cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A178480 For n=0,1,2,... list all products of the first n primes raised to some positive power not exceeding n.

Original entry on oeis.org

1, 2, 6, 12, 18, 36, 30, 60, 120, 90, 180, 360, 270, 540, 1080, 150, 300, 600, 450, 900, 1800, 1350, 2700, 5400, 750, 1500, 3000, 2250, 4500, 9000, 6750, 13500, 27000, 210, 420, 840, 1680, 630, 1260, 2520, 5040, 1890, 3780, 7560, 15120, 5670, 11340, 22680
Offset: 1

Views

Author

M. F. Hasler, May 31 2010

Keywords

Comments

Alternate construction: For n=0,1,2,... write all strings of length n using the first n symbols of the alphabet (""; a; aa,ab,ba,bb; aaa,aab,aac, aba,...), then code / interpret them as "positional" notation of exponents (a=1, b=2, ...) of primes (last digit = least prime), e.g.: acb => [1,3,2] => 5^1 3^3 2^2.
These numbers have the property that, if a prime p divides the number, then all primes less than p also divide it. (But not all such numbers are listed, neither are they listed in increasing order.)

Examples

			The sequence begins: a(1)=1 (empty product); a(2)=2^1;
a(3,...,6)=2^1 3^1, 2^2 3^1, 2^1 3^2, 2^2 3^2;
a(7,...)=2^1 3^1 5^1, 2^2 3^1 5^1, 2^3 3^1 5^1,
________ 2^1 3^2 5^1, 2^2 3^2 5^1, 2^3 3^2 5^1,
________ 2^1 3^3 5^1, 2^2 3^3 5^1, 2^3 3^3 5^1,
________ 2^1 3^1 5^2, 2^2 3^1 5^2, 2^3 3^1 5^2, ...
They correspond to the strings (cf. comment) "" a aa ab ba bb aaa aab aac aba abb abc aca acb acc baa bab bac ...

Crossrefs

Cf. A178483

Programs

  • PARI
    for( L=0,3, forvec( v=vector(L,i,[1,L]), print1( prod( j=1,L,prime(j)^v[L-j+1] )",")))

A178484 For n=1,2,... list all numbers not occurring earlier which can be written as a product of the first n primes raised to some nonnegative power less than n.

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900, 7, 8, 14, 21, 24, 27, 28, 35, 40, 42, 49, 54, 56, 63, 70, 72, 84, 98, 105, 108, 120, 125, 126, 135, 140, 147, 168, 175, 189, 196, 200, 210, 216, 245, 250, 252
Offset: 1

Views

Author

M. F. Hasler, May 31 2010

Keywords

Comments

A condensed version of sequence A178483.
Every positive integer occurs exactly once in this sequence, but depending on its largest prime factor, it may appear later than much larger numbers. E.g. 7=a(29) appears after a(28)=900, and 11=a(257) appears only after a(256)=9261000.
The first n^n terms are the divisors of n#^(n-1), so any term divisible by the k-th prime must appear later than position (k-1)^(k-1). - Charlie Neder, Mar 08 2019

Examples

			n=1 gives a(1) = 1: numbers 2^a with a < 1.
n=2 gives a(2..4) = [2, 3, 6]: numbers 2^a 3^b with a,b < 2.
n=3 gives a(5..28) = [4, 5, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900]: numbers 2^a 3^b 5^c not occurring earlier, with a,b,c < 3.

Links

Crossrefs

Cf. A178480, A178483, A111791.

Programs

  • Mathematica
    DeleteDuplicates@Flatten@Table[Sort[Times @@ (Prime@Range@n^PadLeft[ IntegerDigits[#, n], n]) & /@ (Range[n^n] - 1)], {n, 2, 4}] (* Ivan Neretin, May 02 2019 *)
  • PARI
    { s=0; for( L=1,4, a=[]; forvec( v=vector(L,i,[0,L-1]), bittest(s,t=prod( j=1,L,prime(j)^v[L-j+1] )) & next; s+=1<

A178479 For n=0,1,2,... list all numbers not occurring earlier which can be written as product of the first n primes raised to some nonnegative power not exceeding n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36, 5, 8, 10, 15, 20, 24, 25, 27, 30, 40, 45, 50, 54, 60, 72, 75, 90, 100, 108, 120, 125, 135, 150, 180, 200, 216, 225, 250, 270, 300, 360, 375, 450, 500, 540, 600, 675, 750, 900, 1000, 1080, 1125, 1350, 1500, 1800, 2250, 2700, 3000, 3375
Offset: 1

Views

Author

M. F. Hasler, May 31 2010

Keywords

Comments

Every positive integer occurs exactly once in this sequence, but depending on its largest prime factor, it may appear quite late with respect to larger numbers. E.g. prime(4)=7=a(65) appears after a(4^3)=27000=(2*3*5)^3, prime(5)=11=a(626) appears after a(5^4)=(2*3*5*7)^4=1944810000.
First A000169(n) terms are the divisors of A181555(n), and a(A000169(n))=A181555(n). [From Matthew Vandermast, Oct 31 2010]

Examples

			n=0, n=1 and n=2 give a(1)=1 (empty product), a(2)=2=prime(1)^1,
and a(3..9) = 3, 4, 6, 9, 12, 18, 36: numbers 2^a 3^b with a,b <= 2.
n=3 gives a(10..64) = 5, 8, 10, 12, 15, 18...: numbers 2^a 3^b 5^c not occurring earlier, with a,b,c <= 3.

Crossrefs

Cf. A178480, A178483, A178484, A111791.

Programs

  • PARI
    { s=[]; for( L=0,3, a=[]; forvec( v=vector(L,i,[0,L]), setsearch( s, t=prod( j=1,L,prime(j)^v[L-j+1] )) & next; s=setunion(s,Set(t)); a=concat(a,t)); apply(x->print1(x","),vecsort(a))) }
Showing 1-3 of 3 results.

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