A003959 -id:A003959 - cp's OEIS frontend

A387711 Numbers k for which A003959(k) > 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

4, 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 100, 102, 104, 108, 112, 114, 116, 120, 124, 126, 128, 132, 135, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 174, 176, 180, 184, 186, 188, 189, 192, 196

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959, A387710.
Disjoint union of A387712 and A387713. Positions of nonzero terms in A387715.
Subsequence of A005101, and of A246282.
After the initial 4 also a subsequence of A033942.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387711(n) = (A003959(n)>(2*n));

A387712 Primitive terms of A387711: numbers k for which A003959(k) > 2k, but for all whose proper divisors d|k, dA003959(d) <= 2d.

Original entry on oeis.org

4, 18, 27, 30, 42, 45, 50, 63, 66, 70, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 375, 402, 426, 438, 474, 498, 525, 534, 582, 606, 618, 625, 642, 654, 678, 686, 735, 762, 786, 822, 825, 834, 894, 906, 942, 975, 978, 1002, 1038, 1074, 1078, 1086, 1089, 1146, 1158, 1182, 1194, 1210, 1266, 1274, 1275

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959, A387711, A387713.
Positions of 1's in A387715.
Cf. also A091191, A337372.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A387715(n) = sumdiv(n,d,(A003959(d)>(2*d)));
is_A387712(n) = (1==A387715(n));

{k | A387715(k) == 1}.

A387715 Number of divisors d of n for which A003959(d) > 2*d, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 1, 0, 2, 1, 0, 0, 6, 0, 1, 0, 2, 0, 3, 0, 4, 0, 0, 0, 5, 0, 0, 1, 5, 0, 1, 0, 2, 0, 1, 0, 7, 0, 0, 0, 2, 0, 1, 0, 6, 2, 0, 0, 5, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 8, 0, 0, 0, 4, 0, 1, 0, 4, 0

Offset: 1

Antti Karttunen, Sep 06 2025

Number of terms of A387711 that divide n.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003959, A387711 (positions of terms > 0), A387712 (of 1's), A387713 (of terms > 1).

Cf. also A080224, A337345.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387711(n) = (A003959(n)>(2*n));
A387715(n) = sumdiv(n,d,is_A387711(d));

a(n) = Sum_{d|n} [A003959(d) > 2*d], where [ ] is the Iverson bracket.

A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 4, 4, 10, 2, 12, 6, 8, 1, 16, 4, 18, 4, 12, 10, 22, 2, 16, 12, 8, 6, 28, 8, 30, 1, 20, 16, 24, 4, 36, 18, 24, 4, 40, 12, 42, 10, 16, 22, 46, 2, 36, 16, 32, 12, 52, 8, 40, 6, 36, 28, 58, 8, 60, 30, 24, 1, 48, 20, 66, 16, 44, 24, 70, 4, 72, 36, 32, 18, 60, 24, 78, 4, 16

Offset: 1

Marc LeBrun

Completely multiplicative.

Dirichlet inverse of A097945. - R. J. Mathar, Aug 29 2011

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Vaclav Kotesovec, Graph - the asymptotic ratio (10^7 terms).
Index to divisibility sequences

Cf. A003959, A168065, A168066, A027746, A006093, A027748, A124010.

a003958 1 = 1
a003958 n = product $ map (subtract 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Mar 02 2012

a:= n-> mul((i[1]-1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; Table[ DirichletInverse[ muphi][n], {n, 1, 81}] (* Jean-François Alcover, Dec 12 2011, after R. J. Mathar *)
a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 10 2016 *)

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X+X))[n]) /* Ralf Stephan */

from math import prod
from sympy import factorint
def a(n): return prod((p-1)**e for p, e in factorint(n).items())
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Feb 27 2022

Multiplicative with a(p^e) = (p-1)^e. - David W. Wilson, Aug 01 2001
a(n) = A000010(n) iff n is squarefree (see A005117). - Reinhard Zumkeller, Nov 05 2004
a(n) = abs(A125131(n)). - Tom Edgar, May 26 2014
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4 / (315 * zeta(3)) = 1/(2*A082695) = 0.25725505075419... - Vaclav Kotesovec, Jun 14 2020
Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)). - Ilya Gutkovskiy, Feb 27 2022
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + (p^(1-s) - 2) / (1 - p + p^s)), (with a product that converges for s=2). - Vaclav Kotesovec, Feb 11 2023

Definition reedited (from formula) by Daniel Forgues, Nov 17 2009

A166590 Totally multiplicative sequence with a(p) = p+2 for prime p.

