Displaying 1-10 of 35 results found.
2, 3, 5, 9, 7, 25, 11, 5, 7, 49, 13, 625, 17, 121, 117649, 25, 19, 49, 23, 2401, 1771561, 169, 29, 175, 14641, 289, 55, 14641, 31, 26411, 37, 21, 4826809, 361, 299393809, 2401, 41, 529, 24137569, 11, 43, 13, 47, 28561, 161051, 841, 53, 343, 6311981, 214358881, 47045881, 83521, 59, 3025, 48841, 214358881, 148035889, 961
PROG
(PARI)
A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).
+20
8
1, -2, -3, 0, -5, 3, -7, 0, 0, 10, -11, 0, -13, 14, 5, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -5, -31, 0, 33, 34, 7, 0, -37, 38, 39, 0, -41, -21, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -33, -67, 0, 69, -14, -71, 0, -73, 74, 0, 0, 11, -39, -79, 0, 0, 82
COMMENTS
Because the ratio n / A064989(n) = A319626(n) / A319627(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354366 gives the denominators. See the examples.
EXAMPLE
The ratio a(n)/ A354366(n) for n = 1..22: 1, -2, -3/2, 0, -5/3, 3, -7/5, 0, 0, 10/3, -11/7, 0, -13/11, 14/5, 5/2, 0, -17/13, 0, -19/17, 0, 21/10, 22/7.
PROG
(PARI)
A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
Denominators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).
+20
7
1, 1, 2, 1, 3, 1, 5, 1, 1, 3, 7, 1, 11, 5, 2, 1, 13, 1, 17, 1, 10, 7, 19, 1, 1, 11, 1, 1, 23, 1, 29, 1, 14, 13, 3, 1, 31, 17, 22, 1, 37, 5, 41, 1, 1, 19, 43, 1, 1, 1, 26, 1, 47, 1, 21, 1, 34, 23, 53, 1, 59, 29, 1, 1, 33, 7, 61, 1, 38, 3, 67, 1, 71, 31, 1, 1, 5, 11, 73, 1, 1, 37, 79, 1, 39, 41, 46, 1, 83, 1, 55, 1, 58
COMMENTS
Equally, denominators of Dirichlet inverse of fraction n / A064989(n). See also comments in A354365.
PROG
(PARI)
A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
1, 2, 3, 4, 5, 9, 7, 3, 5, 25, 11, 81, 13, 49, 15625, 9, 17, 25, 19, 625, 117649, 121, 23, 45, 2401, 169, 21, 2401, 29, 4375, 31, 10, 1771561, 289, 14235529, 625, 37, 361, 4826809, 7, 41, 11, 43, 14641, 16807, 529, 47, 125, 2093663, 5764801, 24137569, 28561, 53, 441, 20449, 5764801, 47045881, 841, 59, 343, 61, 961, 1331, 100, 396067447082177
PROG
(PARI)
A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 7, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 14, 1, 4, 15, 2, 1, 6, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 12, 1, 2, 7, 1, 5, 6, 1, 4, 23, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 28, 5, 2, 3, 8, 1, 30, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 105
0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3
PROG
(PARI)
A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 5, 16, 17, 9, 19, 5, 3, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 3, 43, 11, 3, 23, 47, 8, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 3, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 81, 41, 83, 3, 17, 43, 29, 11, 89, 3
Lexicographically earliest infinite sequence such that a(i) = a(j) => A319626(i) = A319626(j) for all i, j.
+20
3
1, 2, 3, 4, 5, 3, 6, 7, 8, 9, 10, 11, 12, 13, 5, 14, 15, 8, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 5, 27, 28, 29, 30, 6, 8, 31, 32, 33, 34, 35, 18, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 24, 47, 48, 49, 50, 51, 9, 52, 53, 54, 55, 56, 29, 57, 58, 59, 13, 60, 61, 62, 63, 22, 64, 10, 33, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 38, 76
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
v330750 = rgs_transform(vector(up_to, n, A319626(n)));
Number of values of k, 1 <= k <= n, with A319626(k) = A319626(n), where A319626(n) gives the numerator of rational valued primorial deflation of n.
+20
3
1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3
PROG
(PARI)
up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
v330751 = ordinal_transform(vector(up_to, n, A319626(n)));
Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).
+10
388
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
COMMENTS
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
FORMULA
Other identities. For all n >= 1:
(End)
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)]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product 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:
(End)
EXAMPLE
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.
MAPLE
local a;
a := 1;
for pf in ifactors(n)[2] do
a := a*ithprime(pf[2]) ;
end do:
a ;
end proc:
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
MATHEMATICA
{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)
PROG
(Haskell)
a181819 = product . map a000040 . a124010_row
(PARI) a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); } \\ Michel Marcus, Nov 16 2015 (Scheme, with memoization-macro definec, two variants)
CROSSREFS
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.
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