A007955 -id:A007955 - cp's OEIS frontend

A334730 a(n) = Product_{d|n} gcd(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

1, 2, 1, 2, 1, 8, 1, 8, 3, 8, 1, 48, 1, 8, 1, 8, 1, 144, 1, 16, 1, 8, 1, 1536, 1, 8, 3, 16, 1, 256, 1, 16, 1, 8, 1, 7776, 1, 8, 1, 512, 1, 256, 1, 16, 9, 8, 1, 3072, 1, 16, 1, 16, 1, 1152, 1, 512, 1, 8, 1, 36864, 1, 8, 9, 16, 1, 256, 1, 16, 1, 256, 1, 2985984, 1, 8, 3, 16, 1, 256, 1

Offset: 1

Jaroslav Krizek, May 09 2020

nonn

			a(6) = gcd(tau(1), pod(1)) * gcd(tau(2), pod(2)) * gcd(tau(3), pod(3)) * gcd(tau(6), pod(6)) = gcd(1, 1) * gcd(2, 2) * gcd(2, 3) * gcd(4, 36) = 1 * 2 * 1 * 4 = 8.

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

Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334662 (Sum_{d|n} gcd(tau(d), pod(d))).

Cf. A000005 (tau(n)), A007955 (pod(n)), A306671 (gcd(tau(n), pod(n))).

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

a[n_] := Product[GCD[DivisorSigma[0, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)

pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

a(p) = 1 for p = odd primes (A065091).

A337324 a(n) is the smallest number m such that gcd(m, tau(m), sigma(m), pod(m)) = n where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 6, 18, 24, 5000, 90, 66339, 56, 288, 3240, 10036224, 60, 582160384, 20412, 16200, 3968, 49030215219, 612, 4637065216, 1520, 142884, 912384, 98881718827959, 480, 7543125, 479232, 3175200, 5824, 26559758051835904, 76950, 25796647321600, 2688, 491774976, 1268973568

Offset: 1

Jaroslav Krizek, Aug 23 2020

nonn

From David A. Corneth, Aug 24 2020: (Start)
a(35) <= 1289027059712000000.
a(36) <= 136064563937280.
a(37) = 207816012706349056.
a(38) <= 1835772101525504.
a(39) <= 418089296461824.
a(40) <= 11698803719536640.
gcd(m, tau(m), sigma(m), pod(m)) = gcd(m, tau(m), sigma(m)) which may ease the search.
(End)

			For n = 6; a(6) = 90 because 90 is the smallest number with gcd(90, tau(90), sigma(90), pod(90)) = gcd(90, 12, 234, 531441000000) = 6.

Cf. A337323 (gcd(n, tau(n), sigma(n), pod(n))).
Cf. A337325 (least m such that gcd(tau(m), sigma(m), pod(m)) = n).
Cf. A000005, A000203, A007955.

[Min([m: m in[1..10^5] | GCD([m, #Divisors(m), &+Divisors(m), &*Divisors(m)]) eq k]): k in [1..10]]

a(n) = {for(i = 1, oo, f = factor(i); if(gcd([i, numdiv(f), sigma(f)]) == n, return(i)))} \\ David A. Corneth, Aug 24 2020

a(11) from Amiram Eldar, Aug 24 2020

More terms from Jaroslav Krizek and David A. Corneth, Aug 24 2020

A346956 Numbers k such that A000203(k) and A007955(k) are both divisible by A187680(k).

Original entry on oeis.org

4, 9, 14, 16, 25, 38, 42, 49, 51, 55, 62, 64, 70, 81, 86, 92, 96, 117, 121, 130, 134, 138, 140, 158, 159, 161, 168, 169, 182, 206, 209, 234, 254, 256, 266, 267, 278, 282, 284, 289, 302, 322, 326, 351, 361, 376, 390, 398, 408, 410, 422, 426, 434, 446, 477, 508, 529, 532, 534, 542, 551, 566, 590

Offset: 1

J. M. Bergot and Robert Israel, Aug 08 2021

nonn

Numbers k such that both the sum s and product p of the divisors of k are divisible by (p mod s).

			a(3) = 14 is a term because A000203(14) = 1+2+7+14 = 24, A007955(14) = 1*2*7*14 = 196, A187680(14) = 196 mod 24 = 4, and both 24 and 196 are divisible by 4.

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

Cf. A000203, A007955, A187680.