Original entry on oeis.org

1, 4, 5, 16, 7, 20, 9, 64, 25, 28, 13, 80, 15, 36, 35, 256, 19, 100, 21, 112, 45, 52, 25, 320, 49, 60, 125, 144, 31, 140, 33, 1024, 65, 76, 63, 400, 39, 84, 75, 448, 43, 180, 45, 208, 175, 100, 49, 1280, 81, 196, 95, 240, 55, 500, 91, 576, 105, 124, 61, 560

Offset: 1

Jaroslav Krizek, Oct 17 2009

From Peter Munn, Feb 16 2024: (Start)
Consider the orthotope with sides given by the prime factors of n (including repetitions). a(n) is the sum of the sizes of all the orthotope's elements (vertices, edges, faces, ..., whole orthotope) with the size of a vertex taken to be 1. See the example.
If, instead, we identify congruent parallel elements (i.e., we use only one element with a given dimension and orientation) we get A003959.
(End)

			For n = 12. 12 = 2 * 2 * 3, so we sum the sizes of the elements of a cuboid with base 2 X 2 and height 3.
  Vertices: 8 of nominal size 1                 8
  Vertical edges: 4 of length 3         12
  Horizontal edges: 8 of length 2       16
   Total edge length:                  ---     28
  Vertical faces: 4 of area 2 * 3       24
  Horizontal faces: 2 of area 2 * 2      8
    Total surface area:                ---     32
  Volume: n = 2 * 2 * 3                        12
                                              ---
  Vertices + lengths + areas + volume:         80
So a(12) = 80.

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

Cf. A001620, A003959.

a166590[n_] := {1}~Join~Rest[Times @@ Power @@@ Transpose[{Plus[First /@ FactorInteger@ #, 2], Last /@ FactorInteger@ #}] & /@ Range@n]; a166590[60] (* Michael De Vlieger, Jan 07 2015 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] += 2); factorback(f); \\ Michel Marcus, Jun 09 2014

for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-2*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2023

from math import prod
from sympy import factorint
def A166590(n): return prod((p+2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 26 2022

Multiplicative with a(p^e) = (p+2)^e.
If n = Product p(k)^e(k) then a(n) = Product (p(k)+2)^e(k).
From Vaclav Kotesovec, Feb 26 2023: (Start)
Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 2*p^(-s)).
Dirichlet g.f.: zeta(s-1) * (1 + 2/(2^s - 4)) * Product_{primes p, p>2} (1 + 2/(p^s - p - 2)).
Let f(s) = Product_{primes p, p>2} (1 + 2/(p^s - p - 2)), then Sum_{k=1..n} a(k) has an average value n^2*(f(2)*(2*log(n) + 3*log(2) + 2*gamma - 1)/(8*log(2)) + f'(2)/(4*log(2))), where f(2) = Product_{primes p, p>2} (1 + 2/(p^2 - p - 2)) = 1.8687850774185607888850727174873699009051478019094666888484965828668606561..., f'(2) = f(2) * Sum_{primes p, p>2} (2*p*log(p) / (-p^3 + 2*p^2 + p - 2)) = -2.563594878667999839768204519417845474796924720924625514292420625983768019... and gamma is the Euler-Mascheroni constant A001620. (End)

More terms from Michel Marcus, Jun 09 2014

A336467 Fully multiplicative with a(2) = 1 and a(p) = A000265(p+1) for odd primes p, with A000265(k) giving the odd part of k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 7, 1, 3, 1, 9, 1, 5, 3, 1, 3, 3, 1, 9, 7, 1, 1, 15, 3, 1, 1, 3, 9, 3, 1, 19, 5, 7, 3, 21, 1, 11, 3, 3, 3, 3, 1, 1, 9, 9, 7, 27, 1, 9, 1, 5, 15, 15, 3, 31, 1, 1, 1, 21, 3, 17, 9, 3, 3, 9, 1, 37, 19, 9, 5, 3, 7, 5, 3, 1, 21, 21, 1, 27, 11, 15, 3, 45, 3, 7, 3, 1, 3, 15, 1, 49, 1, 3, 9, 51, 9, 13, 7, 3

Offset: 1

Antti Karttunen, Jul 22 2020

For the comment here, we extend the definition of the first kind of Cunningham chain (see Wikipedia-article) so that also isolated primes for which neither (p-1)/2 nor 2p+1 is a prime are considered to be in singular chains, that is, in chains of the length one. If we replace one or more instances of any particular odd prime factor p in n with any odd prime q of the same Cunningham chain, so that m = (q^k)*n / p^(e-k), where e is the exponent of p of n, and k <= e is the number of instances of p replaced with q, then it holds that a(m) = a(n), and by induction, the value stays invariant for any number of such replacements. Note also that A001222, but not necessarily A001221 will stay invariant in such changes.

