Felice Russo - cp's OEIS frontend

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

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

A171765 a(n) = 0 if n <= 10; for n >= 11, a(n) = product of digits of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Felice Russo, Oct 11 2010

Cf. A007954.

A171765 := proc(n)
    if n<= 10 then
        0;
    else
        A007954(n);
    end if:
end proc: # R. J. Mathar, Jan 11 2020

Table[If[n<11,0,Times@@IntegerDigits[n]],{n,0,100}] (* Harvey P. Dale, Aug 02 2016 *)

a(n)=if(n<=9,0,n=digits(n); prod(i=1,#n,n[i])) \\ Charles R Greathouse IV, Aug 25 2014

A134997 Number of dihedral primes with n digits.

Original entry on oeis.org

2, 1, 2, 2, 1, 14, 40, 52, 228, 482, 1592, 4758, 15810, 53202, 197429, 725196

Offset: 1

Felice Russo

C. K. Caldwell, The Prime Glossary, Dihedral Prime
C. Rivera, Source

Cf. A038136, A048662, A134996.

a(1) corrected by Patrick Capelle, Feb 06 2008

a(11)-a(16) from Hiroaki Yamanouchi, Sep 04 2014

A134998 Dihedral palindromic primes.

Original entry on oeis.org

2, 5, 11, 101, 181, 18181, 1008001, 1022201, 1055501, 1082801, 1085801, 1180811, 1208021, 1221221, 1250521, 1280821, 1508051, 1520251, 1551551, 1580851, 1802081, 1805081, 1880881, 1881881, 100111001, 100888001, 108101801

Offset: 1

Felice Russo

C. K. Caldwell, The Prime Glossary, Dihedral Prime
C. Rivera, Source

Cf. A134996, A134997.

5 added by Patrick Capelle, Feb 06 2008

A064156 Smallest prime with n decimal digits such that the product of its digits equals n times the sum of its digits, or 0 if no such prime exists.

Original entry on oeis.org

2, 0, 167, 1427, 0, 126241, 1111457, 12222241, 111113543, 0, 0, 111111118273, 0, 0, 111111111126581, 1111111111144841, 0, 111111111111126443, 0, 11111111111122225421, 111111111111111135781, 0, 0, 111111111111111111244561, 1111111111111111121255521, 0, 111111111111111111111113797

Offset: 1

Felice Russo, Sep 14 2001

			167 belongs to the sequence because this is the smallest prime with 3 digits such that 1*6*7=42 and 3*(1+6+7)=42

Max Alekseyev, Table of n, a(n) for n=1..200

NextPrim[ n_ ] := (k = n + 1; While[ ! PrimeQ[ k ], k++ ]; k); Do[ If[ n != 1 && Transpose[ FactorInteger[ n ] ][ [ 1, -1 ] ] <= 10, k = NextPrim[ (10^n - 1)/9 ];
While[ d = IntegerDigits[ k ]; k < 10^n && n*Apply[ Plus, d ] != Apply[ Times, d ], k = NextPrim[ k ] ]; If[ k < 10^n, Print[ k ], Print[ 0 ] ], If[ n == 1, Print[ 2 ], Print[ 0 ] ] ], {n, 1, 9} ]

Corrected and extended by Robert G. Wilson v, Oct 05 2001

a(14), a(20), a(25) and b-file from Max Alekseyev, May 07 2009

A067779 Primes such that the sum of the squares of its digits is equal to the product of its digits.

