A141900 -id:A141900 - cp's OEIS frontend

A142593 Irregular triangle read by rows: row n gives successive exponents in prime factorization of A141900(n).

1, 2, 1, 3, 1, 4, 1, 1, 5, 1, 1, 6, 1, 1, 1, 7, 1, 1, 1, 8, 3, 1, 1, 9, 3, 1, 1, 10, 3, 1, 1, 1, 11, 3, 1, 1, 1, 12, 3, 1, 1, 1, 1, 13, 3, 1, 1, 1, 1, 14, 3, 1, 1, 1, 1, 15, 3, 1, 1, 1, 1, 16, 3, 1, 1, 1, 1, 1, 17, 3, 1, 1, 1, 1, 1, 18, 3, 1, 1, 1, 1, 1, 1, 19, 3, 1, 1, 1, 1, 1, 1, 20, 3, 1, 1

Offset: 1

David Applegate, Tim Kunisky (tkunisky(AT)gmail.com), Gerry Manoim (gerrymanoim(AT)gmail.com) and N. J. A. Sloane, Sep 25 2008

			Triangle begins:
1
2 1
3 1
4 1 1
5 1 1
6 1 1 1
7 1 1 1
8 3 1 1
9 3 1 1
10 3 1 1 1
11 3 1 1 1
12 3 1 1 1 1
Row 8 = [8,3,1,1] because A141900(8) = 2^8*3^3*5*7 = 241920.

David Applegate, Maple program which generates this triangle [The command hpat(n); returns the n-th row in an encoded form.]
N. J. A. Sloane, Rows 1 through 50 of the triangle

Cf. A141586, A141900, A142594.

A142594 Product of exponents in prime factorization of A141900(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 288, 300, 390, 405, 420, 435, 450, 465, 480, 495, 510, 525, 540, 555, 570, 585, 600, 615, 630, 645, 660, 675, 690, 705, 1728, 1764, 1800

Offset: 1

N. J. A. Sloane, Sep 25 2008

Product of entries in row n of triangle in A142593.

A062247 Let f(h) = A141900(h) = 2^h * 3^i * 5^j * ... be the smallest term in A141586 that is divisible by 2^h but not by 2^(h+1). Sequence gives values of h where i increases.

Original entry on oeis.org

0, 2, 8, 24, 26, 48, 80, 120, 168, 242, 288, 360, 528, 728, 840, 960, 1330, 1368, 1680, 1848, 2186, 2208, 2808, 3480, 3720, 4488, 5040, 5328, 6240, 6560, 6888, 7920, 9408

Offset: 1

N. J. A. Sloane, Nov 24 2008

nonn

			Pairs (h,i) where A141900 contains a term 2^h*3^i*... for the first time, with a dash if that value of i never occurs:
h i
0 0
2 1
- 2
8 3
24 4
26 5
48 6
80 7
- 8
120 9
168 10
242 11
288 12
360 13
528 14
728 15
840 16
960 17
1330 18
1368 19
1680 20
...
The missing values of i for h <= 10000 are 2, 8 and 26.

Cf. A141586, A141900.

A141586 Strongly refactorable numbers: numbers n such that if n is divisible by d, it is divisible by the number of divisors of d.

Original entry on oeis.org

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 3360, 4320, 5280, 6240, 6720, 8160, 9120, 10080, 11040, 13440, 13920, 14880, 15840, 17760, 18720, 19680, 20160, 20640, 21600, 22560, 24480, 25440, 27360, 28320, 29280, 32160, 33120, 34080

Offset: 1

J. Lowell, Aug 19 2008

nonn

Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p + 1 }. Then n is in the sequence iff for all primes q in the range 2 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ). - N. J. A. Sloane, Sep 01 2008
All terms > 1 are even. A subsequence of A033950. - N. J. A. Sloane, Aug 27 2008
Contains 480*p for all primes p > 5 (see A109802). - N. J. A. Sloane, Aug 27 2008

			72 qualifies because its divisors are 1,2,3,4,6,8,9,12,18,24,36,72, which have 1,2,2,3,4,4,3,6,6,8,9,12 divisors respectively and all of those numbers are divisors of 72.

Dmitriy Kunisky, German Manoim and N. J. A. Sloane, On strongly refactorable numbers, in preparation.

German Manoim and N. J. A. Sloane, Sep 09 2008, Table of n, a(n) for n = 1..240937 [a large file]

Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594, A100549, A100762, A082725, A135130, A143718, A143719, A143720.

isA141586 := proc(n) local dvs,d ; dvs := numtheory[divisors](n) ; for d in dvs do if not numtheory[tau](d) in dvs then RETURN(false) : fi; od: RETURN(true) ; end: for n from 1 to 100000 do if isA141586(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Aug 26 2008
## A100549: if n = prod_p p^e_p, then pp = largest prime <= 1 + max e_p
with(numtheory):
pp := proc(n) local f,m; option remember; if (n = 1) then return 1; end if; m := 1: for f in op(2..-1,ifactors(n)) do if (f[2] > m) then m := f[2]: end if; end do; prevprime(m+2); end proc;
isA141586 := proc(n) local ff,f,g,p,i; global pp;
ff := op(2..-1,ifactors(n));
for f in ff do
p := f[1];
if (add(floor(log(1+g[2])/log(p)),g in ff) > f[2]) then
return false;
end if;
end do;
for i from 1 to pi(pp(n)) do
p := ithprime(i);
if (n mod p <> 0) then
if (add(floor(log(1+g[2])/log(p)),g in ff) > 0) then
return false;
end if;
end if;
end do;
return true;
end proc; # David Applegate and N. J. A. Sloane, Sep 15 2008

l = {}; For[n = 1, n < 100000, n++, b = DivisorSigma[0, Divisors[n]]; If[Length[Select[b, Mod[n, # ] > 0 &]] == 0, AppendTo[l, n]]]; l (* Stefan Steinerberger, Aug 25 2008 *)
sfnQ[n_]:=AllTrue[DivisorSigma[0,Divisors[n]],Mod[n,#]==0&]; Select[ Range[ 35000],sfnQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 27 2019 *)

is_A141586(n)={ bittest(n,0) & return(n==1); fordiv(n,d,n % numdiv(d) & return);1 } \\ M. F. Hasler, Dec 05 2010

is_A141586 = lambda n: all(number_of_divisors(d).divides(n) for d in divisors(n)) # D. S. McNeil, Dec 05 2010

More terms from German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008

cp's OEIS Frontend

A142593 Irregular triangle read by rows: row n gives successive exponents in prime factorization of A141900(n).

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A142594 Product of exponents in prime factorization of A141900(n).

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A062247 Let f(h) = A141900(h) = 2^h * 3^i * 5^j * ... be the smallest term in A141586 that is divisible by 2^h but not by 2^(h+1). Sequence gives values of h where i increases.

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Examples

Crossrefs

A141586 Strongly refactorable numbers: numbers n such that if n is divisible by d, it is divisible by the number of divisors of d.

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