A085082 -id:A085082 - cp's OEIS frontend

A182860 Number of distinct prime signatures represented among the unitary divisors of n.

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

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

a(n) = number of members m of A025487 such that d(m^k) divides d(n^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

a(n) depends only on prime signature of n (cf. A025487).

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  Since a total of 6 distinct prime signatures appear among the unitary divisors of 60, a(60) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A034444, A064553, A085082, A146289, A146290, A182861, A182862, A212180, A328830.

Table[Length@ Union@ Map[Sort[FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 1, 0, e]] &, Select[Divisors@ n, CoprimeQ[#, n/#] &]], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)

A181819(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); }; \\ From A181819
A182860(n) = numdiv(A181819(n)); \\ Antti Karttunen, Jul 19 2017

a(n) = A000005(A181819(n)).
If the canonical factorization of n into prime powers is Product p^e(p), then the formula for d(n^k) is Product_p (ek + 1). (See also A146289, A146290.)
a(n) = A064553(A328830(n)). - Antti Karttunen, Apr 29 2022

A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 1260, 2520, 27720, 138600, 360360, 831600, 10810800, 75675600, 183783600, 1286485200, 24443218800, 38594556000, 424540116000, 733296564000, 8066262204000, 185524030692000, 1693915062840000, 5380196890068000, 38960046445320000, 166786103592108000

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

In other words, the sequence includes k iff A182860(k) > A182860(m) for all m < k.

The records for the number of distinct prime signatures are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, ... (see the link for more values). - Amiram Eldar, Jul 07 2019

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  This makes a total of 6 distinct prime signatures that appear among the unitary divisors of 60.  Since no positive integer smaller than 60 has more than 4 distinct prime signatures appearing among its unitary divisors, 60 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..60
Amiram Eldar, Table of n, a(n), A182860(a(n)) for n = 1..60
Eric Weisstein's World of Mathematics, Unitary Divisor

Subsequence of A025487, A129912, A181826, A182863. See also A034444, A085082, A182860, A182861.

f[1] = 1; f[n_] := Times @@ (Values[Counts[FactorInteger[n][[;; , 2]]]] + 1); fm = 0; s={}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Jan 19 2019 *)

a(14)-a(26) from Amiram Eldar, Jan 19 2019

A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3

Offset: 1

Matthew Vandermast, Jun 04 2012

nonn

Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172, A212173).

The fraction of the divisors of n which have a given second signature {S} is also a function of n's second signature. For example, if n has second signature {3,2}, it follows that 1/3 of n's divisors are squarefree. Squarefree numbers are represented with 0's in A212172, in accord with the usual OEIS custom of using 0 for nonexistent elements; in comments, their second signature is represented as { }.

			The divisors of 72 represent a total of 5 distinct second signatures (cf. A212172), as can be seen from the exponents >= 2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Hence, a(72) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A212172, A085082, A088873, A181796, A182860, A212173, A212642, A212643, A212644.

Array[Length@ Union@ Map[Sort@ Select[FactorInteger[#][[All, -1]], # >= 2 &] &, Divisors@ #] &, 88] (* Michael De Vlieger, Jul 19 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ This function from Charles R Greathouse IV, Aug 13 2013
A212173(n) = A046523(A057521(n));
A212180(n) = { my(vals = Set()); fordiv(n, d, vals = Set(concat(vals, A212173(d)))); length(vals); }; \\ Antti Karttunen, Jul 19 2017

from sympy import factorint, divisors, prod
def P(n): return sorted(factorint(n).values())
def a046523(n):
    x=1
    while True:
        if P(n)==P(x): return x
        else: x+=1
def a057521(n): return 1 if n==1 else prod(p**e for p, e in factorint(n).items() if e != 1)
def a212173(n): return a046523(a057521(n))
def a(n):
    l=[]
    for d in divisors(n):
        x=a212173(d)
        if not x in l:l+=[x, ]
    return len(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

A212642 a(n) = number of distinct prime signatures represented among divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

1, 3, 4, 5, 6, 6, 7, 9, 8, 12, 10, 9, 15, 14, 10, 18, 18, 10, 11, 21, 15, 22, 16, 12, 24, 20, 26, 22, 13, 27, 25, 19, 30, 28, 21, 14, 30, 30, 28, 34, 34, 27, 15, 33, 35, 37, 20, 38, 40, 33, 31, 16, 36, 40, 46, 15, 28, 30, 42, 46, 39, 43, 17, 39, 45, 55, 25, 35

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

Also, number of divisors of A181800 that are members of A025487.

Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. Let n be any integer with second signature {S}. Then A212180(n) = k and A085082(n) is congruent to j modulo k. Cf. A212643, A212644.

			The divisors of 36 represent a total of 6 distinct prime signatures (cf. A085082), as can be seen from the positive exponents, if any, in the canonical prime factorization of each divisor:
{ }: 1 (multiset of positive exponents is the empty multiset)
{1}: 2 (2^1), 3 (3^1)
{1,1}: 6 (2^1*3^1)
{2}: 4 (2^2), 9 (3^2),
{2,1}: 12 (2^2*3^1), 18 (2^1*3^2)
{2,2}: 36 (2^2*3^2)
Since 36 = A181800(6), a(6) = 6.

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

Cf. A181800, A085082, A212171, A212172, A212643, A212644.

a(n) = A085082(A181800(n)).

A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 6, 5, 7, 7, 6, 8, 9, 9, 9, 11, 12, 4, 10, 13, 10, 15, 7, 11, 15, 14, 18, 10, 12, 17, 18, 9, 21, 13, 15, 13, 19, 22, 14, 24, 16, 20, 14, 21, 26, 19, 10, 27, 19, 25, 16, 15, 23, 30, 24, 5, 21, 16, 30, 22, 30, 23, 16, 25, 34, 29, 9, 27, 22, 33

Offset: 1

Matthew Vandermast, Jun 07 2012

nonn

Also, number of divisors of A181800(n) that are members of A181800.

Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures (cf. A212642). Let m be any integer with second signature {S}. Then A212180(m) = k and A085082(m) is congruent to j modulo k. If {S} is the second signature of A181800(n), then A085082(m) is congruent to A212643(n) modulo a(n).

			The divisors of 72 represent 5 distinct second signatures (cf. A212172), as can be seen from the exponents >=2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Since 72 = A181800(8), a(8) = 5.

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

Cf. A181800, A085082, A212172, A212176, A212642, A212643.

a(n) = A212180(A181800(n)).

Data corrected by Amiram Eldar, Jul 14 2019

A088873 Number of different values of A000005(d) where d is a divisor of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 6, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 6, 2, 3, 3, 6, 2, 4, 2, 5, 5, 3, 2, 8, 3, 5, 3, 5, 2, 6, 3, 6, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 8, 2, 3, 5, 5, 3, 4, 2, 8, 5, 3, 2, 7, 3, 3, 3, 6, 2, 7, 3, 5, 3, 3, 3, 9, 2, 5, 5, 6, 2, 4, 2, 6

Offset: 1

Naohiro Nomoto, Nov 30 2003

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

Cf. A000005.

Differs from A085082 and A181796 for the first time at n=24, where a(24) = 6, while A085082(24) = A181796(24) = 7.

a[n_] := Length@ DeleteDuplicates[DivisorSigma[0, Divisors[n]]]; Array[a, 100] (* Amiram Eldar, Mar 27 2024 *)

a(n) = {vals = Set(); fordiv(n, d, vals = Set(concat(vals, numdiv(d)))); return (length(vals));} \\ Michel Marcus, Jul 14 2013

A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 4, 1, 5, 4, 1, 6, 5, 1, 7, 6, 2, 1, 8, 5, 7, 2, 1, 9, 6, 8, 2, 1, 10, 7, 1, 9, 2, 6, 1, 11, 8, 0, 10, 2, 7, 1, 12, 9, 18, 0, 11, 2, 8, 15, 1, 13, 10, 22, 0, 7, 14, 12, 2, 9, 20, 1, 14, 11, 26, 7, 8, 18, 13, 2, 10, 25, 1, 15, 15, 12, 30, 9

Offset: 1

Matthew Vandermast, Jun 05 2012

Significance of the sequence: Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. For all integers n with second signature {S}, A212180(n) = k and A085082(n) is congruent to j modulo k; see examples.

Note: b(n) = A212642(n); c(n) = A212644(n).

