A038618 -id:A038618 - cp's OEIS frontend

A034303 Zeroless primes that become nonprime if any digit is deleted.

11, 19, 41, 61, 89, 227, 251, 257, 277, 281, 349, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 1117, 1129, 1171, 1187, 1259, 1289, 1423, 1447, 1453, 1471, 1483, 1543, 1553, 1559, 1583, 1621, 1669, 1721, 1741, 1747

Offset: 1

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A034302, A034304, A034305.

Cf. A010051, A038618.

import Data.List (inits, tails)
a034303 n = a034303_list !! (n-1)
a034303_list = filter f $ drop 4 a038618_list where
   f x = all (== 0) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011

Select[Prime[Range[5,300]],Union[PrimeQ[FromDigits/@Table[Delete[ IntegerDigits[ #], n],{n,IntegerLength[#]}]]] == {False} && !MemberQ[ IntegerDigits[#],0]&] (* Harvey P. Dale, Jan 09 2014 *)

Definition corrected by T. D. Noe, Apr 02 2008

Name edited by Jon E. Schoenfield, Feb 07 2022

A057628 Primes such that replacing each digit d with d copies of the digit d produces a prime. Zeros are not allowed.

Original entry on oeis.org

11, 31, 53, 131, 149, 223, 283, 311, 313, 331, 397, 463, 641, 691, 937, 941, 1439, 1511, 1741, 1871, 1949, 1993, 1999, 2111, 2447, 2939, 3163, 3391, 3433, 3499, 3559, 3593, 3659, 3911, 3931, 5227, 5399, 5923, 6163, 6269, 6653, 6719, 7177, 7741, 8389

Offset: 1

G. L. Honaker, Jr., Oct 10 2000

"Replacing each digit d with d copies of the digit d" is the function A048376. Therefore this is the largest subset of A038618 stable under the map A048376.

			E.g. 641 becomes 66666644441 which is also prime.

Cf. A038618, A048376, A057630.

Select[Prime[Range[1500]],PrimeQ[FromDigits[Flatten[Table[#,{#}]&/@ IntegerDigits[#]]]]&&DigitCount[#,10,0]==0&]  (* Harvey P. Dale, Mar 27 2011 *)

is_A057628(n)={vecmin(digits(n)) && is_A057630(n)} \\ M. F. Hasler, Jan 23 2013

More terms from Patrick De Geest, Oct 15 2000.

Offset changed to 1, according to OEIS conventions, by M. F. Hasler, Jan 23 2013

A101987 Product of nonzero digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 6, 18, 3, 21, 4, 12, 28, 15, 45, 6, 42, 7, 21, 63, 24, 72, 63, 1, 3, 7, 9, 3, 14, 3, 21, 27, 36, 5, 35, 18, 42, 21, 63, 8, 9, 27, 63, 81, 2, 12, 28, 36, 18, 54, 8, 10, 70, 36, 108, 14, 98, 16, 48, 54, 21, 3, 9, 21, 9, 63, 84, 108, 45, 135, 126, 63, 189, 72, 216

Offset: 1

Zak Seidov, Jan 29 2005

First differs from A053666 in 26th term.

			a(25) = 63 because the 25th prime is 97 and 9 * 7 = 63.
a(26) = 1 because the 26th prime is 101, but we ignore the 0 and thus have 1 * 1 = 1.

Cf. A051801, A053666.

Cf. A038618, A056709.

a:= n-> mul(`if`(i=0, 1, i), i=convert(ithprime(n), base, 10)):
seq(a(n), n=1..77);  # Alois P. Heinz, Mar 11 2022

Table[Times@@ReplaceAll[IntegerDigits[Prime[n]], 0 -> 1], {n, 80}] (* Alonso del Arte, Feb 28 2014 *)

a(n) = vecprod(select(x->(x>1), digits(prime(n)))); \\ Michel Marcus, Mar 11 2022

from math import prod
from sympy import sieve
def A051801(n): return prod(int(d) for d in str(n) if d != '0')
def a(n): return A051801(sieve[n])
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Mar 11 2022

a(n) = A051801(prime(n)). - Michel Marcus, Mar 11 2022

A106101 Primes with minimal digit = 1.

