A046034 -id:A046034 - cp's OEIS frontend

A118718 Lucky numbers with only prime digits.

3, 7, 25, 33, 37, 73, 75, 223, 235, 237, 273, 327, 357, 535, 537, 553, 577, 723, 727, 735, 777, 2253, 2257, 2277, 2323, 2335, 2355, 2523, 2527, 2557, 2575, 2725, 2755, 2773, 3223, 3235, 3325, 3355, 3727, 3753, 5233, 5235, 5253, 5277, 5325

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

Lucky numbers with digits 2,3,5 or 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..189 from N. J. A. Sloane)

Intersection of A000959 and A046034.

L = Table[2*i + 1, {i, 0,2663}]; For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];Select[L,ContainsOnly[IntegerDigits[#],{2,3,5,7}]&] (* James C. McMahon, Sep 16 2024 *)

Corrected by R. J. Mathar, Aug 21 2006

A154780 Numbers k with d digits such that all digits of k and the last d+1 digits of k^2 are prime.

Original entry on oeis.org

5, 35, 235, 335, 2335, 3335, 23335, 32335, 33335, 72335, 233335, 323335, 333335, 372335, 572335, 723335, 2333335, 2372335, 2723335, 3233335, 3323335, 3333335, 3572335, 3723335, 7233335, 7323335, 7372335, 7572335, 22372335, 23333335

Offset: 1

M. F. Hasler, Jan 23 2009

Any term with d digits is the concatenation of a prime digit and an earlier term (with d-1 digits).

The sequence is infinite since it contains subsequences b(n) = (10^n-1)/3+2 = (5,35,335,3335,...), c(n) = 23*10^n+b(n) = (235,2335,23335,...), d(n) = 3233*10^n+b(n), e(n) = 7233*10^n+b(n) etc.

Subsequence of A046034; contains A153025 as a subsequence.

Select[Range[5,24000000,5],And@@PrimeQ[IntegerDigits[#]]&& And@@ PrimeQ[ Take[ IntegerDigits[#^2],-(IntegerLength[#]+1)]]&] (* Harvey P. Dale, Dec 31 2012 *)

last=[0]; {for( d=1,8, new=[]; forprime( p=0,9, for( k=1,#last, is_A046034((p*10^(d-1)+last[k])^2%10^(d+1)+20*10^d) & new=concat( new, p*10^(d-1)+last[k]))); print1(last=new,","))} /* for slightly more efficient code see A154779 */

For all n, a(n) == (5 mod 10).
For a(n) > 5, a(n) == 35 (mod 100).
For a(n) > 35, a(n) == 235 or 335 (mod 1000).
For a(n) > 335, a(n) == 2335 or 3335 (mod 10^4).

A155990 Numbers prime(k) as k runs through the numbers whose digits are all prime.

Original entry on oeis.org

3, 5, 11, 17, 79, 83, 97, 103, 131, 137, 149, 157, 239, 241, 257, 269, 359, 367, 379, 389, 1399, 1409, 1427, 1433, 1459, 1471, 1483, 1489, 1601, 1607, 1613, 1621, 1747, 1753, 1777, 1787, 2137, 2141, 2153, 2179, 2237, 2239, 2251, 2269, 2377, 2381, 2389, 2399

Offset: 1

Juri-Stepan Gerasimov, Feb 01 2009

Cf. A000040, A046034, A141468.

isA046034 := proc(n) local d ; d := convert(convert(n,base,10),set) ; if d intersect {0,1,4,6,8,9} = {} then true; else false; fi; end: for n from 1 to 3000 do if isA046034(n) then printf("%d,",ithprime(n)) ; fi; od: # R. J. Mathar, Feb 10 2009

a(n) = A000040(A046034(n)). - Michel Marcus, Jan 21 2019

Edited by N. J. A. Sloane, Feb 02 2009

149 inserted by R. J. Mathar, Feb 10 2009

A211683 Numbers > 100 such that all the substrings of length = 2 are primes.

