A053666 -id:A053666 - cp's OEIS frontend

A230041 Primes related to the strictly increasing subsequence of A053666.

2, 3, 5, 7, 19, 29, 37, 47, 59, 79, 89, 199, 269, 359, 379, 389, 479, 499, 599, 797, 887, 997, 1889, 1999, 2689, 2699, 2789, 2999, 3889, 3989, 4789, 4799, 4889, 4999, 6899, 8999, 25999, 27799, 28789, 28979, 29989, 37799, 37889, 39799, 39989, 48799, 48889

Offset: 1

Shyam Sunder Gupta, Oct 06 2013

a(1)=2; a(n+1) is the smallest prime with product of digits > product of digits of a(n).
From Wolfdieter Lang, Oct 31 2014: (Start)
A053666 is sieved as follows:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
2, 3, 5, 7, 1, 3, 7, 9, 6, 18,  3, 21,  4, 12, 28, ...
2, 3, 5, 7, x, x, x, 9, x, 18,  x, 21,  x,  x, 28,
and the related primes are:
2, 3, 5, 7,         19,    29,     37,         47, ...
(End)
------------------------------------------------------

			a(6) = 29, product of digits is 18; a(7) = 37, product of digits is 21 and 21 > 18.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..142

Cf. A000040, A069802, A053666.

a = {}; t = 0; Do[s = Apply[Times, IntegerDigits[Prime[n]]]; If[s > t, t = s; AppendTo[a, Prime[n]]], {n, 1, 10^4}]; a

Edited. Name specified. A000040, A053666 and 'easy' added by Wolfdieter Lang, Oct 31 2014

A069802 Primes related to the nondecreasing subsequence of A053666.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 29, 37, 47, 59, 79, 89, 199, 269, 349, 359, 379, 389, 479, 499, 599, 797, 887, 997, 1889, 1999, 2689, 2699, 2789, 2879, 2897, 2999, 3889, 3989, 4789, 4799, 4889, 4999, 6899, 8699, 8969, 8999, 25999, 27799, 27997, 28789, 28879

Offset: 1

Amarnath Murthy, Apr 13 2002

			A053666 is sieved in the following way:
1  2  3  4  5  6  7  8  9  10 11  12 13  14  15 16 ...
2  3  5  7  1  3  7  9  6  18  3  21  4  12  28 15 ...
2  3  5  7  x  x  7  9  x  18  x  21  x   x  28  x ...
with the related primes
2  3  5  7       17 19     29     37         47    ...
- _Wolfdieter Lang_, Nov 01 2014
-------------------------------------------------------

Cf. A067954, A053666.

t = 0; Do[s = Apply[ Times, IntegerDigits[ Prime[n]]]; If[s >= t, t = s; Print[ Prime[n]]], {n, 1, 10^4}]

Edited and extended by Robert G. Wilson v, Apr 15 2002

Edited, name specified, and cross-reference A053666 and example added by Wolfdieter Lang, Nov 01 2014

A107612 Primes with digital product = 2.

Original entry on oeis.org

2, 211, 2111, 111121, 111211, 112111, 1111211, 1111111121, 1111211111, 1121111111, 111111211111, 111211111111, 2111111111111, 111111111111112111, 111111112111111111, 111111211111111111, 112111111111111111

Offset: 1

Zak Seidov, May 17 2005

Corresponding indices of primes in A107611. Cf. A053666, A101987.

Harvey P. Dale, Table of n, a(n) for n = 1..684

Cf. A053666, A101987, A107611.

Cf. A004022, A107689, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

for i from 0 to 30 do it:=sum(10^j, j=0..i): for k from 0 to i do if isprime(it+10^k) then printf(`%d,`, it+10^k) fi: od:od: (Sellers)

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{2, Table[1, {n}]}]]], PrimeQ[ # ] &], {n, 0, 19}]] (* Robert G. Wilson v, May 19 2005 *)
Select[Flatten[Table[FromDigits/@Permutations[PadRight[{2},n,1]],{n,20}]],PrimeQ]//Sort (* Harvey P. Dale, May 28 2017 *)

A107612(n) = prime(A107611(n)).

More terms from Robert G. Wilson v and James Sellers, May 19 2005

A092518 Primes p with no zero digits such that (the digit product of p) plus p is also prime.

