A225546 -id:A225546 - cp's OEIS frontend

A331288 a(n) = min(n, A225546(n)).

1, 2, 3, 3, 5, 6, 7, 6, 9, 10, 11, 12, 13, 14, 15, 5, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 10, 33, 34, 35, 27, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 20, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 15, 65, 66, 67, 68, 69, 70, 71, 54, 73, 74, 75, 76, 77, 78, 79, 80, 25, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 40

Offset: 1

Antti Karttunen, Jan 20 2020

nonn

For all i, j:

a(i) = a(j) => A331287(i) = A331287(j).

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

Cf. A225546, A225547, A331301, A331287.

Array[If[# == 1, 1, Min[#, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 96] (* Michael De Vlieger, Jan 21 2020 *)

PARI
```
A331288(n) = min(n, A225546(n));
```

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A225546(n) = { my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] = A019565(f[i, 2]); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); }; \\ From A225546
\\ If the following returns 1, then it is certainly true that A225546(p^e) > n (where p^e is one of the divisors of n), thus A225546(n) > n follows:
is_certainly_gt(p,e,n) = { my(b=A019565(e),j=(primepi(p)-1)); if(b>n,1,if((logint(b,2)*j)>logint(n,2),1,0)); };
A331288(n) = if((1==n)||isprime(n),n,my(f=factor(n)); for(i=1,#f~,if(is_certainly_gt(f[i,1],f[i,2],n),return(n))); min(n, A225546(n)));

A331734 a(n) = A033879(A225546(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, 5, 1, 1, -4, 1, 1, 1, 4, 1, -3, 1, -28, 1, 1, 1, -12, 41, 1, -19, -508, 1, 1, 1, 2, 1, 1, 1, 14, 1, 1, 1, -60, 1, 1, 1, -131068, -115, 1, 1, -2, 3281, -39, 1, -8589934588, 1, -51, 1, -1020, 1, 1, 1, -124, 1, 1, -2035, 6, 1, 1, 1, -36893488147419103228, 1, 1, 1, -12, 1, 1, -199, -680564733841876926926749214863536422908

Offset: 1

Antti Karttunen, Feb 02 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..136

Cf. A005117, A033879, A225546, A331733, A331735, A331741.

Cf. A323244, A323174, A324055, A324185, A324546 for other permutations of the deficiency, and also A324574, A324654.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331734(n) = if(issquarefree(n),1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); (2*prod(i=1,u,prime(i)^A048675(prods[i]))) - prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A033879(A225546(n)) = 2*A225546(n) - A331733(n).
For all n, a(A005117(n)) = 1. [It is not known if there are 1's in any other positions. See Jianing Song's Oct 13 2019 comment in A033879.]
For a necessary condition that a(s) would be zero for any square, see A331741.

A331740 Number of prime factors in A225546(n), counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 2, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 4, 4, 33, 4, 9, 512, 7, 1024, 2, 18, 65, 12, 3, 2048, 129, 34, 6, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 5, 20, 10, 130, 513, 65536, 7, 131072, 1025, 10, 2, 36, 19, 262144, 65, 258, 13, 524288, 4, 1048576, 2049, 6

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A000720, A001222, A048675, A225546, A331590.
Cf. also A331309, A331591.
Positions of 1's: A001146.

Array[If[# == 1, 0, PrimeOmega@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 75] (* Michael De Vlieger, Feb 08 2020 *)

A331740(n) = if(1==n,0,my(f=factor(n)); sum(i=1,#f~,hammingweight(f[i,2])*(2^(primepi(f[i,1])-1))));

Additive with a(p^e) = A000120(e) * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n).
a(n) = A001222(A225546(n)).
A331591(n) <= a(n) <= A048675(n).
From Peter Munn, Sep 11 2021: (Start)
a(A001146(m)) = 1.
a(A331590(m, k)) = a(m) + a(k).
For squarefree k, a(k*m^2) = a(k) + a(m) = A048675(k) + a(m).
(End)

A334859 a(n) = A243071(A225546(n)).

Original entry on oeis.org

0, 1, 2, 3, 8, 4, 128, 6, 5, 16, 32768, 12, 2147483648, 256, 32, 7, 9223372036854775808, 10, 170141183460469231731687303715884105728, 48, 512, 65536, 57896044618658097711785492504343953926634992332820282019728792003956564819968, 24, 17, 4294967296, 20, 768

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A334860. Composition of permutations A225546 and A243071, and also of A054429 and A334865.

Cf. A299090, A334871.

a(n) = A243071(A225546(n)).
a(n) = A054429(A334865(n)).
For n >= 1, A000120(a(n)) = A299090(n).
For n > 1, A070939(a(n)) = A334871(n).

