A212180 -id:A212180 - cp's OEIS frontend

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

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

Original entry on oeis.org

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

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

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).

cp's OEIS Frontend

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

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A182860 Number of distinct prime signatures represented among the unitary 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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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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