A038618 -id:A038618 - cp's OEIS frontend

A255964 Five-digit distinct-digit zeroless primes.

12347, 12379, 12437, 12457, 12473, 12479, 12487, 12497, 12539, 12547, 12569, 12583, 12589, 12637, 12647, 12653, 12659, 12689, 12697, 12739, 12743, 12763, 12853, 12893, 12953, 12967, 12973, 12983, 13249, 13259, 13267, 13297, 13457

Offset: 1

Zak Seidov, Mar 12 2015

The last term is a(1610)=98731.

Intersection of A038618 and A074671. - Michel Marcus, Mar 16 2015

Zak Seidov, Table of n, a(n) for n = 1..1610

Cf. A038618 (zeroless primes), A074671 (5-digit distinct-digit primes).

f[n_] := Block[{d = DigitCount@ n}, And[Plus @@ d == 5, Last@ d == 0, Max@ d == 1, PrimeQ@ n]]; Select[Range[10000, 99999], f] (* or *) Select[FromDigits /@ Permutations[Range[1, 9], {5}], PrimeQ] (* Michael De Vlieger, Mar 12 2015 *)

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 *)

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

A350574 Primes p such that p, asc(p), and desc(p) are all distinct primes (where asc(p) and desc(p) are the digits of p in ascending and descending order, respectively) and p is the minimum of all permutations of its digits with this property.

Original entry on oeis.org

131, 197, 373, 419, 571, 593, 617, 839, 919, 1297, 1327, 1429, 1879, 1949, 1993, 2129, 2213, 2591, 3539, 4337, 4637, 4639, 5519, 6619, 8389, 8933, 11491, 11519, 11527, 11597, 11897, 11969, 12757, 12829, 12979, 13649, 13879, 14537, 14737, 14741, 14891

Offset: 1

William Riley Barker, Jan 06 2022

All terms in this sequence are zero-free: If p contained 0 as a digit, then desc(p) would end with the zero digit, making it divisible by 10 (thus composite).

			131 is prime, and so are asc(131) = 113 and desc(131) = 311. Further, these are all distinct primes.
419, asc(419) = 149, and desc(419) = 941 are all distinct primes, and 419 is the smallest permutation of the digits {1,4,9} with this property (149 is not included because asc(149) = 149, so these would not be distinct; 491 is not included because 419 < 491 with the same set of digits).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A009994.

Subsequence of A038618.

d:= n-> convert(n, base, 10):
g:= (n, r)-> parse(cat(sort(d(n), r)[])):
f:= n-> (s-> nops(s)=3 and andmap(isprime, s))({n, g(n, `<`), g(n, `>`)}):
q:= n-> f(n) and n=min(select(f, map(x-> parse(cat(x[])),
        combinat[permute](d(n))))):
select(q, [$1..15000])[];  # Alois P. Heinz, Jan 15 2022

Select[Prime@Range@2000,AllTrue[FromDigits/@{s=Sort[d=IntegerDigits@#],Reverse@s},PrimeQ]&&Min@Most@Rest@Sort@Select[FromDigits/@Permutations[d],PrimeQ]==#&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

import numpy as np
# preliminary functions we will use in building our list
def is_prime(n):
    for d in range(2,int(np.sqrt(n))+1):
        if n % d == 0:
            return False
    return True
def asc(n): # returns integer with digits of n in ascending order
    n_list = [int(digit) for digit in str(n)] # separate digits of n into a list
    n_list.sort() # rearrange numbers in ascending order
    asc_n = int(''.join([str(digit) for digit in n_list])) # concatenate the sorted digits
    return asc_n
def desc(n): # returns integer with digits of n in descending order
    n_list = [int(digit) for digit in str(n)]
    n_list.sort(reverse=True)
    desc_n = int(''.join([str(digit) for digit in n_list]))
    return desc_n
N = 5
# get list of integers n such that n, asc(n), and desc(n) are all distinct primes
condition_1 = [n for n in range(2,10**N)
               if is_prime(n)
               and is_prime(asc(n))
               and is_prime(desc(n))
               and n not in (asc(n),desc(n))]
# refine so that our list includes only the minimum permutation of a given set of digits satisfying condition 1
condition_2 = []
for num in condition_1:
    if asc(num) not in [asc(n) for n in condition_2]:
        condition_2.append(num)
print(condition_2)

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A350574_gen(): # generator of terms
    for l in count(1):
        rlist = []
        for a in combinations_with_replacement('123456789',l):
            s = ''.join(a)
            p, q = int(s), int(s[::-1])
            if p != q and isprime(p) and isprime(q):
                for b in multiset_permutations(a):
                    r = int(''.join(b))
                    if p < r < q and isprime(r):
                        rlist.append(r)
                        break
        yield from sorted(rlist)
A350574_list = list(islice(A350574_gen(),50)) # Chai Wah Wu, Feb 13 2022

A368051 Successive primes building the lexicographically earliest k X k 'Futility Squares' (see the Comment and Example sections for more explanations).

