A077515 -id:A077515 - 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;
}

A077516 Largest n-digit number beginning with n and having n divisors, or 0 if no such number exists.

Original entry on oeis.org

1, 29, 361, 4997, 0, 699964, 0, 89999995, 999887641, 1099999504, 0, 129999999988, 0, 14999999996224, 159999933195664, 1699999999999998, 0, 189999999999999477, 0, 20999999999999999817, 219999996968515384384, 2299999999999999765504, 0, 249999999999999999999988

Offset: 1

Amarnath Murthy, Nov 08 2002

			a(4) = 4989 = 3*1663 has 4 divisors: 1, 3, 1663 and 4989. a(5) = 0 as 13^4 = 28561 < 50000 < 59999 < 83521 = 17^4.

Cf. A077515.

Corrected and extended by Ray Chandler, Aug 15 2003

a(24) from Jon E. Schoenfield, Mar 17 2022

A124100 Sum_(x^iy^jz^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

Original entry on oeis.org

1, 40, 1089, 25160, 531521, 10625640, 204744769, 3844391560, 70827391041, 1286290883240, 23101397290049, 411249127989960, 7269184506192961, 127745926316548840, 2234231991096868929, 38920247688751940360

Offset: 0

Giorgio Balzarotti and Paolo P. Lava, Nov 26 2006

nonn

			a(2) = 1089 because x^2 + y^2 + z^2 + x*y + x*z + y*z = 8^2 + 15^2 + 17^2 + 8*15 + 8*17 + 15*17 = 1089 and x^2 + y^2 = z^2.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 196.

Harvey P. Dale, Table of n, a(n) for n = 0..811
Index entries for linear recurrences with constant coefficients, signature (40, -511, 2040).

Cf. A019682, A020000, A020340-A020342, A020344-A020346, A021664, A021684, A021844, A025942, A077515.

seq(sum(8^(m-n)*sum(15^p*17^(n-p),p=0..n),n=0..m),m=0..N);

LinearRecurrence[{40,-511,2040},{1,40,1089},30] (* Harvey P. Dale, May 25 2025 *)

a(m) = (x^(m+2)*(z-y) + y^(m+2)*(x-z) + z^(m+2)*(y-x))/((x-y)*(y-z)*(z-x)).
From Chai Wah Wu, Sep 24 2016: (Start)
a(n) = 40*a(n-1) - 511*a(n-2) + 2040*a(n-3) for n > 2.
G.f.: 1/((1 - 8*x)*(1 - 15*x)*(1 - 17*x)). (End)
a(n) = 2^(3*n+6)/63 - 15^(n+2)/14 + 17^(n+2)/18. - Vaclav Kotesovec, Sep 25 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

R

A077516 Largest n-digit number beginning with n and having n divisors, or 0 if no such number exists.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A124100 Sum_(x^iy^jz^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

R

A077516 Largest n-digit number beginning with n and having n divisors, or 0 if no such number exists.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A124100 Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A124100 Sum_(x^iy^jz^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).