A052205 -id:A052205 - 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

A110105 a(n) is the number of coverings of 1..n by cyclic words of length n, such that each value from 1 to n appears precisely 3 times. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,1,2,2,2,...,n,n,n}. Repeats of words are not allowed in a given covering.

Original entry on oeis.org

1, 1, 2, 12, 192, 5744, 260904, 16542648, 1395722688, 151232990208, 20468918305536, 3384387717897216, 671260382408564352, 157302245641224362112, 42996605332700377396992, 13558408172347636250832384, 4885584146166061652811300864, 1994958243661170192648338792448

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive.

			a(2)=2 because the two cyclic word coverings are {112, 221} and {111, 222}.
a(3)=12: {111 222 333} {111 223 233} {112 122 333} {112 133 223} {113 122 233} {113 123 223} {113 132 223} {112 132 233} {113 133 222} {122 123 133} {122 132 133} {112 123 233}.

Vaclav Kotesovec, Table of n, a(n) for n = 0..240

Cf. A052205, A110104, A110106, A108242.

RecurrenceTable[{-(-10+n) (-9+n) (-8+n) (-7+n) (-6+n) (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (-49-243 n+243 n^2) a[-11+n]-126 (-9+n) (-8+n) (-7+n) (-6+n) (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) a[-10+n]-2 (-8+n) (-7+n) (-6+n) (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (-130-162 n+243 n^2) a[-9+n]+6 (-7+n) (-6+n) (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (196+1166 n-1458 n^2+243 n^3) a[-8+n]+3 (-6+n) (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (-931-117 n+243 n^2) a[-7+n]+54 (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (61-63 n+9 n^2) a[-6+n]-(-4+n) (-3+n) (-2+n) (-1+n) (-3686-21339 n+38682 n^2-17496 n^3+2187 n^4) a[-5+n]-18 (-3+n) (-2+n) (-1+n) (412+410 n-918 n^2+243 n^3) a[-4+n]+18 (-2+n) (-1+n) (14-1659 n+2867 n^2-1458 n^3+243 n^4) a[-3+n]-6 (-1+n) (-344+680 n-810 n^2+243 n^3) a[-2+n]-3 (118-2013 n+3984 n^2-2916 n^3+729 n^4) a[-1+n]+6 (437-729 n+243 n^2) a[n]==0, a[0]==1, a[1]==1, a[2]==2, a[3]==12, a[4]==192, a[5]==5744, a[6]==260904, a[7]==16542648, a[8]==1395722688, a[9]==151232990208, a[10]==20468918305536}, a, {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2016 *)

Differential equation satisfied by e.g.f.: {( - 6 + 12*t - 138*t^7 - 12*t^12 + 213*t^6 - 92*t^8 - 126*t^9 - 9*t^14 - 170*t^4 + 54*t^2 + 162*t^11 - 72*t^3 + 162*t^5 + 38*t^10)*F(t) + (6 + 54*t^12 + 72*t^4 + 126*t^3 + 54*t^6 - 324*t^9 - 156*t^8 - 42*t^2 - 18*t - 36*t^10 + 594*t^7 - 378*t^5)*(d/dt)F(t) + (-81*t^10 - 9*t^2 + 216*t^8 - 198*t^6 + 72*t^4)*(d^2/dt^2)F(t), F(0) = 1}.
Recurrence satisfied by a(n): {a(0) = 1, a(10) = 20468918305536, a(11) = 3384387717897216, a(12) = 671260382408564352, a(2) = 2, a(3) = 12, a(4) = 192, a(5) = 5744, a(6) = 260904, a(7) = 16542648, a(8) = 1395722688, a(9) = 151232990208, a(1) = 1, 0 = (3*n^12 + 618210450*n^5 + 20779902*n^7 + 4242044664*n^3 + 134970693*n^6 + 4459328640*n + 1971620508*n^4 + 1437004800 + 5794678656*n^2 + 234*n^11 + 8151*n^10 + 167310*n^9 + 2248389*n^8)*a(n) + (25151175*n^5 + 12450*n^8 + 3000165*n^6 + 1919851200*n + 143497300*n^4 + 549556500*n^3 + 1350370080*n^2 + 5*n^10 + 375*n^9 + 1197504000 + 240750*n^7)*a(n + 2) + (-116250876*n^5 - 18*n^10 - 12385923840*n - 711103032*n^4 - 2944635984*n^3 - 7897844736*n^2 - 8622028800 - 1404*n^9 - 48708*n^8 - 989496*n^7 - 13032306*n^6)*a(n + 3) + (-748*n^7 - 24541132*n^3 - 22022*n^6 - 3770459*n^4 - 98660628*n^2 - 219542400 - 366520*n^5 - 223906320*n - 11*n^8)*a(n + 4) + (240408*n^5 + 2653854*n^4 + 18626328*n^3 + 81157896*n^2 + 200675232*n + 215550720 + 6*n^8 + 432*n^7 + 13524*n^6)*a(n + 5) + (84272481*n^3 + 1083375*n^5 + 11978658*n^4 + 27*n^8 + 60885*n^6 + 914771880*n + 1944*n^7 + 987940800 + 368381790*n^2)*a(n + 6) + (6874416*n^2 + 26085888*n + 75330*n^4 + 961740*n^3 + 41057280 + 54*n^6 + 3132*n^5)*a(n + 7) + (-63*n^6 - 58045680 - 92897*n^4 - 1223139*n^3 - 9036160*n^2 - 35519268*n - 3753*n^5)*a(n + 8) + (-1188*n^2 - 13032*n - 47520 - 36*n^3)*a(n + 9) + (634392 + 232902*n + 45*n^4 + 32067*n^2 + 1962*n^3)*a(n + 10) + (-3024 - 540*n - 24*n^2)*a(n + 11) + (-9*n^2 - 1410 - 225*n)*a(n + 12) + 6*a(n + 13)}.
a(n) ~ 3^(n+1/2) * n^(2*n) / (2^n * exp(2*n)). - Vaclav Kotesovec, Feb 28 2016