Original entry on oeis.org

13, 31, 113, 101, 313, 1117, 1009, 1019, 7993, 11113, 10007, 10009, 10039, 37997, 111119, 100003, 100019, 100129, 101207, 939973, 1111151, 1000003, 1000033, 1000037, 1000099, 5033981, 1337911, 11111117, 10000019, 10000079, 10000103, 10000121, 10011163, 11702641, 79931311

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Dec 09 2023

A  k X k 'Futility Square' is a stack of k primes, each one being of length k. The 1st horizontal prime is also the 1st vertical one; the 2nd horizontal prime is also the 2nd vertical one, and so on. The first horizontal prime must be zeroless. The number on the main diagonal (running from top left to bottom right) is also a prime. The k + 1 primes involved in a k X k square must be distinct and the smallest possible not leading to a contradiction.
There might be more than one k X k 'Futility Square' for some k >= 2. For example another such square for k = 2 is
.
  6 7
  7 1
. - David A. Corneth, Dec 23 2023

			Here is the lexicographically earliest 3 X 3 'Futility Square':
.
  1 1 3
  1 0 1
  3 1 3
.
We see that 113, 101, 313 and the diagonal 103 are distinct primes.
Hereunder is the lexicographically earliest 6 X 6 'Futility Square':
.
  1 1 1 1 1 9
  1 0 0 0 0 3
  1 0 0 0 1 9
  1 0 0 1 2 9
  1 0 1 2 0 7
  9 3 9 9 7 3
.
We see that 111119, 100003, 100019, 100129, 101207, 939973 and the diagonal 100103 are distinct primes.
The sequence is formed by the two horizontal primes of the 2 X 2 square [13, 31], then the three horizontal primes of the 3 X 3 square [113, 101, 313], then the four horizontal primes of the 4 X 4 square [1117, 1009, 1019, 7993], etc.

Michael S. Branicky, Table of n, a(n) for n = 1..5049 (through k=101; terms 1..44 from Michael S. Branicky, terms 45..1539 from David A. Corneth)
Éric Angelini, Squaring Primes, Personal blog, December 2023.
Michael S. Branicky, Python program
David A. Corneth, PARI program
Futility Closet, Squaring Words, Futility Closet, December 2023.

Cf. A038618, A091633.

PARI
```
\\ See PARI link
```
Python
```
# See Python link
```

a(28)-a(35) for k=8 from Michael S. Branicky, Dec 23 2023

A385711 Primes whose digits are all distinct and pairwise coprime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 137, 149, 157, 167, 173, 179, 197, 251, 257, 271, 317, 347, 419, 431, 457, 479, 491, 521, 523, 541, 547, 571, 587, 617, 719, 743, 751, 761, 853, 857, 859, 941, 947, 971, 1237, 1259

Offset: 1

Gonzalo Martínez, Jul 07 2025

This sequence has 252 terms, the last being 95471, which have at most 5 digits. This is because each term has at most one even digit and at most 4 odd digits, since gcd(3, 9) = 3.

All terms are in A038618, since if zero is among the digits of a prime p, then p must have at least 3 digits, where at least one of them is greater than 1, say d, and in such a case gcd(0, d) = d ! = 1.

			857 is a term since it is prime and gcd(8, 5) = gcd(5, 7) = gcd(8, 7) = 1.

Gonzalo Martínez, Table of n, a(n) for n = 1..252

Subsequence of A029743.

Cf. A038618.

Select[Prime[Range[10000]], UnsameQ @@ (d = IntegerDigits[#]) && AllTrue[Subsets[d, {2}], CoprimeQ @@ # &] &] (* Amiram Eldar, Jul 13 2025 *)

A386321 Primes without {0, 3} as digits.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 41, 47, 59, 61, 67, 71, 79, 89, 97, 127, 149, 151, 157, 167, 179, 181, 191, 197, 199, 211, 227, 229, 241, 251, 257, 269, 271, 277, 281, 419, 421, 449, 457, 461, 467, 479, 487, 491, 499, 521, 541, 547, 557, 569, 571, 577, 587, 599, 617

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038611 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 0] == 0 && DigitCount[#, 10, 3] == 0 &]

primes_with(, 1, [1, 2, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("12456789"), 41))) # uses function/imports in A385776

A386322 Primes without {0, 4} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038612 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 3, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 0] == 0 && DigitCount[#, 10, 4] == 0 &]

primes_with(, 1, [1, 2, 3, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("12356789"), 41))) # uses function/imports in A385776

A386323 Primes without {0, 5} as digits.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 83, 89, 97, 113, 127, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038613 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 3, 4, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 0] == 0 && DigitCount[#, 10, 5] == 0 &]

primes_with(, 1, [1, 2, 3, 4, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("12346789"), 41))) # uses function/imports in A385776

A386324 Primes without {0, 6} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 83, 89, 97, 113, 127, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038614 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 3, 4, 5, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 0] == 0 && DigitCount[#, 10, 6] == 0 &]

primes_with(, 1, [1, 2, 3, 4, 5, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("12345789"), 41))) # uses function/imports in A385776

cp's OEIS Frontend

A255964 Five-digit distinct-digit zeroless primes.

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Mathematica

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

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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A350574 Primes p such that p, asc(p), and desc(p) are all distinct primes (where asc(p) and desc(p) are the digits of p in ascending and descending order, respectively) and p is the minimum of all permutations of its digits with this property.

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Maple

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Python

A368051 Successive primes building the lexicographically earliest k X k 'Futility Squares' (see the Comment and Example sections for more explanations).

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PARI

Python

Extensions

A385711 Primes whose digits are all distinct and pairwise coprime.

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Mathematica

A386321 Primes without {0, 3} as digits.

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Magma

Mathematica

PARI

Python

A386322 Primes without {0, 4} as digits.

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