Original entry on oeis.org

23, 29, 61, 67, 83, 163, 233, 239, 283, 293, 347, 349, 431, 439, 443, 449, 499, 563, 569, 613, 617, 619, 653, 659, 677, 683, 743, 929, 941, 1123, 1163, 1217, 1231, 1237, 1249, 1289, 1297, 1321

Offset: 1

Ray G. Opao, Apr 06 2004

			a(2) = 29: 29+2(9) = 29+18 = 47 which is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A053666.

Select[Prime[Range[300]],DigitCount[#,10,0]==0&&PrimeQ[#+Times@@ IntegerDigits[ #]]&] (* Harvey P. Dale, Jan 27 2020 *)

dprod(n)=n=digits(n); prod(i=1, #n, n[i])
is(n)=my(d=dprod(n)); d && isprime(n+d) && isprime(n) \\ Charles R Greathouse IV, Dec 27 2013

Definition clarified by Harvey P. Dale, Jan 27 2020

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

A230082 a(n) is the smallest prime that is the first of n consecutive primes whose product of digits is equal and nonzero.

Original entry on oeis.org

2, 1913, 442619, 336737123, 123381165263, 11865678519229

Offset: 1

Shyam Sunder Gupta, Oct 08 2013

			a(3) = 442619, since 442619, 442633 and 442691 are three consecutive primes with product of digits as 1728 and this is the first occurrence of three consecutive primes whose product of digits is equal and nonzero.

Cf. A071613, A007954, A053666.

a(5)-a(6) from Giovanni Resta, Oct 08 2013

A230083 Smaller of two consecutive primes whose product of digits is equal and nonzero.

Original entry on oeis.org

1913, 2819, 6719, 14519, 16319, 18379, 19319, 21419, 29819, 34613, 35617, 35879, 36979, 37379, 37619, 37813, 39119, 45613, 46619, 46919, 49279, 51613, 55313, 56179, 56713, 58613, 62219, 63179, 65479, 66413, 74779, 75913, 76213, 76579, 76679, 79319, 82619

Offset: 1

Shyam Sunder Gupta, Oct 08 2013

			1913 is in the sequence because 1913 and 1931 are consecutive primes and the product of the digits of each = 27.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..4966

Cf. A230082, A007954, A053666.

a = {}; m = 1; s = 1; Do[If[(y = Apply[Times, IntegerDigits[x = Prime[n]]]) == s && s != 0, m = m + 1; If[m > 1, AppendTo[a, Prime[n - 1]]], m = 1]; s = y, {n, 1, 10000}]; a

A230200 Product of digits of n-th palindromic prime.

Original entry on oeis.org

2, 3, 5, 7, 1, 0, 3, 5, 8, 9, 9, 45, 63, 72, 98, 245, 392, 441, 81, 162, 0, 0, 0, 3, 4, 16, 28, 32, 27, 72, 81, 48, 112, 100, 125, 0, 108, 180, 216, 196, 441, 64, 256, 243, 648, 729, 0, 0, 0, 0, 0, 0, 45, 108, 144, 405, 720, 1152, 0, 225, 675, 1575, 648

Offset: 1

Shyam Sunder Gupta, Oct 11 2013

			a(6) = 0, since product of digits of 6th palindromic prime, that is, 101 is 0.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..5953

Cf. A002385, A007954, A053666.

a = {}; Do[z = n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[z], s = Apply[Times, IntegerDigits[z]]; AppendTo[a, s]], {n, 1, 10^5}]; Insert[a, 1, 5]
Times@@IntegerDigits[#]&/@Select[Prime[Range[5000]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 17 2017 *)

A230084 Smallest of three consecutive primes whose product of digits is equal and nonzero.

