A182855 -id:A182855 - cp's OEIS frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

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

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A182853 Squarefree composite integers and powers of squarefree composite integers.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Matthew Vandermast, Jan 04 2011

Numbers that require exactly three iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.

Numbers such that A001221(n) > 1 and A071625(n) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 3. See also A182854, A182855.
Subsequence of A072774 and A182851.
Cf. A120944.

isoka(n) = (omega(n) > 1) && issquarefree(n); \\ A120944
isok(n) = isoka(n) || (ispower(n,,&k) && isoka(k)); \\ Michel Marcus, Jun 24 2017

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A182853(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

(define A182853 (MATCHING-POS 1 1 (lambda (n) (= 3 (A182850 n))))) ;; After the alternative definition of the sequence given by the original author. Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 05 2016

A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Michael De Vlieger, Oct 22 2024

Also, numbers k such that there exists a pair of necessarily composite divisors {d, k/d}, d < k/d, with quality Q, i.e., gcd(d, k/d) > 1 but there exists a prime p | d that does not divide k/d, and also a prime q | k/d that does not divide d.
A178212 is a proper subset.
This sequence is distinct from A123712 since 420 is here.
This sequence is distinct from A182855 since 360 is here.

			a(1) = 60 = 2^2 * 3 * 5, the smallest number such that bigomega(60) > omega(60) > 2. Bigomega(60) = 4, omega(60) = 3.
72 is not in the sequence because it is the product of 2 distinct prime factors.
a(2) = 84 = 2^2 * 3 * 7, since bigomega(84) = 4, omega(84) = 3.
a(3) = 90 = 2 * 3^2 * 5, since bigomega(90) = 4, omega(90) = 3.
a(4) = 120 = 2^3 * 3 * 5, since bigomega(120) = 5, omega(120) = 3.
210 is not in the sequence because it is squarefree.
a(35) = 360 = 2^3 * 3^2 * 5 since bigomega(360) = 6, omega(360) = 3.
a(43) = 420 = 2^2 * 3 * 5 * 7 since bigomega(420) = 5, omega(420) = 4, etc.
.
Table showing pairs of factors of a(n) for select n, such that the pair possesses quality Q (see comments).
    n    a(n)   pair of factors with quality Q.
  -------------------------------------------------------------------
    1     60    6 X 10;
    2     84    6 X 14;
    3     90    6 X 15;
    4    120    6 X 20,  10 X 12;
    5    126    6 X 21;
    6    132    6 X 22;
    7    140   10 X 14;
    8    150   10 X 15;
   17    240    6 X 40,  10 X 24, 12 X 20;
   51    480    6 X 80,  10 X 48, 12 X 40, 20 X 24;
  117    840    6 X 140, 10 X 84, 12 X 70, 14 X 60, 20 X 42, 28 X 30.

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

Cf. A001221, A001222, A013929, A024619, A126706, A178212.

Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]

{a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.

A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244, 245

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

Numbers that require exactly four iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.

Not the same sequence as A177425, which is a proper subsequence. 1260 is the first nonmember of A177425 that belongs to this sequence; its prime signature is (2,2,1,1).

			The prime signature of 12 (2^2*3^1) is (2,1). Since (2,1) contains at least two distinct numbers, and since each number that appears in (2,1) appears exactly as often as any other number that appears, 12 belongs to this sequence.
12 also requires exactly four iterations under the x -> A181819(x) map to reach a fixed point (namely, 2) .  A181819(12) = 6;  A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).

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

Numbers n such that A182850(n) = 4. See also A182853, A182855.

Subsequence of A059404 and A182852.

aQ[n_] := Length[v = Values @ Counts @ FactorInteger[n][[;;,2]]] > 1 && Length @ Union @ v == 1; Select[Range[250], aQ] (* Amiram Eldar, Aug 08 2019 *)

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Original entry on oeis.org

174636000, 206388000, 244490400, 261954000, 269892000, 274428000, 288943200, 291060000, 301644000, 309582000, 343980000, 349272000, 365148000, 366735600, 377848800, 383292000, 404838000, 411642000, 412776000, 422301600, 433414800, 449820000, 452466000, 457380000

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(5) = 174636000.
Positions of 5's in A071625.
Numbers k such that A001221(A181819(k)) = 5.

			174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 has five distinct exponents so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024.

Select[Range[300000000],Length[Union[Last/@FactorInteger[#]]]==5&]

is(n) = #Set(factor(n)[, 2]) == 5 \\ David A. Corneth, Jan 12 2019

a(13)-a(24) from Daniel Suteu, Jan 12 2019

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

Original entry on oeis.org

1, 2, 60, 1801800, 11657093261814000, 7167827541370578634694420017740000, 291943326350524088652207164949980988754136887856059678357800000

Offset: 0

Matthew Vandermast, Jan 05 2011

a(n) = smallest number m such that A181819(m) = A006939(n).  a(n) belongs to A182855 iff n > 1.
Next term has 105 digits.
Smallest number k with A323022(k) = n, where A323022(m) is the number of distinct multiplicities in the prime signature of m. - Gus Wiseman, Jan 03 2019

			The canonical prime factorization of a(3) = 1801800 is 2^3*3^2*5^2*7*11*13. The prime signature of 1801800 is therefore (3,2,2,1,1,1). Note that (3,2,2,1,1,1) contains exactly one number that appears once (3), one number that appears twice (2), and one number that appears three times (1).

David A. Corneth, Table of n, a(n) for n = 0..12
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A006939, A181819, A182855.

Cf. A056239, A059404, A062770, A071625, A118914, A181821, A182857, A323022.

Table[Product[Times@@Prime[i*(i-1)/2+Ceiling[Range[i*(n-i)]/(n-i)]],{i,n-1}],{n,6}] (* Gus Wiseman, Jan 03 2019 *)

a(n) = if(n == 0, return(1)); my(f = matrix(binomial(n+1,2), 2)); f[, 1] = primes(#f~ )~; f[, 2] = Vecrev(concat(vector(n, i, vector(n+1-i, j, i))))~; factorback(f) \\ David A. Corneth, Jan 03 2019

Partial products of A113511.
log a(n) ~ (1/3) n^3 log n. [Charles R Greathouse IV, Jan 13 2012]
A001222(a(n)) = A000292(n). - Gus Wiseman, Jan 03 2019
a(0) = 1; a(n + 1) = A002110(binomial(n + 2, 2)) * a(n). - David A. Corneth, Jan 03 2019

cp's OEIS Frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Scheme

Scheme

Formula

Extensions

A182853 Squarefree composite integers and powers of squarefree composite integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Scheme

A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A324206 Numbers with exactly six distinct exponents in their prime factorization, or six distinct parts in their prime signature.

Views

Author

Keywords

Examples

Links