A100762 -id:A100762 - cp's OEIS frontend

A082725 a(n) = n/A100762(n).

1, 1, 3, 1, 5, 3, 7, 1, 1, 5, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 11, 23, 1, 25, 13, 1, 7, 29, 15, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 21, 43, 11, 5, 23, 47, 1, 49, 25, 51, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 7, 1, 65, 33, 67, 17, 69, 35, 71, 1, 73, 37, 25, 19, 77, 39, 79, 1, 1, 41

Offset: 1

N. J. A. Sloane, Nov 17 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A100762, A100549, A141586.

{1}~Join~Table[Function[{q, P}, n/Times @@ Power @@@ Select[q, First@ # <= P &]] @@ {#, Prime@ PrimePi[1 + Max@ #[[All, -1]] ]} &@ FactorInteger[n], {n, 2, 82}] (* Michael De Vlieger, Nov 13 2018 *)

A100549(n) = if(1==n,1,prime(primepi(1+vecmax(factor(n)[,2]))));
A100762(n) = if(1==n,1,my(u = A100549(n), f=factor(n)); prod(i=1, #f~, if(f[i, 1]<=u, f[i, 1]^f[i, 2], 1)));
A082725(n) = (n/A100762(n)); \\ Antti Karttunen, Nov 11 2018

A100417 Numbers n such that P(n) = P(B(n)), where P() = A100549() and B() = A100762.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 118, 120, 122, 124, 126

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

Cf. A100549, A100762.

# First load the Maple programs from A100549 and A100762
# Bgood = (is pp(n) = pp(B(n))), that is, is B(n) enough to establish pp(n)?
Bgood := proc(n) global pp;
`if`(pp(B(n))=pp(n),true,false);
end proc;
select(Bgood, [$1..200]);

A141758 Elements n of A141586 with property that A100762(n) = n.

Original entry on oeis.org

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 4320, 6720, 13440, 20160, 21600, 40320, 60480, 64800, 120960, 194400, 241920, 302400, 423360, 483840, 604800, 846720, 907200, 1814400, 2721600, 3628800, 5443200, 6350400, 7257600, 10644480, 10886400, 12700800, 18144000

Offset: 1

David Applegate and N. J. A. Sloane, Sep 16 2008

nonn

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..66

A100786 Apply the function n -> A100762(n) to the terms of A141586.

Original entry on oeis.org

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 480, 4320, 480, 480, 6720, 480, 480, 1440, 480, 13440, 480, 480, 1440, 480, 1440, 480, 20160, 480, 21600, 480, 1440, 480, 1440, 480, 480, 480, 1440, 480, 480, 480, 480, 40320, 1440, 480, 1440, 480, 480, 480, 480, 480, 1440

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

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

A100549 Let n = 2^e_2 * 3^e_ * 5^e_ * ... be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, ...}; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 3, 2

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

			If n = 8 = 2^3, a(n) = (largest prime <= 3+1) = 3.
If n = 480 = 2^5*3*5, a(n) = (largest prime <= 1 + max{5,1,1}) = 5.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A100762, A100417, A141586, A082725.

# 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;

{1}~Join~Array[Prime@PrimePi[1 + Max@FactorInteger[#][[All, -1]]] &, 105, 2] (* Michael De Vlieger, Nov 13 2018 *)

a(n) = if (n==1, 1, precprime(1 + vecmax(factor(n)[,2]~))); \\ Michel Marcus, Nov 14 2018

A141757 Even terms in A100933.

Original entry on oeis.org

50, 98, 150, 242, 250, 294, 338, 350, 490, 550, 578, 650, 686, 722, 726, 750, 850, 950, 1014, 1050, 1058, 1078, 1150, 1210, 1274, 1450, 1470, 1550, 1650, 1666, 1682, 1690, 1694, 1734, 1750, 1850, 1862, 1922, 1950, 2050, 2058, 2150, 2166, 2254, 2350, 2366

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

with(numtheory):
# For A100549: if n = prod_p p^e_p, then pp = largest prime <= 1 + max e_p
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;
# For A100762: B = prod_{p <= pp(n)} p^e_p
B := proc(n) local v,f,pv; global pp; option remember;
pv := pp(n);
v := 1:
for f in op(2..-1,ifactors(n)) while f[1] <= pv do
v := v * f[1]^f[2];
end do;
return v;
end proc;
# For A100417: Bgood = (is pp(n) = pp(B(n))), that is, is B(n) enough to establish pp(n)?
Bgood := proc(n) global pp;
`if`(pp(B(n))=pp(n),true,false);
end proc;
# For A100933 and A141757:
t0:=select(not Bgood, [$1..3000]);
t1:=[];
for n from 1 to nops(t0) do
if t0[n] mod 2 = 0 then t1:=[op(t1),t0[n]]; fi; od: t1;

cp's OEIS Frontend

A082725 a(n) = n/A100762(n).

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A100417 Numbers n such that P(n) = P(B(n)), where P() = A100549() and B() = A100762.

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A141758 Elements n of A141586 with property that A100762(n) = n.

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A100786 Apply the function n -> A100762(n) to the terms of A141586.

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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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A100549 Let n = 2^e_2 * 3^e_ * 5^e_ * ... be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, ...}; a(1) = 1 by convention.

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Maple

Mathematica

PARI

A141757 Even terms in A100933.

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Maple