A056709 -id:A056709 - cp's OEIS frontend

A211655 Down-sortable primes: Primes that are also primes after digits are sorted into decreasing order.

2, 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 53, 61, 71, 73, 79, 83, 97, 113, 131, 149, 157, 163, 167, 179, 181, 191, 197, 199, 211, 241, 251, 281, 311, 313, 331, 337, 347, 359, 373, 389, 419, 421, 431, 433, 443, 461, 463, 491, 521, 541, 563, 571, 593, 613, 617, 631, 641, 643, 653

Offset: 1

Francis J. McDonnell, Apr 17 2012

All 1- and 2-digit reversible primes (A007500) are trivially in this sequence. No primes from A056709 are in this sequence. Clearly all absolute primes (A003459) are sortable primes but not all sortable primes are absolute primes. - Alonso del Arte, Oct 08 2013

			131 is prime and after sorting its digits into nonincreasing order we obtain 311, which is prime.
163 is in the sequence because its digits sorted in decreasing order give 631, which is prime. (Note that this is not a reversible prime, since 361 = 19^2.)

Francis J. McDonnell, Table of n, a(n) for n = 1..10000
Francis J. McDonnell, Java Program

Cf. A004186, A211654, A028864, A028867.

Select[Prime[Range[200]], PrimeQ[FromDigits[-Sort[-IntegerDigits[#]]]] &] (* T. D. Noe, Apr 17 2012 *)

A256227 Naught-y numbers (A011540) that after removing all zeros become zeroless primes (A038618).

Original entry on oeis.org

20, 30, 50, 70, 101, 103, 107, 109, 110, 130, 170, 190, 200, 203, 209, 230, 290, 300, 301, 307, 310, 370, 401, 403, 407, 410, 430, 470, 500, 503, 509, 530, 590, 601, 607, 610, 670, 700, 701, 703, 709, 710, 730, 790, 803, 809, 830, 890, 907, 970, 1001, 1003, 1007, 1009, 1010, 1013, 1027, 1030

Offset: 1

Charles R Greathouse IV and Zak Seidov, Mar 19 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A256186 is the intersection of this sequence with A000040.

Cf. A011540, A038618, A056709.

N:= 4: # to produce all terms with <= N digits
ZLO:= proc(d) # produce set of d-digit odd zeroless numbers
       option remember;
       if d = 1 then {1,3,5,7,9}
       else
         map(t -> seq(t+x*10^(d-1),x=1..9), ZLO(d-1))
       fi
end proc:
addzeros:= proc(x,d) # d-digit numbers formed by inserting 0's into x
        local L,n,R;
      L:= convert(x,base,10);
      n:= nops(L);
      R:= map(t -> [op(t),d], combinat[choose](d-1,n-1));
      seq(add(L[i]*10^(r[i]-1),i=1..n), r = R);
    end proc:
Z[1]:= {2,3,5,7}:
for i from 2 to N-1 do Z[i]:= select(isprime,ZLO(i)) od:
`union`(seq(seq(map(addzeros,Z[i],d), i=1..d-1),d=2..N));
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Mar 19 2015

ss={};Do[id=IntegerDigits[p];If[Min[id]<1&&PrimeQ[FromDigits[Delete[id,Position[id,0]]]],ss={ss,p}],{p,20,2000}];Flatten[ss]
Select[Range[1200],DigitCount[#,10,0]>0&&PrimeQ[FromDigits[DeleteCases[ IntegerDigits[ #],0]]]&] (* Harvey P. Dale, Jan 01 2024 *)

is(n)=my(d=digits(n),e=select(x->x,d)); #e<#d && isprime(fromdigits(e)) \\ Charles R Greathouse IV, Mar 19 2015

A159613 Palindromic primes with multiplicative persistence value 1.

Original entry on oeis.org

11, 101, 131, 151, 181, 191, 313, 10301, 10501, 10601, 11311, 11411, 16061, 30103, 30203, 30403, 30703, 30803, 31013, 35053, 38083, 70207, 70507, 70607, 73037, 74047, 90709, 91019, 94049, 1003001, 1008001, 1022201, 1028201, 1035301

Offset: 1

Lekraj Beedassy, Apr 17 2009

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A046501.

