A052091 -id:A052091 - cp's OEIS frontend

A053600 a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

2, 727, 37273, 333727333, 93337273339, 309333727333903, 1830933372733390381, 92183093337273339038129, 3921830933372733390381293, 1333921830933372733390381293331, 18133392183093337273339038129333181

Offset: 1

G. L. Honaker, Jr., Jan 20 2000

			As a triangle:
.........2
........727
.......37273
.....333727333
....93337273339
..309333727333903
1830933372733390381

G. L. Honaker, Jr. and Chris K. Caldwell, Palindromic Prime Pyramids, J. Recreational Mathematics, Vol. 30(3) 169-176, 1999-2000.

Clark Kimberling, Table of n, a(n) for n = 1..200
P. De Geest, Palindromic Prime Pyramid Puzzle by G.L.Honaker,Jr
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 18133...33181 (35-digits)
G. L. Honaker, Jr. & C. K. Caldwell, Palindromic Prime Pyramids
G. L. Honaker, Jr. & C. K. Caldwell, Supplement to "Palindromic Prime Pyramids"
Ivars Peterson, Primes, Palindromes, and Pyramids, Science News.
Inder J. Taneja, Palindromic Prime Embedded Trees, RGMIA Res. Rep. Coll. 20 (2017), Art. 124.
Inder J. Taneja, Same Digits Embedded Palprimes, RGMIA Research Report Collection (2018) Vol. 21, Article 75, 1-47.

Cf. A000040, A002385, A047076, A052205, A034276, A256957, A052091, A052092, A261881.

d[n_] := IntegerDigits[n]; t = {x = 2}; Do[i = 1; While[! PrimeQ[y = FromDigits[Flatten[{z = d[i], d[x], Reverse[z]}]]], i++]; AppendTo[t, x = y], {n, 10}]; t (* Jayanta Basu, Jun 24 2013 *)

from gmpy2 import digits, mpz, is_prime
A053600_list, p = [2], 2
for _ in range(30):
    m, ps = 1, digits(p)
    s = mpz('1'+ps+'1')
    while not is_prime(s):
        m += 1
        ms = digits(m)
        s = mpz(ms+ps+ms[::-1])
    p = s
    A053600_list.append(int(p)) # Chai Wah Wu, Apr 09 2015

A052205 a(n+1) is smallest palindromic prime containing exactly 3 more digits on each end than the previous term, with a(n) as a central substring.

Original entry on oeis.org

2, 1022201, 1051022201501, 1241051022201501421, 1071241051022201501421701, 1051071241051022201501421701501, 1091051071241051022201501421701501901, 1351091051071241051022201501421701501901531

Offset: 1

G. L. Honaker, Jr., Jan 28 2000

Giovanni Resta, Table of n, a(n) for n = 1..26 (full sequence)

Cf. A047076, A053600, A034276, A256957, A052091, A052092.

Shown to be finite by Felice Russo

A052092 Lengths of the palindromic primes from Honaker's sequence A053600.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 19, 23, 25, 31, 35, 41, 45, 49, 55, 59, 63, 69, 75, 81, 87, 93, 99, 105, 109, 113, 119, 125, 129, 133, 139, 145, 151, 157, 161, 167, 173, 179, 185, 191, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297

Offset: 0

Patrick De Geest, Jan 15 2000

Since the terms from a(34) onward are currently only probable primes, the lengths given in this sequence beyond that point are only provisional.

For n > 0, a(n) = a(n-1)+2*m where m is the number of digits of A052091(n). - Chai Wah Wu, Dec 03 2015

Chai Wah Wu, Table of n, a(n) for n = 0..501

Cf. A052091, A053600, A047076.

from sympy import isprime
A052092_list, l, p = [1], 1, 2
for _ in range(100):
    m, ps = 1, str(p)
    s = int('1'+ps+'1')
    while not isprime(s):
        m += 1
        ms = str(m)
        if ms[0] in '268':
            ms = str(int(ms[0])+1) + '0'*(len(ms)-1)
            m = int(ms)
        if ms[0] in '45':
            ms = '7' + '0'*(len(ms)-1)
            m = int(ms)
        s = int(ms+ps+ms[::-1])
    p = s
    l += 2*len(ms)
    A052092_list.append(l) # Chai Wah Wu, Dec 02 2015

Comments from G. L. Honaker, Jr., Mar 30 2000

Edited by T. D. Noe, Oct 30 2008

A034276 Smallest prime that generates a prime pyramid of height n.

Original entry on oeis.org

11, 29, 2, 5, 41, 251, 43, 145577, 51941, 4372877, 26901631, 366636187, 15387286403, 218761753811, 3313980408469

Offset: 1

Felice Russo, Jan 25 2000

Let p be prime; look for the smallest prime in {1|p|1, 3|p|3, 7|p|7, 9|p|9}, where '|' stands for concatenation; repeat until no such prime can be found; then height(p) = number of rows in pyramid.

a(13) > 10^10. - Donovan Johnson, Aug 13 2010

			Example for p=43: 43 3433 334333 93343339 3933433393 939334333939 39393343339393, stop; height(43)=7.

H. Heinz, Patterns in Primes, Illustrating Two Prime Pyramids
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A047076, A053600, A052205, A256957, A052091, A052092.

More terms from Naohiro Nomoto, Jul 14 2001
a(11)-a(12) from Donovan Johnson, Aug 13 2010
a(13) from Chai Wah Wu, Apr 10 2015
a(14)-a(15) from Giovanni Resta, May 15 2020

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A053600 a(1) = 2; for n>=1, a(n+1) is the smallest palindromic prime with a(n) as a central substring.

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A052205 a(n+1) is smallest palindromic prime containing exactly 3 more digits on each end than the previous term, with a(n) as a central substring.

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A052092 Lengths of the palindromic primes from Honaker's sequence A053600.

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Python

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A034276 Smallest prime that generates a prime pyramid of height n.

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Extensions