cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A061537 Product of unitary divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 36, 7, 8, 9, 100, 11, 144, 13, 196, 225, 16, 17, 324, 19, 400, 441, 484, 23, 576, 25, 676, 27, 784, 29, 810000, 31, 32, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 3111696, 43, 1936, 2025, 2116, 47, 2304, 49, 2500, 2601, 2704, 53, 2916
Offset: 1

Views

Author

Labos Elemer, May 15 2001

Keywords

Comments

Also appears to be smallest number m such that A066296(m) = n.

Examples

			For n = 288, unitary divisors = {1, 9, 32, 288}, a(288) = 1 * 9 * 32 * 288 = 82944.

Links

Crossrefs

Cf. A000005, A001221, A007955, A007956, A034444, A048105, A066296.

Programs

  • Maple
    a:= n-> mul(`if`(igcd(d, n/d)=1, d, 1), d=numtheory[divisors](n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 01 2017
  • Mathematica
    Table[Times@@ Select[Divisors[n], GCD[#, n/#]==1 &], {n, 1, 100}] (* Indranil Ghosh, Aug 04 2017 *)
a[n_] := n^(2^(PrimeNu[n]-1)); Array[a, 60] (* Amiram Eldar, Jul 22 2024 *)
  • PARI
    { for (n=1, 1000, s=divisors(n); a=1; for (i=2, length(s), d=s[i]; if (gcd(d, n/d)==1, a*=d)); write("b061537.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 24 2009
  • PARI
    a(n) = n^(2^(omega(n)-1)); \\ Amiram Eldar, Jul 22 2024
  • Python
    from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d)==1)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

Formula

a(n) = n^(A034444(n)/2) = n^(2^(A001221(n)-1)).

Extensions

Corrected and edited by Jaroslav Krizek, Mar 05 2009

A136655 Product of odd divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11, 23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5, 41, 441, 43, 11, 91125, 23, 47, 3, 343, 125, 2601, 13, 53, 729, 3025, 7, 3249, 29, 59, 225, 61, 31, 250047, 1, 4225, 1089
Offset: 1

Views

Author

Jonathan Vos Post, Jun 25 2008

Keywords

Comments

Product of rows of triangle A182469. - Reinhard Zumkeller, May 01 2012

Links

Crossrefs

Cf. A000265, A000593, A007955, A007956, A078701.
Cf. A140210, A140211, A140213, A140214, A140215.
Cf. A125911, A126192.
Cf. A001227, A183063.

Programs

  • Haskell
    a136655 = product . a182469_row  -- Reinhard Zumkeller, May 01 2012
  • Maple
    with(numtheory); f:=proc(n) local t1,i,k; t1:=divisors(n); k:=1; for i in t1 do if i mod 2 = 1 then k:=k*i; fi; od; k; end; # N. J. A. Sloane, Jul 14 2008
  • Mathematica
    Array[Times @@ Select[Divisors@ #, OddQ] &, 66] (* Michael De Vlieger, Aug 03 2017 *)
a[n_] := (oddpart = n/2^IntegerExponent[n, 2])^(DivisorSigma[0, oddpart]/2); Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, if (d[k]%2, d[k], 1)); \\ Michel Marcus, Aug 04 2017
  • Python
    from math import isqrt
from sympy import divisor_count
def A136655(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    return isqrt(m)**d if d&1 else m**(d>>1) # Chai Wah Wu, Jun 27 2025

Formula

a(p) = p if p noncomposite; a(2^n) = 1; a(pq) = p^2 * q^2 when p, q are odd primes.
a(n) = sqrt(n^od(n)/2^ed(n)), where od(n) = number of odd divisors of n = tau(2*n)-tau(n) and ed(n) = number of even divisors of n = 2*tau(n)-tau(2*n). - Vladeta Jovovic, Jun 25 2008
Also a(n) = A007955(A000265(n)). - David Wilson, Jun 26 2008
a(n) = Product_{h == 1 mod 4 and h | n}*Product_{i == 3 mod 4 and i | n}.
a(n) = Product_{j == 1 mod 6 and j | n}*Product_{k == 5 mod 6 and k | n}.
a(n) = A140210(n)*A140211(n). - R. J. Mathar, Jun 27 2008
a(n) = A007955(n) / A125911(n).

