A120736 -id:A120736 - cp's OEIS frontend

A293376 Corresponding values of pod(n)/tau(n) of numbers n from A120736.

1, 1, 9, 16, 9, 25, 288, 49, 972, 121, 41472, 169, 101250, 289, 1119744, 361, 320000, 388962, 529, 1062882, 1229312, 841, 3888000000, 961, 2371842, 3001250, 11609505792, 1369, 4626882, 327680000, 1681, 29274835968, 1849, 7496192, 44286750000, 2209, 65229815808

Offset: 1

Jaroslav Krizek, Oct 07 2017

nonn

Integer of pod(n)/tau(n) of numbers n such that tau(n) = the number of the divisors of n (A000005) divides pod(n) = the product of the divisors of n (A007955).

			For n = 3; A120736(3) = 6; pod(6)/tau(6) = 36/4 = 9.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A007955, A120736, A137927, A291186.

[Numerator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]): n in [1..1000] | Denominator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]) eq 1];

(Times @@ #)/Length@ # &@ Divisors@ # & /@ Select[Range@ 100, Divisible[Times @@ Divisors@ #, DivisorSigma[0, #]] &] (* Michael De Vlieger, Oct 10 2017 *)

a(n) = A291186(A120736(n))/A137927(A120736(n)).

A306671 a(n) = gcd(tau(n), pod(n)) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 1, 1, 1, 4, 1, 4, 3, 4, 1, 6, 1, 4, 1, 1, 1, 6, 1, 2, 1, 4, 1, 8, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 9, 1, 4, 1, 8, 1, 8, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 8, 1, 8, 1, 4, 1, 12, 1, 4, 3, 1, 1, 8, 1, 2, 1, 8, 1, 12, 1, 4, 3, 2, 1, 8, 1, 10, 1, 4, 1, 12, 1, 4

Offset: 1

Jaroslav Krizek, Mar 04 2019

nonn

Sequence of the smallest numbers k such that a(k) = n: 1, 2, 9, 6, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 120, ...

			For n=6: a(6) = gcd(tau(6), pod(6)) = gcd(4, 36) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A007955, A009205, A046642.

[GCD(NumberOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

a(n) = gcd(numdiv(n), vecprod(divisors(n))); \\ Michel Marcus, Mar 04 2019

a(n) = 1 for numbers in A046642.

a(n) = tau(n) for numbers in A120736.

A120737 Numbers k whose number of divisors d(k) is divisible by every prime factor of k.

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 32, 72, 80, 96, 108, 128, 243, 288, 448, 486, 512, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2048, 2560, 2592, 3888, 4032, 4050, 4608, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 8192, 8748, 9408, 10800, 11250

Offset: 1

Leroy Quet, Jun 29 2006

nonn

Numbers k such that A000005(k)/A007947(k) is an integer. A070226 is a subsequence of this sequence. Conjecture: If A000005(k) divides A007947(k) for some k, then A007947(k)/A000005(k)=1. - Ctibor O. Zizka, Feb 05 2009

This sequence contains exactly those positive integers k where 1 is the only positive divisor of k that is coprime to d(k). - Leroy Quet, May 23 2009

			d(32) = 6. 2 is the only prime dividing 32. 2 divides 6, so 32 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..100 from Paolo P. Lava)

Cf. A000005, A070226, A007947, A120736.

isA120737 := proc(n) local d,p; d := numtheory[tau](n) ; p := 2 ; while p <= n do if ( n mod p ) = 0 then if ( d mod p ) <> 0 then RETURN(false) ; fi ; fi ; p := nextprime(p) ; od ; RETURN(true) ; end: for n from 1 to 12000 do if isA120737(n) then printf("%d,",n) ; fi ; od ;
# R. J. Mathar, Aug 17 2006

divQ[n_] := AllTrue[FactorInteger[n][[;; , 1]], Divisible[DivisorSigma[0, n], #] &]; Select[Range[10^4], divQ] (* Amiram Eldar, Nov 08 2020 *)