For example, if some of the odd prime divisors p of n are Sophie Germain primes (in A005384), then replacing any of them with 2p+1 ("safe primes", i.e., the corresponding terms of A005385), gives a new number m, for which a(m) = a(n). And vice versa, the same is true for any safe prime factors > 5 of n (that are in A005385), then replacing any one of them with (p-1)/2 will not affect the result. For example, a(5*11*23*47) = a(11*11*23*23) = a(5^4) = a(11^4) = a(23^4) = 81, as 5, 11, 23 and 47 are in the same Cunningham chain of the first kind.

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

Cf. A000265, A003959, A206787, A336390, A336392, A336651, A336921, A336922, A351461.

Cf. also A335915, A336466 (similar sequences).

A000265(n) = (n>>valuation(n,2));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };

For all n >= 1, A331410(a(n)) = A336921(n).
From Antti Karttunen, Nov 21 2023: (Start)
a(n) = A335915(n) / A336466(n).
a(1) = 1, and for n > 1, a(n) = A000265(A206787(n)) * a(A336651(n)).
(End)

A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)(b+1)...*(k+1).

Original entry on oeis.org

2, 6, 8, 30, 12, 168, 16, 270, 80, 396, 24, 10920, 28, 720, 768, 4590, 36, 31920, 40, 41580, 1408, 1656, 48, 2457000, 312, 2268, 2240, 104400, 60, 5499648, 64, 151470, 3264, 3780, 3456, 76767600, 76, 4680, 4480, 15343020, 84, 19071360, 88, 372600, 353280, 6768

Offset: 1

Amarnath Murthy, Jun 01 2003

Named "Vandiver's arithmetical function" by Sándor (2021), after the American mathematician Harry Schultz Vandiver (1882-1973). - Amiram Eldar, Jun 29 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
József Sándor, On Vandiver's arithmetical function - I, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38.
Harry S. Vandiver, Problem 116, American Mathematical Monthly, Vol. 11, No. 2 (1904), pp. 38-39.

Cf. A027750, A000005, A003959, A007955.
Cf. A057643 (LCM instead of product).
Cf. A299436 (exp).

a020696 = product . map (+ 1) . a027750_row'
-- Reinhard Zumkeller, Mar 28 2015

a:= n-> mul(d+1, d=numtheory[divisors](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 30 2022~

Table[Times @@ (Divisors[n] + 1), {n, 43}] (* Ivan Neretin, May 27 2015 *)

a(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1));} \\ Michel Marcus, Jun 12 2013

from math import prod
from sympy import divisors
def A020696(n): return prod(d+1 for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 30 2022

a(p) = 2(p+1), a(p^2) = 2(p+1)(p^2+1) for primes p.
a(n) = Product_{k = 1..A000005(n)} (A027750(n,k) + 1). - Reinhard Zumkeller, Mar 28 2015
a(n) = Product_{d|n} (d+1). - Amiram Eldar, Jun 29 2022

Edited by Don Reble, Jun 05 2003

A064476 For an integer k with prime factorization p_1p_2p_3* ... p_m let k = (p_1+1)(p_2+1)(p_3+1)* ... (p_m+1) (A064478); sequence gives k such that k is divisible by k.

Original entry on oeis.org

1, 6, 12, 36, 72, 144, 216, 432, 864, 1296, 1728, 2592, 5184, 7776, 10368, 15552, 20736, 31104, 46656, 62208, 93312, 124416, 186624, 248832, 279936, 373248, 559872, 746496, 1119744, 1492992, 1679616, 2239488, 2985984, 3359232, 4478976

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 06 2001

Could be generalized by defining x* = (p_1+v)*(p_2+v) .. (p_m+v) where v is any integer.

It is not difficult to show that these numbers have the form 2^i*3^j with j <= i <= 2j. Hence 1 is the only odd term; also if k|k* then k*|k**. The values of i and j are given in A064514 and A064515. - Vladeta Jovovic and N. J. A. Sloane, Oct 07 2001

			12 is in the sequence because 12 = 2 * 2 * 3, so 12* is 3 * 3 * 4 = 36 and 36 is divisible by 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..50 from Harry J. Smith)

Cf. A064478, A064514, A064515, A064518, A064522, A003959.

function p2p3(stop:integer): array; var c,i,j,x: integer; b: boolean; ar: array; begin ar := alloc(array,stop); x := 0; c := 0; b := c < stop; while b do i := x; j := x - i; while b and i >= j do if i <= 2*j then ar[c] := (2^i * 3^j,i,j); inc(c); b := c < stop; end; dec(i); inc(j); end; inc(x); end; return sort(ar, comparefirst); end; function comparefirst(x,y: array): integer; begin return y[0] - x[0]; end; function a064476(maxarg: integer); var j: integer; ar: array; begin ar := p2p3(maxarg); for j := 0 to maxarg - 1 do write(ar[j][0]," "); end; end; a064476(35);