Includes A188061.

filter:= proc(n) local d,s,p,r;
  d:= numtheory:-divisors(n);
  s:= convert(d,`+`);
  p:= convert(d,`*`);
  r:= p mod s;
  r <> 0 and p mod r = 0 and s mod r = 0
end proc:
select(filter, [$1..1000]);

okQ[n_] := Module[{d, s, p, m},
  d = Divisors[n];
  s = Total[d];
  p = Times @@ d;
  m = Mod[p, s];
  If[m == 0, False, Divisible[s, m] && Divisible[p, m]]];
Select[Range[1000], okQ] (* Jean-François Alcover, May 16 2023 *)

isok(k) = my(d=divisors(k), s=vecsum(d), p=vecprod(d), m=p % s); (m>0) && !(s%m) && !(p%m); \\ Michel Marcus, Aug 09 2021

A174900 a(n) = the smallest numbers k such that A007955(k) = n, or 0 if there is no such k, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 2, 3, 0, 5, 0, 7, 4, 0, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 0, 0, 23, 0, 0, 0, 9, 0, 29, 0, 31, 0, 0, 0, 0, 6, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 0, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 8, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 0, 0, 83

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = the largest numbers k such that A007955 (k) = n, or 0 if there is no such k.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007955.

With[{m = 100}, s[n_] := n^(DivisorSigma[0, n]/2); (FirstPosition[Array[s, m], #] & /@ Range[m]) /. Missing["NotFound"] -> 0 // Flatten] (* Amiram Eldar, Aug 06 2024 *)

More terms from Amiram Eldar, Aug 06 2024

A174901 a(n) = the smallest numbers k such that A007955(k) >= n, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

A174932 a(n) = Sum_{d|n} A007955(d) * A000027(n/d) = Sum_{d|n} A007955(d) * (n/d), where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 4, 6, 16, 10, 54, 14, 96, 45, 130, 22, 1860, 26, 238, 270, 1216, 34, 6048, 38, 8300, 504, 550, 46, 335688, 175, 754, 864, 22484, 58, 811050, 62, 35200, 1188, 1258, 1330, 10095048, 74, 1558, 1638, 2576920, 82, 3113586, 86, 86372, 92070, 2254, 94, 255478416

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)*(4/1) + b(2)*(4/2) + b(4)*(4/4) = 1*4 + 2*2 + 8*1 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A007955 (product of divisors), A322671.

[&+[&*Divisors(d)*(n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Jan 05 2020

a(n)={n*sumdiv(n, d, vecprod(divisors(d))/d)} \\ Andrew Howroyd, Jan 05 2020

from math import isqrt
from sympy import divisor_count, divisors
def A174932(n): return n*sum(isqrt(d)**(c-2) if (c:=divisor_count(d)) & 1 else d**(c//2-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 25 2022

a(n) = n*A322671(n). - Andrew Howroyd, Jan 05 2020

A174935 a(n) = Sum_{k<=n} A007955(k) * A000027(k) = Sum_{k<=n} A007955(k) * k, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 5, 14, 46, 71, 287, 336, 848, 1091, 2091, 2212, 22948, 23117, 25861, 29236, 45620, 45909, 150885, 151246, 311246, 320507, 331155, 331684, 8294308, 8297433, 8315009, 8334692, 8949348, 8950189, 33250189, 33251150, 34299726, 34335663, 34374967, 34417842, 397214898

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)*1 + b(2)*2 + b(3)* 3 + b(4)*4 = 1*1 + 2*2 + 3*3 + 8*4 = 46.

Cf. A000027, A007955.

a[n_] := Sum[k^(DivisorSigma[0, k]/2 + 1), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Aug 06 2024 *)

More terms from Amiram Eldar, Aug 06 2024

A174936 a(n) = Sum_{d|n} A007955(d) * A007955(n/d), where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 4, 6, 20, 10, 84, 14, 160, 63, 220, 22, 3648, 26, 420, 480, 2368, 34, 11988, 38, 16480, 924, 1012, 46, 671424, 275, 1404, 1620, 44800, 58, 1621860, 62, 70656, 2244, 2380, 2520, 20190816, 74, 2964, 3120, 5154240, 82, 6226836, 86, 172480, 183870, 4324, 94, 510973440, 735, 251500

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)*b(4/1) + b(2)*b(4/2) + b(4)*b(4/4) = 1*8 + 2*2 + 8*1 = 20.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A007955.