Original entry on oeis.org

11353, 13513, 15313, 15331, 31153, 31513, 31531, 33151, 35311, 51133, 53113, 1125221, 1212251, 1212521, 1221251, 1252211, 1512221, 2115221, 2122151, 2122511, 2151221, 2152211, 2215211, 2221511, 2251121, 2251211, 5122121

Offset: 1

Felice Russo, Feb 06 2002

			An eight-digit term is 11224121, a ten-digit term is 1111111843.
11353 belongs to the sequence because 1^2+1^2+3^2+5^2+3^2=45=1*1*3*5*3

forprime(p=2,6e6, n=p; sd=0; pd=1; while(n>0,d=divrem(n,10); n=d[1]; sd=sd+d[2]*d[2]; pd=pd*d[2]); if(sd==pd,print1(p,",")))

Edited and extended by Klaus Brockhaus Feb 11 2002

A063898 Smallest k > 0 such that k + F_n are all primes, where F_n is the n-th Fermat number.

Original entry on oeis.org

2, 2, 2, 14, 14, 14, 66746, 475424, 12124166, 14899339904

Offset: 0

Felice Russo, Aug 29 2001

Is this sequence finite?

			For j=3 a(3)=2 because 257+2, 17+2, 5+2, 3+2 are all primes.
For j=4 a(4)=14 because 65537+14, 257+14, 17+14, 5+14, 3+14 are all primes.

Cf. A000215 (Fermat numbers).

okprimep(mink, vecf) = {for (i=1, #vecf, if (! isprime(mink + vecf[i]), return (0));); return (1);}
a(n) = {mink = 1; vecf = vector(n+1, i, 2^(2^(i-1)) + 1); while (! okprimep(mink, vecf), mink++); mink;} \\ Michel Marcus, Sep 28 2013

a(10) from Donovan Johnson, Oct 12 2008

A064153 Values of n that are not solution of pr(x)=n where pr(x) is the prime partition function of x.

Original entry on oeis.org

8, 11, 13, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Felice Russo, Sep 13 2001

Cf. A000607.

More terms from Michel Marcus, Sep 28 2013

A072947 Primes in A053065.

Original entry on oeis.org

2, 191373251117, 37291913732511172331, 1391311131071018979716153433729191373251117233141475967738397103109127137149

Offset: 1

Felice Russo, Aug 20 2002

nonn

The next term is too large to include.

Cf. A053065.

A073958 Fibonacci numbers for which the number of prime factors (with multiplicity) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 233, 377, 610, 987, 1597, 2584, 4181, 10946, 17711, 28657, 75025, 121393, 514229, 1346269, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 165580141, 433494437

Offset: 1

Felice Russo, Sep 03 2002

The prime Fibonacci numbers, A005478, are a subsequence.

			Example: 8=2*2*2, the number of prime factors is equal to 3, a Fibonacci number.

Charles R Greathouse IV, Table of n, a(n) for n = 1..328

Cf. A000045.

With[{fibs=Fibonacci[Range[0,50]]},Rest[Select[fibs,MemberQ[fibs, PrimeOmega[#]]&]]] (* Harvey P. Dale, Oct 27 2011 *)

isFibonacci(n)=my(k=n^2);k+=((k + 1) << 2);issquare(k) || (n > 0 && issquare(k-8))
select(n->isFibonacci(bigomega(n)), vector(99,i,fibonacci(i+1))) \\ Charles R Greathouse IV, Jun 17 2013

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User: Felice Russo

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

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A171765 a(n) = 0 if n <= 10; for n >= 11, a(n) = product of digits of n.

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PARI

A134997 Number of dihedral primes with n digits.

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A134998 Dihedral palindromic primes.

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A064156 Smallest prime with n decimal digits such that the product of its digits equals n times the sum of its digits, or 0 if no such prime exists.

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Mathematica

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A067779 Primes such that the sum of the squares of its digits is equal to the product of its digits.

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PARI

Extensions

A063898 Smallest k > 0 such that k + F_n are all primes, where F_n is the n-th Fermat number.

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PARI

Extensions

A064153 Values of n that are not solution of pr(x)=n where pr(x) is the prime partition function of x.

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Extensions

A072947 Primes in A053065.

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A073958 Fibonacci numbers for which the number of prime factors (with multiplicity) is a Fibonacci number.

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