a064476 n = a064476_list !! (n-1)
a064476_list = filter (\x -> a003959 x `mod` x == 0) [1..]
-- Reinhard Zumkeller, Feb 28 2013

diQ[n_]:=Divisible[Times@@(#+1&/@Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[n]]),n]; Select[Range[4500000],diQ] (* Harvey P. Dale, Aug 16 2011 *)
With[{max = 5*10^6}, Select[Flatten[Table[2^i*3^j, {j, 0, Log[6, max]}, {i, j, 2*j}]] // Sort, # <= max &]] (* Amiram Eldar, Mar 29 2025 *)

ns(n)= { local(f,p=1); f=factor(n); for(i=1, matsize(f)[1], p*=(1 + f[i, 1])^f[i, 2]); return(p) }
{ n=0; for (m=1, 10^9, if (ns(m)%m == 0, write("b064476.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 15 2009

from sympy import integer_log
def A064476(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(max(0,min((i<<1)+1,(x//3**i).bit_length())-i) for i in range(integer_log(x,3)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 26 2025

Sum_{n>=1} 1/a(n) = 72/55. - Amiram Eldar, Mar 29 2025

More terms from Vladeta Jovovic, Oct 07 2001

A063441 a(n) = sigma(n) * mu(n).

Original entry on oeis.org

1, -3, -4, 0, -6, 12, -8, 0, 0, 18, -12, 0, -14, 24, 24, 0, -18, 0, -20, 0, 32, 36, -24, 0, 0, 42, 0, 0, -30, -72, -32, 0, 48, 54, 48, 0, -38, 60, 56, 0, -42, -96, -44, 0, 0, 72, -48, 0, 0, 0, 72, 0, -54, 0, 72, 0, 80, 90, -60, 0, -62, 96, 0, 0, 84, -144, -68, 0, 96, -144, -72, 0, -74, 114, 0, 0, 96, -168, -80, 0, 0, 126, -84, 0, 108

Offset: 1

Jason Earls, Jul 23 2001

			n=6: divisors of 6 are = [1, 2, 3, 6] and 1 * mu(6) + 2 * mu(6) + 3 * mu(6) + 6 * mu(6) = 12.

Harry J. Smith, Table of n, a(n) for n=1..2000

a[n_] := DivisorSigma[1, n] MoebiusMu[n]; Array[a, 90] (* Jean-François Alcover, Dec 05 2015 *)

j=[]; for(n=1,200,j=concat(j,sumdiv(n,d,d*moebius(n)))); j

a(n)=if(n<1,0,direuler(p=2,n,1-X-p*X)[n]) /* Ralf Stephan, Jul 07 2013 */

{ for (n=1, 2000, write("b063441.txt", n, " ", direuler(p=2, n, 1-X-p*X)[n]) ) } \\ Harry J. Smith, Aug 21 2009

a(n) = sum_{d|n} d * mu(n).
a(n) = A000203(n) * A008683(n).
a(n) = A003959(n) * A008683(n) if n is squarefree, 0 otherwise. - Ralf Stephan, Mar 26 2004
Multiplicative with a(p^e) = -p-1, if e = 1, 0 otherwise. - Mitch Harris, Jun 27 2005, sign flipped by R. J. Mathar, May 29 2011
sum(n>0, a(n)/n^s) = product(p prime, 1-p^(-s)-p^(1-s) ). - Ralf Stephan, Jul 07 2013

A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 12, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 36, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1

Offset: 1

Ilya Gutkovskiy, Mar 03 2020

			a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.

a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

a(1) = 1; a(n) = -Sum_{d|n, dA001221(n/d) * A003557(n/d) * a(d).
a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).
a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - Antti Karttunen, Oct 29 2021
Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - Amiram Eldar, Dec 07 2023

cp's OEIS Frontend

A387711 Numbers k for which A003959(k) > 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A387712 Primitive terms of A387711: numbers k for which A003959(k) > 2*k, but for all whose proper divisors d|k, dA003959(d) <= 2*d.

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A387715 Number of divisors d of n for which A003959(d) > 2*d, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

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A166590 Totally multiplicative sequence with a(p) = p+2 for prime p.

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A336467 Fully multiplicative with a(2) = 1 and a(p) = A000265(p+1) for odd primes p, with A000265(k) giving the odd part of k.

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PARI

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A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)*(b+1)*...*(k+1).

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Haskell

Maple

Mathematica

PARI

A387712 Primitive terms of A387711: numbers k for which A003959(k) > 2k, but for all whose proper divisors d|k, dA003959(d) <= 2d.

A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)(b+1)...*(k+1).

A064476 For an integer k with prime factorization p_1p_2p_3* ... p_m let k = (p_1+1)(p_2+1)(p_3+1)* ... (p_m+1) (A064478); sequence gives k such that k is divisible by k.