Extensions

More terms from N. J. A. Sloane, Jul 14 2008
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A240694 Partial products of divisors of n, cf. A027750.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 8, 1, 5, 1, 2, 6, 36, 1, 7, 1, 2, 8, 64, 1, 3, 27, 1, 2, 10, 100, 1, 11, 1, 2, 6, 24, 144, 1728, 1, 13, 1, 2, 14, 196, 1, 3, 15, 225, 1, 2, 8, 64, 1024, 1, 17, 1, 2, 6, 36, 324, 5832, 1, 19, 1, 2, 8, 40, 400, 8000, 1, 3, 21, 441, 1, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 10 2014

Keywords

Comments

Triangle read by rows in which row n lists the partial products of divisors of n. - Omar E. Pol, Apr 12 2014

Examples

			.    n |  n-th row of A240694     |  n-th row of A027750
.  ----+--------------------------+---------------------
.    1 |  1                       |  1
.    2 |  1, 2                    |  1, 2
.    3 |  1, 3                    |  1, 3
.    4 |  1, 2, 8                 |  1, 2, 4
.    5 |  1, 5                    |  1, 5
.    6 |  1, 2, 6, 36             |  1, 2, 3, 6
.    7 |  1, 7                    |  1, 7
.    8 |  1, 2, 8, 64             |  1, 2, 4, 8
.    9 |  1, 3, 27                |  1, 3, 9
.   10 |  1, 2, 10, 100           |  1, 2, 5, 10
.   11 |  1, 11                   |  1, 11
.   12 |  1, 2, 6, 24, 144, 1728  |  1, 2, 3, 4, 6, 12
.   13 |  1, 13                   |  1, 13 .

Links

Crossrefs

Cf. A000005 (row lengths), A007955, A020639, A027750, A240698.

Programs

  • Haskell
    a240694 n k = a240694_tabf !! (n-1) !! (k-1)
a240694_row n = a240694_tabf !! (n-1)
a240694_tabf = map (scanl1 (*)) a027750_tabf
  • Mathematica
    Table[FoldList[Times,Divisors[n]],{n,30}]//Flatten (* Harvey P. Dale, Jul 29 2021 *)
  • PARI
    row(n) = my(d=divisors(n)); vector(#d, k, prod(i=1, k, d[i])); \\ Michel Marcus, Jan 24 2022

Formula

T(n,1) = 1, T(n,k) = T(n,k-1) * A027750(n,k), 1 < k <= n.
T(n,1) = 1;
T(n,2) = A020639(n), n > 1;
T(n,A000005(n)) = A007955(n);
T(n,A000005(n)-1) = A007956(n), n > 1.

A034288 Product of proper divisors is larger than for any smaller number.

Original entry on oeis.org

1, 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
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Cf. A034287, A034090, A034091, A067128, A140999.
Indices of records of A007956.

Programs

  • Mathematica
    maxTerm = 10^6; record = 0; Reap[For[n = 1, n <= maxTerm, n++, p = Times @@ Most[Divisors[n]]; If[p > record, record = p; Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Aug 01 2013 *)
DeleteDuplicates[Table[{n,Times@@Most[Divisors[n]]},{n,11000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Nov 21 2024 *)

A111398 Numbers which are the cube roots of the product of their proper divisors.

Original entry on oeis.org

1, 24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 102, 104, 105, 110, 114, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 258, 266, 273, 282, 285, 286, 290, 296, 297
Offset: 1

Views

Author

Ant King, Nov 11 2005

Keywords

Comments

This sequence is actually the sequence of 4-multiplicatively perfect numbers all of whose elements (>1) have prime signature {7}, {1,3} or {1,1,1}.

Links

Crossrefs

Cf. A048945, A111399. Essentially the same as A030626.
Cf. A030626, A007956.