isok(k) = Mod(numdiv(k), k)^eulerphi(k) == 0; \\ Michel Marcus, May 11 2019

More terms from R. J. Mathar, Aug 17 2006

Name simplified by Jon E. Schoenfield, Mar 03 2019

A306682 a(n) = gcd(sigma(n), pod(n)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 4, 1, 4, 3, 1, 1, 3, 1, 2, 1, 4, 1, 12, 1, 2, 1, 56, 1, 72, 1, 1, 3, 2, 1, 1, 1, 4, 1, 10, 1, 48, 1, 4, 3, 4, 1, 4, 1, 1, 9, 2, 1, 24, 1, 8, 1, 2, 1, 24, 1, 4, 1, 1, 1, 144, 1, 2, 3, 16, 1, 3, 1, 2, 1, 4, 1, 24, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

See A324527(n) = the smallest numbers k such that a(k) = n.

			For n=6: a(6) = gcd(tau(6), pod(6)) = gcd(4, 36) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A000203, A007955, A009205, A046642, A120736, A324526, A324527.

[GCD(SumOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

a(n) = my(d=divisors(n)); gcd(vecsum(d), vecprod(d)); \\ Michel Marcus, Mar 05 2019

a(n) = 1 for numbers in A014567.

a(n) = tau(n) for numbers in A324526.

A324528 a(n) = lcm(tau(n), pod(n)) where tau(k) = the number of divisors of k (A000005) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 2, 6, 24, 10, 36, 14, 64, 27, 100, 22, 1728, 26, 196, 900, 5120, 34, 5832, 38, 24000, 1764, 484, 46, 331776, 375, 676, 2916, 65856, 58, 810000, 62, 98304, 4356, 1156, 4900, 10077696, 74, 1444, 6084, 2560000, 82, 3111696, 86, 255552, 182250, 2116, 94

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

			For n=4: a(4) = lcm(tau(4), pod(4)) = lcm(3, 8) = 24.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A007955, A046642, A120736.

[LCM(NumberOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

Table[LCM[DivisorSigma[0,n],Times@@Divisors[n]],{n,50}] (* Harvey P. Dale, Aug 14 2019 *)

a(n) = my(d=divisors(n)); lcm(vecprod(d), #d); \\ Michel Marcus, Mar 05 2019

a(n) = pod(n) for numbers n in A120736.

a(n) = tau(n) * pod(n) for numbers n in A046642.

A137927 a(n) = the largest divisor of A000005(n) that is coprime to n. (A000005(n) = the number of positive divisors of n.).

Original entry on oeis.org

1, 1, 2, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 5, 2, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4, 3, 2, 1, 2, 3, 4, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 3, 2, 1, 2, 5, 3, 3, 4, 3, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 2, 7, 4, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1

Offset: 1

Leroy Quet, Feb 23 2008

nonn

Apparently also the denominator of A007955(n)/A000005(n). See A291186. - Jaroslav Krizek, Sep 05 2017

			20 has 6 positive divisors. The divisors of 6 are 1,2,3,6. The divisors of 6 that are coprime to 20 are 1 and 3. 3 is the largest of these; so a(20) = 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A137926, A007955, A120736, A291899.

A137927 := proc(n)
    local a;
    a := 1 ;
    for d in numtheory[divisors](numtheory[tau](n)) do
        if igcd(d,n) = 1 then
            a := max(a,d) ;
        end if:
    end do:
    a ;
end proc:
seq(A137927(n),n=1..100) ; # R. J. Mathar, Sep 22 2017

Table[Select[Divisors[Length[Divisors[n]]], GCD[ #, n] == 1 &][[ -1]], {n, 1, 80}] (* Stefan Steinerberger, Mar 09 2008 *)

a(n) = my(d=divisors(numdiv(n))); forstep(k=#d, 1, -1, if (gcd(d[k], n) == 1, return (d[k]))); \\ Michel Marcus, Sep 22 2017; corrected Jun 13 2022

