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 1-2 of 2 results.

A349899 Least number in A349898 divisible by the n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 22, 52, 136, 190, 1610, 12760, 35464, 196840, 2112320, 4093600, 22789360, 288608320
Offset: 1

Views

Author

Charles R Greathouse IV, Dec 04 2021

Keywords

Comments

Least value of A349194(k) divisible by the n-th prime for some k.
Does a(n) exist for each n? Is the sequence increasing?

Crossrefs

Cf. A349194, A349898.

Programs

  • PARI
    smooth(P,lim)=my(v=List([1]), nxt=vector(#P,i,1), indx, t); while(1, t=vecmin(vector(#P, i, v[nxt[i]]*P[i]), &indx); if(t>lim, break); if(t>v[#v], listput(v, t)); nxt[indx]++); Vec(v);
has(n)=my(v=apply(k->[k,k],select(k->n%k==0,[2..10]))); while(#v, my(u=List()); for(i=1,#v, my(sd=v[i][1],a=v[i][2]); for(k=0,9, my(nsd=sd+k,t=nsd*a); if(n%t==0, if(n==t, return(1)); listput(u,[nsd,t])))); v=Set(u)); 0
a(n)=if(n<5,return(prime(n))); my(P=primes(n+1),p=P[n],L=p,v=smooth(P,L),x); while(1, for(i=x+1,#v, if(has(p*v[i]), return(p*v[i]))); x=#v; v=smooth(P,L*=2))
\\ This program is heuristic and fails if a(n) is divisible by a prime >= prime(n+2). If this sequence is strictly increasing, "primes(n+1)" can be replaced with "primes(n)" for better speed and memory use.

A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 49, 56, 63, 70, 77
Offset: 1

Views

Author

Malo David, Nov 10 2021

Keywords

Comments

The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Nov 23 2021

Examples

			For n=256, a(256) = 2*(2+5)*(2+5+6) = 182.

Links

Crossrefs

Cf. A055642, A284001 (binary analog), A349190 (fixed points).
Cf. A007953 (sum of digits), A059995 (floor(n/10)).
Cf. A349278 (similar, with the last digits).
Cf. A349898, A349899.

Programs

  • Magma
    f:=func; [f(n):n in [1..100]]; // Marius A. Burtea, Nov 23 2021
  • Mathematica
    Table[Product[Sum[Part[IntegerDigits[n],j],{j,i}],{i,Length[IntegerDigits[n]]}],{n,74}] (* Stefano Spezia, Nov 10 2021 *)
  • PARI
    a(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ Michel Marcus, Nov 10 2021
  • PARI
    first(n)=if(n<9,return([1..n])); my(v=vector(n)); for(i=1,9,v[i]=i); for(i=10,n, v[i]=sumdigits(i)*v[i\10]); v \\ Charles R Greathouse IV, Dec 04 2021
  • Python
    from math import prod
from itertools import accumulate
def a(n): return prod(accumulate(map(int, str(n))))
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 10 2021

Formula

For n>10: a(n) = a(A059995(n))*A007953(n) where A059995(n) = floor(n/10).
In particular, for n<100: a(n) = floor(n/10)*A007953(n)
From Bernard Schott, Nov 23 2021: (Start)
a(n) = 1 iff n = 10^k, k >= 0 (A011557).
a(n) = 2 iff n = 10^k + 1, k >= 0 (A000533 \ {1}).
a(n) = 3 iff n = 10^k + 2, k >= 0 (A133384).
a(n) = 5 iff n = 10^k + 4, k >= 0.
a(n) = 7 iff n = 10^k + 6, k >= 0. (End)
From Marius A. Burtea, Nov 23 2021: (Start)
a(A002275(n)) = n! = A000142(n), n >= 1.
a(A090843(n - 1)) = (2*n - 1)!! = A001147(n), n >= 1.
a(A097166(n)) = (3*n - 2)!!! = A007559(n).
a(A093136(n)) = 2^n = A000079(n).
a(A093138(n)) = 3^n = A000244(n). (End)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.