Programs

  • Mathematica
    Select[Range[300],Surd[Times@@Most[Divisors[#]],3]==#&] (* Harvey P. Dale, Nov 16 2015 *)
  • PARI
    isok(n) = {prd = 1; fordiv(n, d, prd = prd*d); prd == n^4;} \\ Michel Marcus, Oct 04 2013

Formula

1 together with numbers with 8 divisors. - Vladeta Jovovic, Nov 12 2005

Extensions

More terms from Michel Marcus, Oct 04 2013

A191906 The remainder of (product of proper divisors of n) mod (sum of proper divisors of n).

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 1, 3, 2, 0, 0, 0, 4, 6, 4, 0, 9, 0, 4, 10, 8, 0, 0, 5, 10, 1, 0, 0, 36, 0, 1, 3, 14, 9, 41, 0, 16, 5, 0, 0, 0, 0, 16, 12, 20, 0, 44, 7, 6, 9, 36, 0, 54, 4, 0, 11, 26, 0, 0, 0, 28, 33, 8, 8, 66, 0, 42, 15, 10, 0, 81, 0, 34, 39, 16, 1, 72, 0, 10, 9, 38, 0, 84, 16, 40, 21
Offset: 2

Views

Author

Juri-Stepan Gerasimov, Jun 19 2011

Keywords

Examples

			a(2) = 1 mod 1 = 0;
a(3) = 1 mod 1 = 0;
a(4) = 2 mod 3 = 2.

Links

Crossrefs

Cf. A001065, A007956, A187680.

Programs

  • Maple
    A007956 := n -> mul(i, i=op(numtheory[divisors](n) minus {1, n}));
A001065 := proc(n) numtheory[sigma](n)-n ; end proc:
A191906 := proc(n) A007956(n) mod A001065(n) ; end proc:
seq(A191906(n),n=2..90) ; # R. J. Mathar, Jun 25 2011
  • Mathematica
    Table[With[{pd=Most[Divisors[n]]},Mod[Times@@pd,Total[pd]]],{n,2,90}] (* Harvey P. Dale, Nov 24 2021 *)
  • PARI
    A191906(n) = { my(m=1,s=0); fordiv(n, d, if(dAntti Karttunen, Jul 11 2019

Formula

a(n) = A007956(n) mod A001065(n).

A061538 Product of all divisors of n, divided by product of unitary divisors; or equivalently product of non-unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 8, 3, 1, 1, 12, 1, 1, 1, 64, 1, 18, 1, 20, 1, 1, 1, 576, 5, 1, 27, 28, 1, 1, 1, 1024, 1, 1, 1, 7776, 1, 1, 1, 1600, 1, 1, 1, 44, 45, 1, 1, 110592, 7, 50, 1, 52, 1, 2916, 1, 3136, 1, 1, 1, 3600, 1, 1, 63, 32768, 1, 1, 1, 68, 1, 1, 1, 26873856, 1, 1, 75, 76, 1, 1, 1
Offset: 1

Views

Author

Labos Elemer, May 15 2001

Keywords

Examples

			For n = 16: only {1,16} are unitary, while {2,4,8} are non-unitary divisors, so a(16) = 64.
When all divisors are unitary, then A048105 is 0 and the corresponding terms here are equal to 1.

Links

Crossrefs

Cf. A000005, A007955, A007956, A034444, A048105.

Programs

  • Mathematica
    Table[Times @@ Select[Divisors@ n, ! CoprimeQ[#, n/#] &], {n, 79}] (* Michael De Vlieger, Mar 17 2017 *)
a[n_] := n^((DivisorSigma[0, n] - 2^PrimeNu[n]) / 2); Array[a, 80] (* Amiram Eldar, Jul 22 2024 *)
  • PARI
    { for (n=1, 1000, s=divisors(n); a=1; for (i=2, length(s), d=s[i]; if (gcd(d, n/d)!=1, a*=d)); write("b061538.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 24 2009
  • PARI
    a(n) = {my(f = factor(n)); n^((numdiv(f) - 2^omega(f))/2);} \\ Amiram Eldar, Jul 22 2024

Formula

a(n) = n^(A048105(n)/2) = n^((A000005(n) - A034444(n))/2).

