A330749 -id:A330749 - cp's OEIS frontend

A337375 a(n) = A330749(A005940(1+n)).

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 5, 6, 3, 2, 1, 1, 3, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 7, 5, 15, 6, 5, 6, 3, 2, 1, 1, 1, 1, 5, 6, 9, 4, 1, 1, 3, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 11, 7, 7, 5, 35, 30, 15, 6, 7, 5, 15, 12, 5, 6, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 7, 5

Offset: 0

Antti Karttunen, Sep 11 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A005940, A064989, A330749, A337376, A337377.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A330749(n) = gcd(n, A064989(n));
A337375(n) = A330749(A005940(1+n));

a(n) = A330749(A005940(1+n)).

A355693 Dirichlet inverse of A330749, gcd(n, A064989(n)), where A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, -1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 0, 1, 1, -3, -2, -1, 1, 1, 0, -1, 0, -1, 0, 2, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 1, -1, 1, 0, 0, 1, 0, -1, 0, 1, 3, -1, 1, -1, 1, 2, 0, -5, 0, -1, 0, 0, 1, -1, -1, 1, 1, 1, 0, -1, 3, 1, 0, 1, 1, 1, 0

Offset: 1

Antti Karttunen, Jul 18 2022

sign

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

Cf. A064989, A330749.

Cf. also A354365, A354366.

A330749(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)); gcd(n, factorback(f)); };
memoA355693 = Map();
A355693(n) = if(1==n,1,my(v); if(mapisdefined(memoA355693,n,&v), v, v = -sumdiv(n,d,if(dA330749(n/d)*A355693(d),0)); mapput(memoA355693,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA330749(n/d) * a(d).

A319626 Primorial deflation of n (numerator): Let f be the completely multiplicative function over the positive rational numbers defined by f(p) = A034386(p) for any prime number p; f constitutes a permutation of the positive rational numbers; let g be the inverse of f; for any n > 0, a(n) is the numerator of g(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 14, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 69

Offset: 1

Rémy Sigrist, Sep 25 2018

See A319627 for the corresponding denominators.
The restriction of f to the natural numbers corresponds to A108951.
The function g is completely multiplicative over the positive rational numbers with g(2) = 2 and g(q) = q/p for any pair (p, q) of consecutive prime numbers.
The ratio A319626(n)/A319627(n) can be viewed as a "primorial deflation" of n (see also A329900), with the inverse operation being n = A108951(A319626(n)) / A108951(A319627(n)), where A319627(k) = 1 for all k in A025487. - Daniel Suteu, Dec 29 2019

			f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 21.

Daniel Suteu, Table of n, a(n) for n = 1..10000

A left inverse of A108951. Coincides with A329900 on A025487.

Cf. A006530, A053585, A064989, A181815, A307035, A319627, A319630, A329902, A330749, A330750 (rgs-transform), A330751 (ordinal transform).

Array[#1/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &, 120] (* Michael De Vlieger, Aug 27 2020 *)

a(n) = my (f=factor(n)); numerator(prod(i=1, #f~, my (p=f[i,1]); (p/if (p>2, precprime(p-1), 1))^f[i,2]))

a(n) = n / gcd(n, A064989(n)) = n / A330749(n).
a(n) <= n with equality iff n belongs to A319630.
A006530(a(n)) = A006530(n).
A053585(a(n)) = A053585(n).
From Antti Karttunen, Dec 29 2019: (Start)
a(A108951(n)) = n.
a(A025487(n)) = A329900(A025487(n)) = A181815(n).
Many of the formulas given in A329900 apply here as well:
a(n!) = A307035(n), a(A002182(n)) = A329902(n), and so on.
(End)

"Primorial deflation" prefixed to the name by Antti Karttunen, Dec 29 2019

A346095 a(n) = gcd(A324886(n), A064989(A324886(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 9, 1, 25, 1, 1, 1, 1, 1, 5, 49, 1, 1, 1, 1, 7, 1, 1, 1, 1, 121, 625, 1, 1, 1, 7, 1, 11, 1, 1, 7, 1, 1, 5, 143, 2401, 1, 1, 1, 1, 169, 1, 1, 1, 1, 343, 1, 1, 1331, 1, 17, 1, 1, 1, 1, 161051, 1, 175, 1, 1, 41503, 1, 169, 1, 1, 49, 35, 1, 1, 121, 19, 1, 1, 1, 1, 49, 24137569

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Cf. A006530, A324886, A329044, A330749, A346096, A346097.

A330749(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)); gcd(n, factorback(f)); };
A346095(n) = A330749(A324886(n)); \\ Rest of program given in A324886.

a(n) = A330749(A324886(n)) = gcd(A324886(n), A329044(n)) = gcd(A324886(n), A064989(A324886(n))).
a(n) = A324886(n) / A346096(n).
a(n) = A329044(n) / A346097(n).
a(n) mod A006530(n) > 0, for all n > 1.

A330742 a(n) = n / gcd(A309639(n), A319626(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 29 2019

nonn

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

Cf. A309639, A319626, A330692, A330741, A330749.

A330742(n) = (n/gcd(A309639(n), A319626(n)));

a(n) = n / A330741(n) = n / gcd(A309639(n), A319626(n)).

A083259 a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 2, 3, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 96, 1

Offset: 1

Labos Elemer, May 09 2003

nonn

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

Cf. A007395, A002808, A046523, A071364, A083255-A083260, A322361, A330749.

A071364(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ From A071364
A083259(n) = gcd(A071364(n),n); \\ Antti Karttunen, Jan 22 2020

A330741 a(n) = gcd(A309639(n), A319626(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 29 2019

nonn

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

Cf. A309639, A319626, A330742, A330749.

A330741(n) = gcd(A309639(n), A319626(n));

a(n) = gcd(A309639(n), A319626(n)).

a(n) = n / A330742(n).

A330750 Lexicographically earliest infinite sequence such that a(i) = a(j) => A319626(i) = A319626(j) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 29 2019

nonn

Restricted growth sequence transform of A319626, where A319626(n) = n / A330749(n) = n / gcd(n, A064989(n)).

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

Cf. A064989, A319626, A330749.

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)};
A319626(n) = (n / gcd(n, A064989(n)));
v330750 = rgs_transform(vector(up_to, n, A319626(n)));
A330750(n) = v330750[n];

cp's OEIS Frontend

A337375 a(n) = A330749(A005940(1+n)).

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A355693 Dirichlet inverse of A330749, gcd(n, A064989(n)), where A064989 shifts the prime factorization one step towards lower primes.

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A346095 a(n) = gcd(A324886(n), A064989(A324886(n))).

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A330742 a(n) = n / gcd(A309639(n), A319626(n)).

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A083259 a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.

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A330741 a(n) = gcd(A309639(n), A319626(n)).

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A330750 Lexicographically earliest infinite sequence such that a(i) = a(j) => A319626(i) = A319626(j) for all i, j.

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