A331301 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A225546(n)) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 5, 6, 7, 3, 8, 3, 9, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 7, 22, 23, 24, 19, 3, 25, 26, 27, 3, 28, 3, 29, 30, 31, 3, 13, 32, 33, 34, 35, 3, 36, 37, 38, 39, 40, 3, 41, 3, 42, 43, 10, 44, 45, 3, 46, 47, 48, 3, 36, 3, 49, 50, 51, 52, 53, 3, 54, 17, 55, 3, 56, 57, 58, 59, 60, 3, 61, 62, 63, 64, 65, 66, 27, 3, 67, 68, 69, 3, 70, 3, 71, 72

Offset: 1

Antti Karttunen, Jan 21 2020

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A064179(i) = A064179(j),
  a(i) = a(j) => A064547(i) = A064547(j),
  a(i) = a(j) => A302777(i) = A302777(j),
  a(i) = a(j) => A331308(i) = A331308(j),
  a(i) = a(j) => A331287(i) = A331287(j),
  a(i) = a(j) => A331592(i) = A331592(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A019565, A225546, A305801, A331174, A331288.

Cf. also A064179, A064547, A302777, A331308, A331287, A331592.

up_to = 10000;
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; };
Aux331301(n) = if((n%2)&&isprime(n),0,A331288(n)); \\ Needs also code from A331288.
v331301 = rgs_transform(vector(up_to, n, Aux331301(n)));
A331301(n) = v331301[n];

A331308 a(n) = min(d(n), d(A225546(n))), where d gives the number of divisors of n, A000005.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 21 2020

nonn

This is not equal to A000005(A331288(n)). The first difference is at n=100, where a(100) = 6, while A000005(A331288(100)) = 9. Note that A225546(100) = 243 and d(243) = 6 < d(100) = 9.

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

Cf. A000005, A225546, A331288, A331309, A331592.

a(n) = min(A000005(n), A331309(n)).

A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

Numbers k for which A001221(k) = A331591(k).
Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.
If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.
k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

			There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Cf. A000120, A046523, A225546, A267116, A329332.
Sequences with related definitions: A001221, A331591, A331592.
Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.
Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

A331735 a(n) = A009194(A225546(n)) = gcd(A225546(n), sigma(A225546(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Cf. A000203, A009194, A225546, A331733, A331734.

Array[GCD[#, DivisorSigma[1, #]] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Feb 12 2020 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331735(n) = if(issquarefree(n),1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); gcd(prod(i=1,u,prime(i)^A048675(prods[i])), prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1))));

a(n) = A009194(A225546(n)) = gcd(A225546(n), A331733(n)).

A334865 a(n) = A156552(A225546(n)).

Original entry on oeis.org

0, 1, 3, 2, 15, 7, 255, 5, 6, 31, 65535, 11, 4294967295, 511, 63, 4, 18446744073709551615, 13, 340282366920938463463374607431768211455, 47, 1023, 131071, 115792089237316195423570985008687907853269984665640564039457584007913129639935, 23, 30, 8589934591, 27, 767

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A334866. Composition of permutations A156552 and A225546, and also of A054429 and A334859.

Cf. A334871.

a(n) = A156552(A225546(n)).
a(n) = A054429(A334859(n)).
For n > 1, A070939(a(n)) = A334871(n).

A331736 The largest odd divisor of A225546(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 9, 1, 1, 3, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 3, 81, 1, 9, 3, 1, 1, 1, 5, 1, 1, 1, 27, 1, 1, 1, 3, 1, 1, 1, 3, 9, 1, 1, 5, 6561, 81, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 1, 9, 15, 1, 1, 1, 3, 1, 1, 1, 27, 1, 1, 81, 3, 1, 1, 1, 5, 25, 1, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 5, 1, 6561, 9, 243, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Feb 02 2020

Antti Karttunen, Table of n, a(n) for n = 1..1680

Cf. A000265, A005117, A008833, A052928, A225546.

Array[#/2^IntegerExponent[#, 2] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Feb 12 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A331736(n) = { my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] = A019565((f[i, 2]>>1)<<1); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); };

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331736(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=2,e); for(i=2,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=2,u,prime(i)^A048675(prods[i])));

Multiplicative, with a(prime(i)^j) = A000265(A019565(j))^A000079(i-1).
Equally, with a(prime(i)^j) = A019565(A052928(j))^A000079(i-1).
a(n) = A000265(A225546(n)).
a(n) = A225546(A008833(n)).

cp's OEIS Frontend

A331288 a(n) = min(n, A225546(n)).

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A331734 a(n) = A033879(A225546(n)).

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A331740 Number of prime factors in A225546(n), counted with multiplicity.

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A334859 a(n) = A243071(A225546(n)).

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A331301 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A225546(n)) for all other n, except for odd primes p, f(p) = 0.

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A331308 a(n) = min(d(n), d(A225546(n))), where d gives the number of divisors of n, A000005.

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A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

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A331735 a(n) = A009194(A225546(n)) = gcd(A225546(n), sigma(A225546(n))).

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A334865 a(n) = A156552(A225546(n)).

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