Original recurrence corrected by Vaclav Kotesovec, following a suggestion of Matthew House, Feb 28 2016

More terms from Vaclav Kotesovec, Feb 28 2016

A256957 Smallest palindromic prime that generates a palindromic prime pyramid of height n.

Original entry on oeis.org

11, 131, 2, 5, 10301, 16361, 10281118201, 35605550653, 7159123219517, 17401539893510471, 3205657651567565023, 14736384418081448363741

Offset: 1

Felice Russo, Jan 25 2000

Start with a palindromic prime p; look for smallest palindromic prime that has previous term as a centered substring and has 2 more digits (i.e., one more digit at each end); repeat until no such palindromic prime can be found; then height(p) = number of rows in pyramid. Each row of pyramid must be the smallest prime that can be used. Then a(n) = smallest value of p that generates a pyramid of height n.

			a(1) = 11.
a(4) = 5:
5
151
31513
3315133, stop;
height(5)=4.
a(6)=16362:
16361
1163611
311636113
33116361133
3331163611333
333311636113333, stop;
height(16361)=6.

G. L. Honaker, Jr. and Chris K. Caldwell, Palindromic prime pyramids
Ivars Peterson's MathTrek, Primes, Palindromes, and 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. A034276, A052205, A053600.

Added a(10)-a(11) and corrected a(4) -  Chai Wah Wu, Apr 09 2015
Entry revised by N. J. A. Sloane, Apr 13 2015
a(12) from Michael S. Branicky, Oct 28 2024

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

A110106 a(n) is the number of coverings of 1..n by cyclic words of length 3n, such that each value from 1 to n appears precisely twice. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,2,2,...,n,n}. Repeats of words are allowed in a given covering.

Original entry on oeis.org

1, 6, 3960, 24151680, 577882166400, 38039350155206400, 5605398331566095462400, 1614162682147590619140096000, 824800497779996439355497811968000

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive.

			a(1)=6: {123, 132} {112, 233} {113, 322} {133, 122} {123, 123} {132, 132}.

Cf. A052205, A110104, A110105, A108242.

RecurrenceTable[{(40320 + 328752*n + 78732*n^7 + 6561*n^8 + 1816668*n^3 + 1818369*n^4 + 1102248*n^5 + 398034*n^6 + 1063116*n^2) * a[n] + (-161280 - 508608*n - 453600*n^3 - 173340*n^4 - 34992*n^5 - 2916*n^6 - 661104*n^2) * a[n + 1] + (12432 + 20070*n + 12114*n^2 + 3240*n^3 + 324*n^4) * a[n + 2] - 2*a[n + 3] == 0, a[0] == 1, a[1] == 6, a[2] == 3960}, a, {n, 0, 15}] (* Vaclav Kotesovec, Oct 24 2023 *)

Differential equation satisfied by F(t)=sum a(n) t^(3n)/(3n!) {F(0) = 1, (6*t^2-12*t^5+t^8)*F(t) + (-4*t^6-2+16*t^3)*(d/dt)F(t) + 4*t^4*(d^2/dt^2)F(t)};
recurrence satisfied by a(n): {(40320 + 328752*n + 78732*n^7 + 6561*n^8 + 1816668*n^3 + 1818369*n^4 + 1102248*n^5 + 398034*n^6 + 1063116*n^2)*a(n) + (-161280 - 508608*n - 453600*n^3 - 173340*n^4 - 34992*n^5 - 2916*n^6 - 661104*n^2)*a(n+1) + (12432 + 20070*n + 12114*n^2 + 3240*n^3 + 324*n^4)*a(n+2) - 2*a(n+3), a(1) = 6, a(0) = 1, a(2) = 3960}.
a(n) ~ 2^n * 3^(4*n + 1/2) * n^(4*n) / exp(4*n). - Vaclav Kotesovec, Oct 24 2023

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.

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A256957 Smallest palindromic prime that generates a palindromic prime pyramid of height n.

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

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A110106 a(n) is the number of coverings of 1..n by cyclic words of length 3n, such that each value from 1 to n appears precisely twice. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,2,2,...,n,n}. Repeats of words are allowed in a given covering.

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A376130 a(1) = 3; for n > 1, a(n) is the smallest palindromic prime containing exactly 3 more digits on each end than a(n-1), with a(n-1) as a central substring.

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