More terms from Stefan Steinerberger, Mar 09 2008

A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

Original entry on oeis.org

1, 6, 30, 66, 102, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, 672, 690, 714, 840, 870, 924, 930, 966, 1122, 1320, 1410, 1428, 1518, 1590, 1638, 1722, 1770, 1890, 1932, 2130, 2226, 2280, 2310, 2346, 2370, 2670, 2730, 2760, 2838, 2970, 2982, 3102, 3162, 3210, 3360, 3444, 3486, 3498, 3570, 3720, 3780, 3948, 3990

Offset: 1

Ilya Gutkovskiy, Oct 19 2016

nonn

Intersection of A003601, A120736 and A145551.
Numbers k such that A000005(k)|A007955(k), A000203(k)|A007955(k) and A000005(k)| A000203(k).
Numbers k such that A000005(k)|A062981(k), A072861(k)|A062758(k) and A245656(k) = 1.

			a(2) = 6 because 6 has 4 divisors {1,2,3,6}, 1*2*3*6/4 = 9, 1*2*3*6/(1 + 2 + 3 + 6) = 3 and (1 + 2 + 3 + 6)/4 = 3 are integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..20000
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Divisor Function.
Wikipedia, Arithmetic number.

Cf. A000005, A000203, A003601, A007955, A062758, A062981, A072861, A120736, A145551, A245656.

with(numtheory): P:=proc(q) local a,b,k,n;for n from 1 to q do
a:=divisors(n); b:=mul(a[k],k=1..nops(a));
if type(sigma(n)/tau(n),integer) and type(b/sigma(n),integer) and
type(b/tau(n),integer) then print(n); fi;
od; end: P(10^5); # Paolo P. Lava, Oct 20 2016

Select[Range[4000], Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[1, #1]] && Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[0, #1]] && Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] & ]

A291186 a(n) = numerator of (pod(n) / tau(n)).

Original entry on oeis.org

1, 1, 3, 8, 5, 9, 7, 16, 9, 25, 11, 288, 13, 49, 225, 1024, 17, 972, 19, 4000, 441, 121, 23, 41472, 125, 169, 729, 10976, 29, 101250, 31, 16384, 1089, 289, 1225, 1119744, 37, 361, 1521, 320000, 41, 388962, 43, 42592, 30375, 529, 47, 127401984, 343, 62500, 2601

Offset: 1

Jaroslav Krizek, Sep 05 2017

pod(n) = the product of the divisors of n (A007955), tau(n) = the number of the divisors of n (A000005).

			For n = 4; pod(4) / tau(4) = 8 / 3; a(n) = 8.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A137927 (denominator).

Cf. A000005, A007955, A120736.

[Numerator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]): n in [1..1000]];

f:= proc(n) local D; D:= numtheory:-divisors(n); numer(convert(D,`*`)/nops(D)) end proc:
map(f, [$1..100]); # Robert Israel, Sep 14 2017

Table[Numerator[Apply[Times, Divisors@ n]/DivisorSigma[0, n]], {n, 51}] (* Michael De Vlieger, Sep 05 2017 *)

a(n) = my(d=divisors(n)); numerator(prod(k=1, #d, d[k])/#d); \\ Michel Marcus, Sep 05 2017

a(n) = numerator of (A007955(n) / A000005(n)).

A048751 Composites k whose product of divisors divided by number of divisors is an integer.

Original entry on oeis.org

6, 8, 9, 10, 12, 14, 18, 22, 24, 26, 30, 34, 36, 38, 40, 42, 46, 54, 56, 58, 60, 62, 66, 70, 72, 74, 78, 80, 82, 84, 86, 88, 90, 94, 96, 102, 104, 106, 108, 110, 114, 118, 120, 122, 126, 128, 130, 132, 134, 136, 138, 142, 146, 150, 152, 154, 156, 158, 166, 168, 170

Offset: 1

Enoch Haga, Dec 11 1999

Sequence is identical to A120736 except that it does not include terms 1 and 2, which are not composite. Michel Marcus, Jun 06 2014

			For k=8, product of divisors is 8*4*2*1=64; number of divisors = 4; 64/4 = 16 (an integer), so 8 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048747, A048752.