Original entry on oeis.org

111, 113, 117, 119, 131, 137, 171, 173, 179, 197, 231, 237, 297, 311, 313, 317, 319, 371, 373, 379, 411, 413, 417, 419, 431, 437, 471, 473, 479, 531, 537, 597, 611, 613, 617

Offset: 1

Hieronymus Fischer, Jun 08 2012

Only numbers > 100 are considered, since all 2-digit primes are trivial members. See A069488 for the sequence with prime terms > 100.
The sequence is infinite (for example, consider the continued concatenation of ‘11’ or of ‘13’: 111, 1111, 11111, ..., 131, 1313, 13131, ... are members).
Infinitely many terms are palindromic.

			a(2)=113, since all substrings of length = 2 are primes (11 and 13).
a(10)=197, since all substrings of length = 2 (19, 97) are primes.

Hieronymus Fischer, Table of n, a(n) for n = 1..5000

Cf. A019546, A035232, A039996, A046034, A069488, A085823, A131648, A211681, A211682.

A211686 Prime numbers > 10000 such that all the substrings of length >= 4 are primes (substrings with leading '0' are considered to be nonprime).

Original entry on oeis.org

11093, 11171, 11933, 12011, 12239, 12377, 12791, 12917, 13037, 13217, 13613, 14519, 14591, 14813, 14831, 15233, 15791, 16073, 16091, 16217, 16673, 16691, 17333, 17417, 17477, 18233, 18311, 18713, 18719, 18731, 19013, 19319, 19739, 19973, 21319

Offset: 1

Hieronymus Fischer, Jun 08 2012

Only numbers > 10000 are considered, since all 4-digit primes are trivial members.
By definition, each term of the sequence with more than 5 digits is built up by an overlapped union of previous terms, i.e., a(254)=182339 has the two embedded previous terms a(26)=18233 and a(208)=82339.
The sequence is finite, the last term is 934919 (n=263). Proof of finiteness: Let p be a number with more than 6 digits. By the argument above, each 6-digit substring must be a previous term. The only 6-digit terms are 182339, 349199, 432713, 487793, 511933, 654799, 782339, 787793, 917333, 934919 (n=254..263, see b-file). As can be directly verified, none of them can be extended to a 7-digit number with the desired property.

			a(1)=11093, since all substrings of length >= 4 are primes (1109, 1093, and 11093).
a(263)=934919, all substrings of length >= 4 (9349, 3491, 4919, 93491, 34919 and 934919) are primes.

Hieronymus Fischer, Table of n, a(n) for n = 1..263

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

sspQ[n_]:=Module[{idn=IntegerDigits[n],s1,s2},s1=FromDigits[Most[idn]];s2=FromDigits[Rest[idn]];IntegerLength[s1]==IntegerLength[s2]==4 && AllTrue[{s1,s2},PrimeQ]]; Select[Prime[Range[1230,9592]],sspQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* The program generates all 253 five-digit terms of the sequence *)(* Harvey P. Dale, May 11 2018 *)

A213308 Numbers with exactly one nonprime substring (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

1, 4, 6, 8, 9, 13, 17, 22, 25, 27, 29, 31, 32, 33, 35, 43, 47, 52, 55, 57, 59, 67, 71, 72, 75, 77, 79, 83, 97, 137, 173, 223, 233, 237, 313, 317, 337, 353, 379, 523, 537, 673, 733, 737, 773, 797, 1373, 3137, 3373, 3733, 3797

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 5-digit number has at least 2 nonprime substrings. Thus, each number with more than 5 digits has >= 2 nonprime substrings, too. Consequently, there is a boundary b<10^4, such that all numbers > b have at least 2 nonprime substrings.

The first term is a(1)=1=A213302(1). The last term is a(51)=3797=A213300(1).

			a(1)=1, since 1 has one nonprime substring.
a(51)=3797, since the only nonprime substring of 3797 is 9.

Hieronymus Fischer, Table of n, a(n) for n = 1..51

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

A213309 Numbers with exactly 2 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

11, 12, 15, 19, 20, 21, 24, 26, 28, 30, 34, 36, 38, 39, 41, 42, 45, 50, 51, 54, 56, 58, 61, 62, 63, 65, 70, 74, 76, 78, 82, 85, 87, 89, 92, 93, 95, 113, 131, 179, 197, 227, 229, 231, 232, 235, 239, 253, 257, 271, 273, 277, 283

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 6-digit number has at least 4 nonprime substrings. Thus, each number with more than 6 digits has >= 4 nonprime substrings, too. Consequently, there is a boundary b<10^5, such that all numbers > b have more than 2 nonprime substrings.