Cf. A056709. - R. J. Mathar, Apr 21 2009

isA002113 := proc(n) local dgs,i ; dgs := convert(n,base,10) ; for i from 1 to nops(dgs)/2 do if op(i,dgs) <> op(-i,dgs) then RETURN(false): fi; od: RETURN(true) ; end: ndigs := proc(n) max(1, ilog10(n)+1) ; end: A031346 := proc(n) local p,nred; p := 0 ; nred := n ; while ndigs(nred) > 1 do nred := mul(d, d=convert(nred,base,10) ) ; p := p+1 ; od: p ; end: isA159613 := proc(n) RETURN(isprime(n) and isA002113(n) and A031346(n) = 1) ; end: for n from 1 to 100000 do p := ithprime(n) ; if isA159613(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Apr 21 2009

palpQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn] && Length[ NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==2]; Select[Prime[ Range[82000]],palpQ] (* Harvey P. Dale, Dec 16 2011 *)

A002385 INTERSECT A046501. - R. J. Mathar, Apr 21 2009

Extended by R. J. Mathar, Apr 21 2009

A243825 Numbers n such that every divisor greater than 1 contains the digit 0.

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901, 1907, 2003, 2011, 2017, 2027, 2029

Offset: 1

Barbara W. Waddell and Robert G. Wilson v, Jun 11 2014

This is an example of a composite number in the sequence which demonstrates that A056709 is a proper subsequence. - R. J. Mathar, Jun 13 2014

			The divisors of 10201 are {1, 101 & 10201}. Except for 1 each has a 0 in its decimal expansion.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A062634, A062653, A062667, A062669, A062671, A062673, A062675, A062677, A062679.

Supersequence of A056709 (primes) and A243819 (composites).

[m:m in [2..2100] |  #[d:d in Divisors(m)|0 in Intseq(d)] eq #Divisors(m)-1]; // Marius A. Burtea, Nov 08 2019

fQ[n_, dgt_] := Union[ MemberQ[#, dgt] & /@ IntegerDigits@ Rest@ Divisors@ n][[1]]; Select[ Range[2, 2030], fQ[#, 0] &]

A267277 Zeroless primes p such that p*(product of digits of p)+(sum of digits of p) is also prime.

Original entry on oeis.org

11, 13, 17, 19, 31, 37, 43, 47, 61, 73, 79, 83, 223, 227, 263, 281, 283, 463, 643, 683, 821, 827, 881, 1117, 1231, 1259, 1291, 1321, 1361, 1367, 1433, 1471, 1543, 1567, 1583, 1597, 1619, 1637, 1657, 1699, 1723, 1741, 1753, 1777, 1933, 1951, 1973

Offset: 1

Emre APARI, Jan 12 2016

Zeroless means that the decimal expansion has no digit "0", so no element of A056709 is in the sequence.

If we define a function "n*times products of digits plus sum of digits", f(n) = n*A007954(n) + A007953(n), then iterating the function starting at 217421 generates a chain of at least 4 primes: 217421 -> 24351169 -> 157795575151 -> 1522234189034803183.

			19 => 19*1*9+1+9 = 181 (is prime).
821 => 821*8*2*1+8+2+1 = 13147 (is prime).
2357 => 2357*2*3*5*7+2+3+5+7 = 494987 (is prime).
99995999 => 99995999*(9^7)*5+9*7+5 = 2391388816705223 (is prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007953, A007954, A038618, A056709.

isA267277 := proc(n)
    local pdgs ;
    if isprime(n) then
        pdgs := A007954(n) ;
        if pdgs <> 0 then
            isprime(n*pdgs+A007953(n)) ;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 400 do
    if isA267277(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jan 16 2016

Select[Prime@ Range@ 480, And[Last@ DigitCount@ # == 0, PrimeQ[Function[k, # Times @@ k + Total@ k]@ IntegerDigits@ #]] &] (* Michael De Vlieger, Jan 12 2016 *)

A273460 Numbers n such that sum of the divisors of n (except 1 and n) is equal to the product of the digits of n.

Original entry on oeis.org

98, 101, 103, 107, 109, 307, 329, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901

Offset: 1

Michel Lagneau, May 23 2016

Or numbers n such that A048050(n) = A007954(n).

Most of the terms are primes which have at least one 0 among their digits (A056709). The composite numbers of the sequence are 98, 329, 3383, 4343, 5561, 6623, 12773, 17267, 21479, 57721, 129383, 136259, 142943, 172793, 246959, 256631, 292571,...

			sigma(98) - 98 - 1 = 171 - 98 - 1 = 72 and 8*9 = 72 so 98 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007954, A048050, A056709, A069675.

with(numtheory):
for n from 1 to 3000 do:
  q:=convert(n,base,10):n0:=nops(q):
  pr:=product('q[i]', 'i'=1..n0):p:=sigma(n)-n-1:
   if p=pr
    then
    printf(`%d, `,n):
    else
   fi:
od:

Do[If[DivisorSigma[1, n]-n-1==Apply[Times, IntegerDigits[n]], Print[n]], {n, 2000}]
Select[Range[2,2000],Total[Most[Rest[Divisors[#]]]]==Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Jul 20 2019 *)

A309566 a(n) is the least prime that can be written as a sequence of primes separated by n single zeros, and where every 0-splitting is prime.