Original entry on oeis.org

11, 13, 17, 19, 31, 41, 61, 71, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 419, 421, 431, 461, 491, 521, 541, 571, 613, 617, 619, 631, 641, 661, 691, 719, 751, 761, 811

Offset: 1

Zak Seidov, May 07 2005

Subsequence of A038618. - Michel Marcus, Oct 23 2013

Cf. A038618.

Select[Prime[Range[200]], Min[IntegerDigits[ # ]]==1&]

A386320 Primes without {0, 2} as digits.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 419, 431, 433, 439

Offset: 1

Jason Bard, Jul 18 2025

Intersection of A038604 and A038618.

Cf. A000040, A385776.

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

Select[Prime[Range[120]], DigitCount[#, 10, 0] == 0 && DigitCount[#, 10, 2] == 0 &]

primes_with(, 1, [1, 3, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("13456789"), 41))) # uses function/imports in A385776

A220488 Primes that have both prime digits (2,3,5,7) and nonprime digits (1,4,6,8,9), without digits "0".

Original entry on oeis.org

13, 17, 29, 31, 43, 47, 59, 67, 71, 79, 83, 97, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 193, 197, 211, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 311, 313, 317, 331, 347, 349, 359, 367, 379, 383, 389, 397, 421, 431, 433, 439

Offset: 1

Omar E. Pol, Feb 01 2013

For similar sequences see A152426 and A152427.

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

Cf. A018252, A019546, A034844, A038618, A087363, A092621, A092626, A152312, A152313, A152426, A152427.

pdnpdQ[n_]:=Module[{idn=IntegerDigits[n],p,z=DigitCount[n,10,0]},p=Count[ idn,?PrimeQ];p>0&&z==0&&Length[idn]>p]; Select[ Prime[ Range[ 150]], pdnpdQ] (* _Harvey P. Dale, Oct 01 2013 *)

A228139 Primes such that the product of their digits subtracted from the prime number is another prime.

Original entry on oeis.org

23, 29, 41, 43, 47, 83, 89, 127, 149, 181, 223, 227, 229, 241, 251, 263, 271, 277, 293, 347, 349, 367, 383, 389, 419, 431, 433, 439, 457, 479, 487, 541, 587, 631, 641, 643, 647, 653, 659, 673, 677, 743, 761, 853, 857, 859, 863, 883, 887, 1123, 1229, 1279, 1297, 1423, 1459, 1489, 1523

Offset: 1

Will Gosnell, Aug 12 2013

			23 is a member since 23-(2*3)=17. 29 is a member since 29-(2*9)=11.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Subsequence of A038618.

pdsQ[n_]:=Module[{pr=Times@@IntegerDigits[n]},pr!=0&&PrimeQ[n-pr]]; Select[Prime[Range[300]],pdsQ] (* Harvey P. Dale, Jul 29 2017 *)

dprod(n)=my(v=digits(n));prod(i=1,#v,v[i])
is(n)=my(d=dprod(n)); d>0 && isprime(n) && isprime(n-d) \\ Charles R Greathouse IV, Aug 12 2013

a(3), a(10), a(15)-a(57) from Charles R Greathouse IV, Aug 12 2013

A259315 Nonprimes containing no zeros in decimal representation.

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 72, 74, 75, 76, 77, 78, 81, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 111

Offset: 1

Reinhard Zumkeller, Jun 24 2015

Intersection of A052382 and A018252;

A168046(a(n)) * (1 - A010051(a(n))) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A018252, A052382, A168046, A038618.

a259315 n = a259315_list !! (n-1)
a259315_list = filter ((== 0) . a010051') a052382_list

Select[Range[150],!PrimeQ[#]&&DigitCount[#,10,0]==0&] (* Harvey P. Dale, Sep 18 2024 *)

isok(m) = !isprime(m) && vecmin(digits(m)); \\ Michel Marcus, Jan 23 2022

A346206 Primes p, with k digits, such that the Sum_{i=1..k} (p without its i-th digit)/(its i-th digit) is a prime.

