A081988 -id:A081988 - cp's OEIS frontend

A066725 Primes whose product of digits + 1 is also prime.

2, 11, 23, 29, 41, 43, 47, 61, 67, 89, 149, 163, 167, 211, 223, 227, 229, 233, 251, 257, 263, 269, 281, 349, 367, 383, 419, 431, 433, 439, 463, 491, 521, 523, 569, 587, 613, 617, 631, 643, 659, 661, 673, 761, 769, 821, 827, 857, 883, 887, 929, 941, 967, 1123

Offset: 1

Joseph L. Pe, Jan 15 2002

			2 * 2 * 3 + 1 = 13, a prime; since 223 is also prime, it is a term of the sequence.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Primes in A081988.

Cf. A011540, A084354.

Select[Prime[Range[200]],PrimeQ[Times@@IntegerDigits[#]+1]&] (* Harvey P. Dale, May 09 2012 *)

isok(p)={ isprime(p) && isprime(vecprod(digits(p)) + 1) } \\ Harry J. Smith, Mar 19 2010

A084354 MINUS A011540. - R. J. Mathar, Aug 26 2007

Terms a(35)-a(54) from Harry J. Smith, Mar 19 2010

A084979 Palindromes such that the product of the digits + 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 11, 22, 44, 66, 111, 121, 141, 161, 212, 232, 242, 272, 292, 323, 343, 383, 414, 464, 474, 545, 565, 616, 626, 636, 656, 747, 838, 848, 878, 898, 929, 969, 1111, 1221, 1441, 1661, 2112, 2222, 2332, 2552, 2772, 2882, 3223, 3883, 4114, 4444, 4554

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 21 2003

			383 is a term since 3*8*3 = 72, 72+1 = 73 is prime.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..10928

Cf. A081988.

Select[ Range[4663], FromDigits[ Reverse[ IntegerDigits[ # ]]] == # && PrimeQ[1 + Times @@ IntegerDigits[ # ]] & ]
Parallelize[While[True,If[PalindromeQ[n]&&PrimeQ[1+Product[Part[IntegerDigits[n],k],{k,1,Length[IntegerDigits[n]]}]],Print[n]];n++];n] (* J.W.L. (Jan) Eerland, Dec 27 2021 *)

from math import prod
from sympy import isprime
from itertools import count, islice, product
def cond(n): return isprime(prod(map(int, str(n))) + 1)
def pals(): # generator of palindromes as strings
    digits = "0123456789"
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]:
                yield int(left + mid + right)
def agen(): yield from filter(cond, pals())
print(list(islice(agen(), 51))) # Michael S. Branicky, Aug 22 2022

a(n) >> n^k, where k = log_3(10) = 2.0959.... - Charles R Greathouse IV, Aug 02 2010

Edited, corrected and extended by Robert G. Wilson v, Jun 21 2003

Formula by Charles R Greathouse IV, Aug 02 2010

A244748 Numbers k such that (product of digits of k)^2 + 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 11, 12, 14, 16, 21, 22, 23, 25, 27, 28, 32, 38, 41, 44, 45, 46, 49, 52, 54, 58, 61, 64, 66, 69, 72, 78, 82, 83, 85, 87, 94, 96, 111, 112, 114, 116, 121, 122, 123, 125, 127, 128, 132, 138, 141, 144, 145, 146, 149, 152, 154, 158, 161, 164, 166, 169, 172, 178, 182, 183

Offset: 1

Derek Orr, Jul 12 2014

A number k is a term of this sequence iff A007954(k)^2 is in A006093.

This sequence is infinite. With any number a(n), you can add infinitely many 1's to its decimal representation. E.g., 85 is in this sequence, so 185, 815, 851, 1185, 1815, 18115, etc. are terms as well.

			(7*2)^2 + 1 = 197 is prime. Thus 72 is a term of this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A081988, A007954.

for(n=1,10^3,d=digits(n);if(ispseudoprime(prod(i=1,#d,d[i])^2+1),print1(n,", ")))

Corrected by Jens Kruse Andersen, Jul 13 2014

A244607 Numbers k such that (product of digits of k) - 1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 13, 14, 16, 18, 22, 23, 24, 26, 27, 29, 31, 32, 34, 36, 38, 41, 42, 43, 45, 46, 48, 54, 56, 61, 62, 63, 64, 65, 67, 68, 69, 72, 76, 81, 83, 84, 86, 89, 92, 96, 98, 113, 114, 116, 118, 122, 123, 124, 126, 127, 129, 131, 132, 134, 136, 138, 141, 142, 143, 145, 146

Offset: 1

Derek Orr, Jul 01 2014

This sequence is infinite. With any number a(n), you can add infinitely many 1's to its decimal representation. E.g., 32 is in this sequence, so 321, 312, 3211, 32111, 31121, 11321, etc. are also terms of this sequence.

A number k is a term of this sequence iff A007954(k) is in A008864.

			3*2 - 1 = 5 is prime. Thus 32 is a term of this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A007954, A008864, A089377, A081988.

for(n=1,10^3,d=digits(n);p=prod(i=1,#d,d[i]);if(isprime(p-1),print1(n,", ")))

Wrong term removed by Jens Kruse Andersen, Jul 13 2014

A245017 Numbers k such that (product of digits of k) + 1 and (product of digits of k)^2 + 1 are both prime.

Original entry on oeis.org

1, 2, 4, 6, 11, 12, 14, 16, 21, 22, 23, 25, 28, 32, 41, 44, 49, 52, 58, 61, 66, 82, 85, 94, 111, 112, 114, 116, 121, 122, 123, 125, 128, 132, 141, 144, 149, 152, 158, 161, 166, 182, 185, 194, 211, 212, 213, 215, 218, 221, 224, 229, 231, 236, 242, 245, 251, 254, 263, 279, 281, 292

Offset: 1

Derek Orr, Jul 12 2014

A number k is a term of this sequence iff A007954(k) and A007954(k)^2 are both in A006093.

This sequence is infinite. With any number a(n), you can add infinitely many 1's to its decimal representation. E.g., 82 is in this sequence, so 821, 812, 1182, 18112, 81211, etc. are also terms of this sequence.

			(9*4) + 1 = 37 is prime and (9*4)^2 + 1 = 1297 is prime. Thus 94 is a term of this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A007954, A009994, A081988, A244748.

bpQ[n_]:=Module[{c=Times@@IntegerDigits[n]},AllTrue[{c+1,c^2+1},PrimeQ]]; Select[Range[300],bpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 09 2019 *)

for(n=1, 10^3, d=digits(n); p=prod(i=1, #d, d[i]); if(ispseudoprime(p+1) && ispseudoprime(p^2 + 1), print1(n,", ")))

cp's OEIS Frontend

A066725 Primes whose product of digits + 1 is also prime.

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A084979 Palindromes such that the product of the digits + 1 is prime.

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A244748 Numbers k such that (product of digits of k)^2 + 1 is prime.

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A244607 Numbers k such that (product of digits of k) - 1 is prime.

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A245017 Numbers k such that (product of digits of k) + 1 and (product of digits of k)^2 + 1 are both prime.

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