Original entry on oeis.org

307, 130307, 309370307, 30281172370306703

Offset: 1

Michel Marcus, Aug 08 2019

			a(1) = 307 = A309101(1).
130307 is a term since 3, 7, 13, 307, 1303, 130307 are all prime.

Carlos Rivera, Puzzle 970. A309566

Subsequence of A309101.

Cf. A000040 (primes), A038618 (zeroless primes), A056709 (primes with zeros).

a(4) from Daniel Suteu and Giovanni Resta, Aug 09 2019

A321151 Primes that yield squares after deletion of their zero digits.

Original entry on oeis.org

409, 1021, 1069, 1201, 1609, 2089, 3061, 5209, 9601, 10069, 10369, 18049, 20089, 20809, 37021, 37201, 40009, 44089, 44809, 50329, 50929, 52009, 59029, 59209, 60889, 62401, 70921, 79201, 96001, 100069, 100609, 101449, 102001, 102769, 103069, 104161, 106129, 106801, 108769, 109321

Offset: 1

Marius A. Burtea, Nov 23 2018

Subsequence of A056709. The squares divisible by 2, 3 or 5 cannot be obtained. Most of the squares obtained seem to be squares of prime or semiprime numbers. Among the first 399 squares obtained are 289 squares of prime numbers, 104 squares of semiprimes and 6 other squares; these squares were obtained by testing the prime numbers up to 10^7. Deletion of the zero digits from primes up to 10^8 yields 929 squares of prime numbers, 506 squares of semiprimes and 47 other squares. Do similar results occur when larger primes are considered?
From David A. Corneth, Nov 26 2018: (Start)
Terms can be obtained by listing squares coprime to 30, inserting zeros between digits, and testing the primality of the resulting numbers.
Records for omega(s) where s is a square producing a term occur at terms 409, 50929, 10713481, 3601722361, 1531869148081, 807916258118689. (End)

			      409 is prime and   49 =  7^2 is a square.
     9601 is prime and  961 = 31^2 is a square.
    20809 is prime and  289 = 17^2 is a square.
    10069 is prime and  169 = 13^2 is a square.
   103069 is prime and 1369 = 37^2 is a square.
  1030069 is prime and 1369 = 37^2 is a square.

David A. Corneth, Table of n, a(n) for n = 1..17124 (terms < 10^10)
David A. Corneth, PARI program

Cf. A000040, A000290, A052041, A056709, A256186.

aQ[n_] := PrimeQ[n] && IntegerQ[Sqrt[FromDigits[Select[IntegerDigits[n], #!=0  &]]]]; Select[Range[100000], aQ] (* Amiram Eldar, Nov 25 2018 *)

isok(p) = isprime(p) && issquare(fromdigits(select(x->x, digits(p)))); \\ Michel Marcus, Nov 26 2018

\\ See Corneth link \\ David A. Corneth, Nov 26 2018

A201014 Composite numbers (include 0) whose product of digits is 0.

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 102, 104, 105, 106, 108, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 302, 303, 304, 305, 306, 308, 309, 310, 320

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A056709 with respect to A011540. Subsequence of A199978 (nonprime numbers (including 0) whose multiplicative digital root is 0).

			Number 102 is in sequence because 1*0*2=0.

Cf. A056709 (primes whose product of digits is 0), A034052 (numbers whose product of digits is 0).

Join[{0},Select[Range[400],CompositeQ[#]&&DigitCount[#,10,0]>0&]] (* Harvey P. Dale, Jul 27 2022 *)

cp's OEIS Frontend

A211655 Down-sortable primes: Primes that are also primes after digits are sorted into decreasing order.

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Mathematica

A256227 Naught-y numbers (A011540) that after removing all zeros become zeroless primes (A038618).

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Maple

Mathematica

PARI

A159613 Palindromic primes with multiplicative persistence value 1.

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A243825 Numbers n such that every divisor greater than 1 contains the digit 0.

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Magma

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A267277 Zeroless primes p such that p*(product of digits of p)+(sum of digits of p) is also prime.

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Maple

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A273460 Numbers n such that sum of the divisors of n (except 1 and n) is equal to the product of the digits of n.

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Maple

Mathematica

A309566 a(n) is the least prime that can be written as a sequence of primes separated by n single zeros, and where every 0-splitting is prime.

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A321151 Primes that yield squares after deletion of their zero digits.

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