Original entry on oeis.org

442619, 2483219, 6325619, 7567919, 7886519, 9883673, 9962219, 11117123, 14669519, 15446819, 17958419, 21337279, 23623129, 26453671, 26872919, 27234419, 27536519, 27948343, 32638213, 32964341, 33539783, 33813419, 34277819, 34554719, 35732381, 37571519

Offset: 1

Shyam Sunder Gupta, Oct 08 2013

			442619 is in the sequence because 442619,442633 and 442691 are consecutive primes and the product of the digits of each = 1728.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..1713

Cf. A230082, A230083, A007954, A053666.

a = {}; m = 1; s = 1; Do[ If[(y = Apply[Times, IntegerDigits[x = Prime[n]]]) == s && s != 0, m = m + 1; If[m > 2, AppendTo[a, Prime[n - 2]]], m = 1]; s = y, {n, 1, 100000}]; a
Transpose[Select[Partition[Prime[Range[23*10^5]],3,1],Times@@ IntegerDigits[ #[[1]]]==Times@@IntegerDigits[#[[2]]] == Times@@ IntegerDigits[#[[3]]]>0&]][[1]] (* Harvey P. Dale, Apr 05 2016 *)
pdeQ[{a_,b_,c_}]:=Module[{un=Union[Times@@@IntegerDigits[{a,b,c}]]},un != {0} && Length[un]==1]; Select[Partition[Prime[Range[23*10^5]],3,1],pdeQ][[All,1]] (* Harvey P. Dale, Feb 01 2022 *)

A370848 Lesser of two consecutive primes such that the product of its digits is also prime and the sum of the digits of the other is composite.

Original entry on oeis.org

13, 17, 31, 71, 113, 1151, 11131, 112111, 113111, 131111, 1111211, 1111711, 11111117, 11111171, 71111111, 115111111, 1111111121, 1111115111, 1115111111, 1117111111, 1151111111, 1711111111, 11111111113, 11113111111, 31111111111, 111113111111, 111511111111, 1111171111111

Offset: 1

Claude H. R. Dequatre, Mar 03 2024

			13 is a term because 13 is prime, the product of its digits is 3 which is also prime and the sum of the digits of 17, the next prime to 13, is 8 which is composite.
23 is not a term because the product of its digits is 6 which is not prime.
131 is not a term because although it is prime and the product of its digits is 3 which is also prime, the sum of the digits of 137, the next prime to 131, is 11 which is not composite.

Hugo Pfoertner, Table of n, a(n) for n = 1..3701

Cf. A000040, A002808, A007605, A053666.
Cf. A370850.
Except for the first, all terms of this sequence are in A370851.

Select[Prime[Range[5*10^6]],PrimeQ[Apply[Times,IntegerDigits[#]]]&&CompositeQ[Total[IntegerDigits[NextPrime[#]]]]&] (* James C. McMahon, Mar 03 2024 *)

isok(p)=my(x=vecprod(digits(p)),y=sumdigits(nextprime(p+1)));isprime(x) && !isprime(y);
forprime(p=2,20000,if(isok(p),print1(p", ")))

a370848(maxdigits=20) = {my (L=List()); for (n=2, maxdigits, my (r=(10^n-1)/9, d=digits(r)); foreach ([2,3,5,7], s, for (k=1, #d, my (dd=d); dd[k]=s; my(q=fromdigits(dd)); if (ispseudoprime(q) && ! isprime(sumdigits(nextprime(q+1))), listput(L,q))))); vecsort(Vec(L))};
a370848() \\ Hugo Pfoertner, Mar 03 2024

from itertools import count, islice
from sympy import isprime, nextprime
def A370848_gen(): # generator of terms
    for l in count(1):
        k = (10**l-1)//9
        for m in range(l):
            a = 10**m
            for j in (1,2,4,6):
                p = k+a*j
                if isprime(p) and not isprime(sum(map(int,str(nextprime(p))))):
                    yield p
A370848_list = list(islice(A370848_gen(),20)) # Chai Wah Wu, Mar 25 2024

a(17)-a(21) from Michel Marcus, Mar 03 2024

a(22)-a(28) from Hugo Pfoertner, Mar 03 2024

cp's OEIS Frontend

A230041 Primes related to the strictly increasing subsequence of A053666.

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A069802 Primes related to the nondecreasing subsequence of A053666.

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A107612 Primes with digital product = 2.

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A092518 Primes p with no zero digits such that (the digit product of p) plus p is also prime.

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A101987 Product of nonzero digits of n-th prime.

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A230082 a(n) is the smallest prime that is the first of n consecutive primes whose product of digits is equal and nonzero.

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A230083 Smaller of two consecutive primes whose product of digits is equal and nonzero.

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Mathematica

A230200 Product of digits of n-th palindromic prime.

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