Original entry on oeis.org

11, 21673, 27367, 32611, 33311, 41141, 48821, 82781, 171263, 211441, 243433, 323443, 343243, 449699, 632623, 663661, 727271, 772127, 847871, 882881, 944969, 1129699, 1192699, 1193939, 1262633, 1334341, 1342433, 1343423, 1361441, 1388641, 1399193, 1461883, 1613441

Offset: 1

Michel Marcus, Jul 10 2021

			21673 gives 1673/2 + 2673/1 + 2173/6 + 2163/7 + 2167/3 = 4903; so 21673 is a term.

Michel Marcus, Table of n, a(n) for n = 1..441
Carlos Rivera, Puzzle 1045. One nice puzzle from Paolo Lava, The Prime Puzzles and Problems Connection.

Subsequence of A038618 (zeroless primes).

subs(d, j) = {my(x=""); for (k=1, #d, if (j != k, x = concat(x, d[k]));); eval(x);}
isok(p) = {my(d=digits(p), res);  if (isprime(p) && vecmin(d), res = sum(j=1, #d, subs(d, j)/d[j]); (denominator(res)==1) && isprime(res););}

from sympy import isprime, primerange
from fractions import Fraction
def ok(p):
    s = str(p)
    if '0' in s or len(s) == 1: return False
    f = sum(Fraction(int(s[:i]+s[i+1:]), int(s[i])) for i in range(len(s)))
    return f.denominator == 1 and isprime(f.numerator)
def aupto(lim): return [p for p in primerange(1, lim+1) if ok(p)]
print(aupto(1620000)) # Michael S. Branicky, Jul 11 2021

A134873 Primes p with the property that the sum of the digits of the product of the digits of p is also a prime number.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 43, 71, 73, 113, 127, 131, 137, 151, 173, 211, 223, 257, 271, 277, 281, 311, 317, 431, 457, 523, 541, 547, 557, 577, 727, 757, 821, 853, 1117, 1151, 1171, 1187, 1217, 1223, 1277, 1427, 1451, 1481, 1511, 1523

Offset: 1

Erich Leistenschneider (el(AT)erichl.net), Feb 01 2008

			2531 is a member of this sequence because it is a prime number and the product of its digits is 2*5*3*1 = 30 and the sum of the digits of this result is 3+0 = 3, which is also a prime number.

Erich Leistenschneider, Table of n, a(n) for n = 1..4095
Erich Lestenschneider's Article about this sequence (in Portuguese).
Erich Leistenschneider, Program used to generate the sequence (Linux)
Erich Leistenschneider, First 4095 numbers of the sequence

Subsequence of A038618 (zeroless primes).

a:=proc(n) local dn,pr,dpr: dn:=convert(n,base,10): pr:=mul(dn[i],i=1..nops(dn)): dpr:=convert(pr,base,10): if isprime(n)=true and isprime(add(dpr[j],j= 1..nops(dpr)))=true then n else end if end proc: seq(a(n),n=1..1600); # Emeric Deutsch, Mar 01 2008

Select[Prime[Range[300]],PrimeQ[Total[IntegerDigits[Times@@ IntegerDigits[#]]]]&] (* Harvey P. Dale, Dec 15 2011 *)

isok(p) = isprime(p) && isprime(sumdigits(vecprod(digits(p)))); \\ Michel Marcus, Jan 16 2019

cp's OEIS Frontend

A034303 Zeroless primes that become nonprime if any digit is deleted.

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A057628 Primes such that replacing each digit d with d copies of the digit d produces a prime. Zeros are not allowed.

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A101987 Product of nonzero digits of n-th prime.

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Maple

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A106101 Primes with minimal digit = 1.

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A386320 Primes without {0, 2} as digits.

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Magma

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A220488 Primes that have both prime digits (2,3,5,7) and nonprime digits (1,4,6,8,9), without digits "0".

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Mathematica

A228139 Primes such that the product of their digits subtracted from the prime number is another prime.

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Mathematica

PARI

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A259315 Nonprimes containing no zeros in decimal representation.

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