The first term is a(1)=11=A213302(2). The last term is a(130)=37337=A213300(2).

			a(1)=11, since 11 has 2 nonprime substrings.
a(130)= 37337, since there are 2 nonprime substrings (33 and 337).

Hieronymus Fischer, Table of n, a(n) for n = 1..130

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

A213310 Numbers with exactly 3 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

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, 117, 123, 127, 132, 133, 135, 139, 153, 157, 167, 171, 172, 175, 177, 193, 211, 213, 217, 222, 225, 230, 234, 236, 238, 241

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 6-digit number has at least 4 nonprime substrings. Thus, each number with more than 6 digits has >= 4 nonprime substrings, too. Consequently, there is a boundary b<10^5, such that all numbers > b have more than 3 nonprime substrings.

The first term is a(1)=10=A213302(3). The last term is a(310)=73373=A213300(3).

			a(1)=10, since 10 has 3 nonprime substrings (0, 1, 10).
a(310)= 73373, since there are 3 nonprime substrings (33, 7337 and 73373).

Hieronymus Fischer, Table of n, a(n) for n = 1..310

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

A213311 Numbers with exactly 4 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

103, 107, 111, 112, 115, 119, 122, 125, 129, 130, 134, 136, 138, 143, 147, 151, 152, 155, 159, 163, 170, 174, 176, 178, 183, 191, 192, 195, 199, 202, 203, 205, 207, 212, 215, 219, 220, 221, 224, 226, 228, 242, 245, 250

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 6-digit number has at least 4 nonprime substrings, and each 4-digit number has at least 1 nonprime substring. Thus, each 10-digit number has at least 5 nonprime substrings. Consequently, there is a boundary b, such that all numbers >= b have more than 4 nonprime substrings.

The first term is a(1)=103=A213302(4). The last term is a(653)=373379=A213300(4).

			a(1) = 103, since 103 has 4 nonprime substrings (0, 03, 1, 10).
a(653) = 373379, since there are 4 nonprime substrings (9, 33, 3379, 7337).

Hieronymus Fischer, Table of n, a(n) for n = 1..653

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

A213312 Numbers with exactly 5 nonprime substrings (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

101, 102, 105, 109, 110, 114, 116, 118, 120, 121, 124, 126, 128, 141, 142, 145, 149, 150, 154, 156, 158, 161, 162, 165, 181, 182, 185, 187, 189, 190, 194, 196, 198, 200, 201, 204, 206, 208, 209, 210, 214, 216, 218, 240

Offset: 1

Hieronymus Fischer, Aug 26 2012

The sequence is finite. Proof: Each 7-digit number has at least 6 nonprime substrings. Thus, each number with more than 7 digits has >= 6 nonprime substrings, too. Consequently, there is a boundary b<10^6, such that all numbers > b have more than 5 nonprime substrings.

The first term is a(1)=101=A213302(5). The last term is a(1330)=831373=A213300(5).

			a(1)=101, since 101 has 5 nonprime substrings (0, 01, 1, 1, 10).
a(1330)= 831373, since there are 5 nonprime substrings (1, 8, 831, 8313, 31373).

Hieronymus Fischer, Table of n, a(n) for n = 1..1330

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

cp's OEIS Frontend

A118718 Lucky numbers with only prime digits.

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A154780 Numbers k with d digits such that all digits of k and the last d+1 digits of k^2 are prime.

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PARI

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A155990 Numbers prime(k) as k runs through the numbers whose digits are all prime.

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Maple

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A211683 Numbers > 100 such that all the substrings of length = 2 are primes.

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A211686 Prime numbers > 10000 such that all the substrings of length >= 4 are primes (substrings with leading '0' are considered to be nonprime).

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A213308 Numbers with exactly one nonprime substring (substrings with leading zeros are considered to be nonprime).

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A213309 Numbers with exactly 2 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213310 Numbers with exactly 3 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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A213311 Numbers with exactly 4 nonprime substrings (substrings with leading zeros are considered to be nonprime).

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