Extensions

Corrected and edited by Jaroslav Krizek, Mar 05 2009

A007624 Numbers m such that the product of proper divisors of m = m^k, k>1.

Original entry on oeis.org

12, 18, 20, 24, 28, 30, 32, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 147, 148, 150, 152, 153, 154
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000005, A007956.

Programs

  • Mathematica
    ppdQ[n_]:=Module[{c=Log[n,Times@@Most[Divisors[n]]]},c>1&&IntegerQ[c]]; Select[Range[2,160],ppdQ] (* Harvey P. Dale, Mar 06 2012 *)
Select[Range[154], EvenQ[(d = DivisorSigma[0,#])] && d > 4 &] (* Amiram Eldar, Nov 20 2019 *)

Formula

d(n) > 4 and even.
a(n) = n + O(n/log n). - Charles R Greathouse IV, Oct 23 2015

A048741 Product of aliquot divisors of composite n (1 and primes omitted).

Original entry on oeis.org

2, 6, 8, 3, 10, 144, 14, 15, 64, 324, 400, 21, 22, 13824, 5, 26, 27, 784, 27000, 1024, 33, 34, 35, 279936, 38, 39, 64000, 74088, 1936, 2025, 46, 5308416, 7, 2500, 51, 2704, 157464, 55, 175616, 57, 58, 777600000, 62, 3969, 32768, 65, 287496, 4624, 69
Offset: 1

Views

Author

Enoch Haga

Keywords

Examples

			The third composite number is 8, for which the product of aliquot divisors is 4*2*1 = 8, so a(3)=8.

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed., pages 10, 23. New York: Dover, 1966. ISBN 0-486-21096-0.

Links

Crossrefs

This is A007956 omitting the 1's.
Cf. A002808, A007422, A007956, A048740.

Programs

  • Mathematica
    Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Times @@ Select[ Divisors[ Composite[n]], # < Composite[n] & ], {n, 1, 60} ]
pd[n_] := n^(DivisorSigma[0, n]/2 - 1); pd /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Sep 07 2019 *)

Formula

a(n) = A007956(A002808(n)). - Michel Marcus, Sep 07 2019

Extensions

a(33) inserted by Amiram Eldar, Sep 07 2019

A064116 Composite numbers whose sum of aliquot divisors as well as product of aliquot divisors is a perfect square.

Original entry on oeis.org

12, 75, 76, 124, 147, 153, 176, 243, 332, 363, 477, 507, 524, 575, 688, 867, 892, 963, 1075, 1083, 1421, 1532, 1573, 1587, 1611, 1916, 2032, 2075, 2224, 2299, 2401, 2421, 2523, 2572, 2883, 2891, 3100, 3479, 3776, 3888, 4107, 4336, 4527, 4961, 4975, 5043
Offset: 1

Views

Author

Shyam Sunder Gupta, Sep 09 2001

Keywords

Examples

			75 is a term because the sum of the aliquot divisors of 75 = 1 + 3 + 5 + 15 + 25 = 49 = 7^2 and the product of the aliquot divisors of 75 = 1*3*5*15*25 = 75^2.

Links

Crossrefs

Intersection of A048699 and A064499.
Cf. A001065, A007956.

Programs

  • Mathematica
    Do[d = Delete[ Divisors[n], -1]; If[ !PrimeQ[n] && IntegerQ[ Sqrt[ Apply[ Plus, d]]] && IntegerQ[ Sqrt[ Apply[ Times, d]]], Print[n]], {n, 2, 10^4} ]
spsQ[n_]:=Module[{d=Most[Divisors[n]]},CompositeQ[n]&&AllTrue[{Sqrt[ Total[ d]],Sqrt[Times@@d]},IntegerQ]]; Select[Range[5100],spsQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 14 2018 *)
  • PARI
    isok(k) = { my(s=sigma(k) - k); s>1 && issquare(s) && issquare(vecprod(divisors(k)[1..-2])) } \\ Harry J. Smith, Sep 07 2009

Extensions

More terms from Robert G. Wilson v, Oct 05 2001
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