			4 is the smallest integer with second signature {2}, and its divisors represent 3 distinct prime signatures and 2 distinct second signatures. 1 = 3 mod 2. Since 4 = A181800(2), a(2) = 1. For all integers m with second signature {2}, A085082(m) is congruent to 1 modulo 2.
10800 is the smallest integer with second signature {4,3,2}, and its divisors represent 28 distinct prime signatures and 14 distinct second signatures. 0 = 28 mod 14.  Since 10800 = A181800(39), a(39) = 0. For all integers m with second signature {4,3,2}, A085082(m) is congruent to 0 modulo 14.

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

Cf. A085082, A212171, A212172, A212180, A212642, A212644.

a(n) = A212642(n)-A212644(n), reduced modulo A212644(n).

A182861 Number of distinct prime signatures represented among the unitary divisors of A025487(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 4, 2, 3, 4, 6, 2, 4, 4, 6, 2, 4, 6, 4, 5, 3, 6, 2, 4, 8, 4, 8, 4, 6, 2, 4, 8, 4, 8, 4, 4, 6, 2, 6, 4, 9, 3, 8, 4, 8, 4, 6, 6, 2, 8, 4, 6, 12, 4, 8, 4, 8, 4, 6, 6, 2, 8, 4, 10, 12, 4, 6, 8, 4, 8, 6, 8, 4, 6, 9, 6, 3, 2, 8, 4, 10, 12, 4

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

a(n) = number of members m of A025487 such that d(m^k) divides d(A025487(n)^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  Since a total of 6 distinct prime signatures appear among the unitary divisors of 60, and since 60 = A025487(13), a(13) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A000005, A025487, A034444, A085082, A146289, A146290, A181820, A182860, A182862.

a(n) = A000005(A181820(n)).

If the canonical factorization of n into prime powers is Product p^e(p), then the formula for d(n^k) is Product_p (ek + 1). (See also A146289, A146290.)

More terms from Amiram Eldar, Jun 20 2019

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 32, 12, 64, 24, 36, 128, 48, 72, 256, 96, 144, 30, 512, 192, 216, 288, 60, 1024, 384, 432, 576, 120, 2048, 768, 864, 180, 1152, 240, 1296, 4096, 1536, 1728, 360, 2304, 480, 2592, 8192, 3072, 3456, 720, 900, 4608, 960, 5184, 1080, 16384

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

The number of second signatures represented by the divisors of A181800(n) equals the number of prime signatures represented among the divisors of a(n). Cf. A212172, A212644.

A permutation of A025487.

			6 (whose prime factorization is 2*3) is the largest squarefree divisor of 144 (whose prime factorization is 2^4*3^2). Since 144 = A181800(10), and 144/6 = 24, a(10) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A003557, A007947, A085082, A181800, A212172, A212642, A212644.

Other permutations of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.

a(n) = A003557(A181800(n)).

A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 7, 4, 6, 6, 9, 7, 7, 9, 11, 10, 8, 12, 9, 13, 5, 10, 13, 9, 15, 14, 15, 9, 14, 16, 10, 18, 19, 17, 13, 18, 10, 19, 11, 16, 21, 12, 15, 24, 19, 17, 22, 16, 22, 12, 23, 24, 6, 19, 20, 29, 21, 21, 26, 22, 25, 13, 30, 27, 11, 26, 25, 19, 34

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Also the number of members of A025487 that divide A025487(n).

			5 members of A025487 divide A025487(6) = 12 (namely, 1, 2, 4, 6 and 12); therefore, a(6) = 5.

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

Cf. A085082, A025487, A181822, A322584.
Rearrangement of A115728, A115729 and A238690.
A116473(n) is the number of times n appears in the sequence.

lpsQ[n_] := n == 1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]); lps = Select[Range[6000], lpsQ]; c[n_] := Count[Divisors[n], ?(MemberQ[lps, #] &)]; c /@ lps  (* _Amiram Eldar, Jan 21 2024 *)

a(n) = A085082(A025487(n)) = A085082(A181822(n)).

a(n) = A322584(A025487(n)). - Amiram Eldar, Jan 21 2024

cp's OEIS Frontend

A182860 Number of distinct prime signatures represented among the unitary divisors of n.

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A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

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A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n.

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A212642 a(n) = number of distinct prime signatures represented among divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

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A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

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A088873 Number of different values of A000005(d) where d is a divisor of n.

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A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

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A182861 Number of distinct prime signatures represented among the unitary divisors of A025487(n).

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