Select[Range[200],CompositeQ[#]&&IntegerQ[(Times@@Divisors[#])/ DivisorSigma[ 0,#]]&] (* Harvey P. Dale, Aug 21 2021 *)

isok(n) = (n!=1) && ! isprime(n) && (d = divisors(n)) && ((prod(i=1, #d, d[i]) % numdiv(n)) == 0); \\ Michel Marcus, Jun 05 2014

is(n)=my(f=factor(n)); n>5 && !isprime(n) && if(gcd(f[,2])%2, n^(numdiv(f)/2), sqrtint(n)^numdiv(f))%numdiv(f)==0 \\ Charles R Greathouse IV, Jun 06 2014

Corrected by Michel Marcus, Jun 05 2014

A291899 Numbers n such that (pod(n)/tau(n)) > (pod(k)/tau(k)) for all k < n.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080, 12600

Offset: 1

Jaroslav Krizek, Oct 10 2017

nonn

pod(n) = the product of the divisors of n (A007955), tau(n) = the number of the divisors of n (A000005).
Contains all members of A002182 except 2. - Robert Israel, Nov 09 2017
Is this the same as A034288 except for 3? - Georg Fischer, Oct 09 2018
From David A. Corneth, Oct 11 2018: (Start)
Various methods exist to find terms for this sequence, possibly combinable:
- Brute force; checking every positive integer up to some bound.
- Finding terms based on the prime signature.
- Relating to that, the number of divisors.
- Finding terms based on the GCD of some earlier found terms.
- ... (?)
There seems to be a method that helps finding terms < 10^150 for the similar A034287. (End)

			6 is a term because pod(6)/tau(6) = 36/4 = 9 > pod(k)/tau(k) for all k < 6.

David A. Corneth, Table of n, a(n) for n = 1..200 (First 126 terms by Robert Israel)
David A. Corneth, Conjectured first 1171 terms

Cf. A000005, A002182, A034287, A034288, A007955, A120736, A137927, A291186, A067128.

a:=1; S:=[a]; for n in [2..60] do k:=0; flag:= true; while flag do k+:=1; if &*[d: d in Divisors(a)] / #[d: d in Divisors(a)] lt &*[d: d in Divisors(k)] / #[d: d in Divisors(k)] then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

f:= proc(n) local t; t:= numtheory:-tau(n); simplify(n^(t/2))/t end proc:
N:= 20000: # to get all terms <= N
Res:= NULL: m:= 0:
for n from 1 to N do
  v:= f(n);
  if v > m then Res:= Res, n; m:= v fi
od:
Res; # Robert Israel, Nov 09 2017

With[{s = Array[Times @@ Divisors@ # &, 12600]}, Select[Range@ Length@ s, Function[m, AllTrue[Range[# - 1], m > s[[#]]/DivisorSigma[0, #] &]][s[[#]]/DivisorSigma[0, #]] &]] (* Michael De Vlieger, Oct 10 2017 *)
DeleteDuplicates[Table[{n,Times@@Divisors[n]/DivisorSigma[0,n]},{n,13000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

Numbers n such that (A007955(n)/A000005(n)) > (A007955(k)/A000005(k)) for all k < n.

Numbers n such that (A291186(n)/A137927(n)) > (A291186(k)/A137927(k)) for all k < n.

cp's OEIS Frontend

A293376 Corresponding values of pod(n)/tau(n) of numbers n from A120736.

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A306671 a(n) = gcd(tau(n), pod(n)) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

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A120737 Numbers k whose number of divisors d(k) is divisible by every prime factor of k.

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A306682 a(n) = gcd(sigma(n), pod(n)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

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A324528 a(n) = lcm(tau(n), pod(n)) where tau(k) = the number of divisors of k (A000005) and pod(n) = the product of divisors of k (A007955).

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A137927 a(n) = the largest divisor of A000005(n) that is coprime to n. (A000005(n) = the number of positive divisors of n.).

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A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

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