[&+[&*Divisors(d)*&*Divisors(n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Jan 05 2020

a(n)={sumdiv(n, d, vecprod(divisors(d))*vecprod(divisors(n/d)))} \\ Andrew Howroyd, Jan 05 2020

Terms a(31) and beyond from Andrew Howroyd, Jan 05 2020

A174940 a(n) = Sum_{d|n} A007955(d) * A008683(n/d) = Sum_{d|n} A007955(d) * mu(n/d), where A007955(m) = number of divisors of m.

Original entry on oeis.org

1, 1, 2, 6, 4, 32, 6, 56, 24, 94, 10, 1686, 12, 188, 218, 960, 16, 5772, 18, 7894, 432, 472, 22, 329992, 120, 662, 702, 21750, 28, 809648, 30, 31744, 1076, 1138, 1214, 10070172, 36, 1424, 1506, 2551944, 40, 3111034, 42, 84694, 90876, 2092, 46, 254471232, 336, 124780

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)*mu(4/1) + b(2)*mu(4/2) + b(4)*mu(4/4) = 1*0 + 2*(-1) + 8*1 = 6.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A008683 (mu), A007955 (product of divisors).

[&+[&*Divisors(d)*MoebiusMu(n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Jan 05 2020

a[n_] := Sum[ MoebiusMu[n/d] * Times @@ Divisors[d], {d, Divisors[n]} ]; Table[ a[n], {n, 1, 30} ] (* Jean-François Alcover, Jan 09 2013 *)

a(n)={sumdiv(n, d, vecprod(divisors(d))*moebius(n/d))} \\ Andrew Howroyd, Jan 05 2020

Moebius transform of A007955. - Andrew Howroyd, Jan 05 2020

Terms a(31) and beyond from Andrew Howroyd, Jan 05 2020

A265310 Least positive k such that the product of divisors of n (A007955) divides k!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 9, 13, 14, 10, 12, 17, 15, 19, 15, 14, 22, 23, 16, 15, 26, 15, 21, 29, 20, 31, 16, 22, 34, 14, 21, 37, 38, 26, 20, 41, 28, 43, 33, 15, 46, 47, 24, 21, 25, 34, 39, 53, 27, 22, 28, 38, 58, 59, 25, 61, 62, 21, 24, 26, 44, 67, 51, 46, 28, 71, 27, 73

Offset: 1

Gionata Neri, Dec 06 2015

nonn

Conjecture: a(n) = n if and only if n is prime, 2*prime, 1, 8 or 9.

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

Cf. A007955.

A265310:= proc(n) local F,f,tau,a,p,k;
      F:= ifactors(n)[2];
      tau:= mul(1+f[2],f=F);
      k:= 1;
      for f in F do
        a:= f[2]*tau/2;
        p:= f[1];
        while add(floor(k/p^j),j=1..ilog[p](k)) < a do k:= p*(1+floor(k/p)) od;
      od;
      k
end proc:
map(A265310, [$1..100]); # Robert Israel, Dec 07 2015

Table[k = 1; While[! Divisible[k!, Times @@ Divisors@ n], k++]; k, {n, 73}] (* Michael De Vlieger, Dec 06 2015 *)

a007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2));
a(n) = {k=1; while(k, if(k! % a007955(n)==0, return(k)); k++)}
vector(100, n, a(n)) \\ Altug Alkan, Dec 06 2015

cp's OEIS Frontend

A334730 a(n) = Product_{d|n} gcd(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

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A337324 a(n) is the smallest number m such that gcd(m, tau(m), sigma(m), pod(m)) = n where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

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A346956 Numbers k such that A000203(k) and A007955(k) are both divisible by A187680(k).

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A174900 a(n) = the smallest numbers k such that A007955(k) = n, or 0 if there is no such k, where A007955(m) = product of divisors of m.

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A174901 a(n) = the smallest numbers k such that A007955(k) >= n, where A007955(m) = product of divisors of m.

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A174932 a(n) = Sum_{d|n} A007955(d) * A000027(n/d) = Sum_{d|n} A007955(d) * (n/d), where A007955(m) = product of divisors of m.

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A174935 a(n) = Sum_{k<=n} A007955(k) * A000027(k) = Sum_{k<=n} A007955(k) * k, where A007955(m) = product of divisors of m.

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A174936 a(n) = Sum_{d|n} A007955(d) * A007955(n/d), where A007955(m) = product of divisors of m.

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