A109864 -id:A109864 - cp's OEIS frontend

A225903 The smallest number beginning with n whose distinct prime factors are the first n primes.

16, 24, 30, 420, 50820, 60060, 7147140, 87297210, 9369900540, 103515091680, 11030826957150, 126152548291770, 13387011595197240, 143910374648370330, 15372244564712285250, 162945792385950223650, 17304843151387913751630, 1876614101750511535732320

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 21 2013

a(3)=30 is the only term with fewer than 1000 digits whose superscripts are all 1.

Though counterexamples are possible, it appears that the sequence is strictly increasing (confirmed for n < 350, and counterexamples are increasingly unlikely statistically thereafter).

			For a(6), the number 60060 = 2^2 * 3 * 5 * 7 * 11 * 13. The only number smaller whose factors contains the first 6 primes is 30030, which does not begin with 6.

Christian N. K. Anderson, Table of n, a(n) for n = 1..349
Christian N. K. Anderson, Table of n, the first digits of a(n), the number of digits of a(n), and all factors of a(n) whose superscript is >1

Cf. A000040, A002110.
Cf. A086558, A109864, A088104
Cf. A018802, A077515, A077727

a[n_] := Block[{p = Prime[n], ba = Product[Prime@k, {k, n}], d = IntegerDigits@ n, mu = 1}, While[d != Take[IntegerDigits[mu*ba], Length@d] || Max[ First /@ FactorInteger[mu]] > p, mu++]; mu*ba]; Array[a, 20] (* Giovanni Resta, May 27 2013 *)

library(gmp); primes<-function(n) { x=as.bigz(rep(2,n)); for(i in 2:n) x[i]=nextprime(x[i-1]); as.vector(x[1:n]) }
newmin<-function(b,d) { if(d>length(b)) return();
    while(1) { b[d]=b[d]+1; if((x=prod(pr^b))>v) return()
        if(substr(x,1,ndig(i))==as.character(i)) { v<<-x; return() }
        if(b[d]==2) {b[d]=1; newmin(b,d+1); b[d]=2 }
        newmin(b,d+1)
    }
}
y=as.bigz(rep(0,50))
for(i in 1:50) {
    pr=primes(i); b=rep(1,i)
    while(substr((v=prod(pr^b)),1,ndig(i))!=as.character(i)) b[1]=b[1]+1;
    while(b[1]>1) { b[1]=b[1]-1; newmin(b,2) }
    if(y[i]>v) y[i]=v;
}

A109865 Consider Pi = 3.1415926535897932384626433832795... On taking the absolute successive differences between successive digits ignoring the decimal point one gets the array shown in Comments (below). Sequence contains the first term of each row (the first diagonal).

Original entry on oeis.org

3, 2, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0

Offset: 0

Amarnath Murthy, Jul 09 2005

3...1...4...1...5...9...2...6...5...3...5...8...9...7...9...3...2...3...8...4
..2...3...3...4...4...7...4...1...2...2...3...1...2...2...6...1...1...5...4
....1...0...1...0...3...3...3...1...0...1...2...1...0...4...5...0...4...1
......1...1...1...3...0...0...2...1...1...1...1...1...4...1...5...4...3
........0...0...2...3...0...2...1...0...0...0...0...3...3...4...1...1
..........0...2...1...3...2...1...1...0...0...0...3...0...1...3...0
............2...1...2...1...1...0...1...0...0...3...3...1...2...3
Call this the Absolute Successive Difference function of an irrational number k denoted by ASDE(k). Then ASDE(Pi) = 3.211002110001... Subsidiary sequences: For e, phi, 2^(1/2), 3^(1/2), Euler's constant and other important irrational numbers can be included.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000796.

Digits := 200: asde := proc(n,L) local b,L2,i,j; b := L ; for i from 1 to n do L2 := [] ; for j from 1 to nops(b)-1 do L2 := [op(L2),abs(op(j+1,b)-op(j,b))] ; od: b := L2 ; od: op(1,b) ; end: A109864 := proc(n) local piL,i ; piL := [] ; for i from 1 to n+1 do piL := [op(piL), floor(Pi*10^(i-1)) mod 10] ; od: asde(n,piL) ; end: seq( A109864(n),n=0..100) ; # R. J. Mathar, Feb 11 2008

First[#]&/@Module[{nn=110,pi},{pi=RealDigits[Pi,10,nn][[1]]};NestList[ Abs[ Differences[ #]]&,pi,nn-1]] (* Harvey P. Dale, Jun 14 2016 *)

More terms from R. J. Mathar, Feb 11 2008

Definition clarified by Harvey P. Dale, Jun 14 2016

A088277 Let f(n) be the n-th palindrome in A089743. Then a(n) is the smallest palindromic prime that begins with f(n).

Original entry on oeis.org

11, 2, 3, 5, 7, 919, 11, 33533, 77377, 9902099, 101, 1114111, 1212121, 131, 1411141, 151, 1611161, 1712171, 181, 191, 303050303, 313, 32323, 3331333, 3439343, 353, 3635363, 373, 383, 3931393, 7073707, 7177717, 727, 737171737, 74747, 757

Offset: 1

Amarnath Murthy, Sep 29 2003

			f(7) = 33, the 7th palindrome that is the beginning of a palindromic prime, so a(7) = A109864(33) = 33533.

Cf. A082769, A089743, A109864.

Edited and extended by David Wasserman, Jul 28 2005

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A225903 The smallest number beginning with n whose distinct prime factors are the first n primes.

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A109865 Consider Pi = 3.1415926535897932384626433832795... On taking the absolute successive differences between successive digits ignoring the decimal point one gets the array shown in Comments (below). Sequence contains the first term of each row (the first diagonal).

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A088277 Let f(n) be the n-th palindrome in A089743. Then a(n) is the smallest palindromic prime that begins with f(n).

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