) = (42^487*27-1093)/41 (791 digits).
a(19) is the smallest prime of the form (15964*19^k-1)/3 if it exists, otherwise (probably) (904*19^110984-1)/3 (141924 digits), a(21) is the smallest prime of the form 16*21^k+335 if it exists, otherwise (probably) (51*21^479149-1243)/4 (633542 digits).

Richard N. Smith, Table of n, a(n) for n = 2..16
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, preprint, 2014.
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics, Volume 25, 2016 - Issue 3.
C. K. Caldwell, The Prime Glossary, minimal prime
Richard N. Smith, List of all known terms
Top PRPs, Search for 8*13^n+183
Top PRPs, Search for (73*17^n-9)/16
Top PRPs, Search for (106*23^n-7)/11
Wikipedia, Minimal prime (recreational mathematics)

Cf. A071062 (base 10 minimal primes), A110600 (base 12 minimal primes).

Cf. A293142 (largest non-repunit permutable prime), A317689 (largest non-repunit circular prime), A103443 (largest left-truncatable prime), A023107 (largest right-truncatable prime), A323137 (largest two-sided prime), A084738 (smallest repunit prime), A186995 (smallest weakly prime).

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.

cp's OEIS Frontend

A071073 Minimal "multiples of 3" set in base 10.

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A071072 Minimal "multiples of 4" set in base 10.

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A163753 At least one prime occurs as a substring of the digits of n.

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Haskell

A110615 Minimal set of composite-strings in base 12 in the sense of A071070.

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

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A131480 Minimal set of prime-strings in Roman numerals.

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A225673 Number of integers whose sum of substrings = n.

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R

A320725 Prime numbers such that all other numbers obtained from all permutations of all subsets of the digits are nonprime.

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Mathematica

A326609 Largest minimal prime in base n (written in base 10).

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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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