A101459 -id:A101459 - cp's OEIS frontend

A101081 Number of distinct prime factors of (prime p concatenated p times).

2, 2, 3, 3, 6, 6, 6, 3, 6, 8, 5, 7, 7, 8, 6, 6, 6, 10, 5, 5, 7, 10, 10, 9, 7, 7, 8, 11, 9, 8, 9, 14, 8, 9, 9, 11, 8, 10, 9, 10

Offset: 1

Parthasarathy Nambi, Jan 21 2005

			The number of distinct prime factors of 22 is 2.
The number of distinct prime factors of 333 is 2.
The number of distinct prime factors of 55555 is 3.
Then the number of distinct prime factors of 7777777 is 3.
a(16) comes from 53 * 107 * 1659431 * 1325815267337711173 * 47198858799491425660200071 * 9090909090909090909090909090909090909090909090909091. a(17) comes from 59 * 1889 * 2559647034361 * 1090805842068098677837 * 4411922770996074109644535362851087 * 4340876285657460212144534289928559826755746751. a(18) comes from 61 * 733 * 4637 * 81131 * 329401 * 974293 * 1360682471 * 106007173861643 * 7061709990156159479 * 11205222530116836855321528257890437575145023592596037161. Concerning a(19) = 67*(100^67-1)/99 = 67 * 493121 * 79863595778924342083 * 25648528130160606364784685146362888405160909090909090909090909090911655761903925151545569377605545379749607 (C107). - _Robert G. Wilson v_, Jan 27 2005

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method.

Cf. A101459.

f[n_] := Length[ FactorInteger[ FromDigits[ Flatten[ Table[ IntegerDigits[ Prime[n]], {Prime[n]}] ]]]]; Table[ f[n], {n, 15}] (* Robert G. Wilson v, Jan 27 2005 *)
Table[PrimeNu[FromDigits[Flatten[IntegerDigits/@PadRight[{},p,p]]]],{p,Prime[Range[33]]}] (* Harvey P. Dale, Apr 06 2023 *)

a(11)-a(15) from Ray Chandler, Jan 25 2005
a(16)-a(18) from Robert G. Wilson v, Jan 27 2005
Corrected and extended to a(33) by D. S. McNeil, Jan 07 2011
a(34)-a(40) from Sean A. Irvine, Nov 06 2024

A126959 a(k) = k! * lim_{n->oo} card({ i*j; i=1..k, j=1..n })/n.

Original entry on oeis.org

1, 3, 12, 58, 352, 2376, 19296, 168912, 1670976, 18000000, 219916800, 2781561600, 39605760000, 584889984000, 9253091635200, 154909552896000, 2834240274432000, 52918877491200000, 1074184895250432000

Offset: 1

M. F. Hasler, Mar 19 2007, Mar 22 2007

nonn

a(k) = k! card { i*j, i<=k, j<=k# } / k# where k# = lcm(1,2,3...,k) a(k)/(k+1)! <= 1/2 for all k.

			a(2)=3/2 since #{ i*j, i=1..2, j=1..2 } / 2 = #{ 1,2, 2,4 } / 2 = #{1,2,4} / 2.
a(3)=2 since #{ i*j, i=1..3, j=1..6 } / 6 = #{ 1,2,3,4,5,6, 2,4,6,8,10,12, 3,6,9,12,15,18 } / 6 = #{ 1,2,3,4,5,6,8,9,10,12,15,18 } / 6.

A. A. Buchstab, "Asymptotic estimates of a general number-theoretic function", Mat. Sbornik 44 (1937), 1239-1246.

M. F. Hasler, Table of n, a(n) for n = 1..36
N. G. de Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Proc. Neder. Akad. Wetensch, 1950.
"Counting Integers and their Multiplicities" on mersenneforum.org

Cf. A101459, A027424.

p:=proc(n) option remember;local s,t,i,j: s:=1; t:={}:
for i from n-1 by -1 to 1+n/(min@op@eval@numtheory[factorset])(n) do
t := t union { ilcm(n,i)/n };
t := select( x-> numtheory[divisors](x) intersect t = { x }, t ):
for j in combinat[powerset](t) do s := s+(-1)^nops(j)/ilcm(op(j)) od:
od; s/n end:
A126959 := k -> k!*add( p(n), n=1..k);

p(n)={ local( cnt=lcm(vector(n-1,j,j)), L=vector(cnt,j,n*j), s=cnt ); forstep( i=n-1, n/factor(n)[1,1]+1, -1, forstep( j=lcm(n,i)/n, #L, lcm(n,i)/n, if( L[j] && (L[j] % i == 0), L[j]=0; cnt--)); s+=cnt ); s/#L/n } a=vector(16); a[1]=1; for( k=2, #a, a[k]=k*a[k-1]+k!*p(k));

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A101081 Number of distinct prime factors of (prime p concatenated p times).

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A126959 a(k) = k! * lim_{n->oo} card({ i*j; i=1..k, j=1..n })/n.

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