A322813 -id:A322813 - cp's OEIS frontend

A322819 a(n) = A000593(A122111(n)).

1, 1, 1, 4, 1, 4, 1, 6, 13, 4, 1, 6, 1, 4, 13, 8, 1, 24, 1, 6, 13, 4, 1, 8, 40, 4, 31, 6, 1, 24, 1, 12, 13, 4, 40, 32, 1, 4, 13, 8, 1, 24, 1, 6, 31, 4, 1, 12, 121, 78, 13, 6, 1, 48, 40, 8, 13, 4, 1, 32, 1, 4, 31, 14, 40, 24, 1, 6, 13, 78, 1, 48, 1, 4, 124, 6, 121, 24, 1, 12, 57, 4, 1, 32, 40, 4, 13, 8, 1, 48, 121, 6, 13, 4, 40, 14, 1, 240

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000593, A122111, A322813.

A000593(n) = sigma(n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322819(n) = A000593(A122111(n));

a(n) = A000593(A122111(n)).

A322865 a(n) = A000265(A122111(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 9, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 27, 3, 25, 5, 1, 15, 1, 11, 9, 3, 27, 21, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 81, 45, 9, 5, 1, 35, 27, 7, 9, 3, 1, 21, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 33, 1, 3, 75, 5, 81, 15, 1, 11, 49, 3, 1, 21, 27, 3, 9, 7, 1, 35, 81, 5, 9, 3, 27, 13, 1, 135, 25, 63, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A122111, A322813, A322819, A322820, A322826, A322865, A322867.

Array[#/2^IntegerExponent[#, 2] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)

A000265(n) = (n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322865(n) = A000265(A122111(n));

a(n) = A000265(A122111(n)).
a(n) = A122111(A322820(n)).
A000005(a(n)) = A322813(n).
A000203(a(n)) = A322819(n).
A122111(a(n)) = A322820(n).
A000120(a(n)) = A322867(n).

A324117 Number of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A001227(A156552(n)) = A000005(A322993(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 3, 4, 1, 2, 2, 4, 2, 4, 1, 4, 1, 2, 2, 4, 2, 4, 1, 2, 4, 4, 1, 2, 1, 2, 2, 8, 1, 2, 2, 3, 4, 2, 1, 2, 3, 2, 4, 6, 1, 2, 1, 4, 2, 6, 2, 4, 1, 4, 2, 2, 1, 4, 1, 4, 2, 4, 2, 4, 1, 2, 4, 4, 1, 6, 4, 8, 8, 8, 1, 6, 3, 4, 6, 12, 4, 4, 1, 3, 4, 4, 1, 6, 1, 2, 4

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4473
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001227, A156552, A246277, A322813, A322993, A324105, A324115, A324118.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A322993(n) = A156552(2*A246277(n));
A324117(n) = if(1==n, 0, numdiv(A322993(n)));

a(1) = 0; for n > 1, a(n) = A000005(A322993(n)) = A000005(A156552(2*A246277(n))) = A324105(2*A246277(n)).

A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 5, 1, 6, 1, 3, 4, 2, 1, 5, 7, 2, 8, 3, 1, 6, 1, 9, 4, 2, 7, 10, 1, 2, 4, 5, 1, 6, 1, 3, 8, 2, 1, 9, 11, 12, 4, 3, 1, 13, 7, 5, 4, 2, 1, 10, 1, 2, 8, 14, 7, 6, 1, 3, 4, 12, 1, 15, 1, 2, 16, 3, 11, 6, 1, 9, 17, 2, 1, 10, 7, 2, 4, 5, 1, 13, 11, 3, 4, 2, 7, 14, 1, 18, 8, 19, 1, 6, 1, 5, 16

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Restricted growth sequence transform of A052126, or equally, of A322820.
For all i, j:
  A300226(i) = A300226(j) => a(i) = a(j),
  a(i) = a(j) => A322813(i) = A322813(j),
  a(i) = a(j) => A322819(i) = A322819(j).
For all i, j > 1:
  a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A001222, A006530, A052126, A305800, A300226, A319994, A319996, A322813, A322819, A322820.

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
v322826 = rgs_transform(vector(up_to,n,A052126(n)));
A322826(n) = v322826[n];

A322820 a(n) = A052126(n) * A006530(A052126(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 8, 9, 4, 1, 8, 1, 4, 9, 16, 1, 18, 1, 8, 9, 4, 1, 16, 25, 4, 27, 8, 1, 18, 1, 32, 9, 4, 25, 36, 1, 4, 9, 16, 1, 18, 1, 8, 27, 4, 1, 32, 49, 50, 9, 8, 1, 54, 25, 16, 9, 4, 1, 36, 1, 4, 27, 64, 25, 18, 1, 8, 9, 50, 1, 72, 1, 4, 75, 8, 49, 18, 1, 32, 81, 4, 1, 36, 25, 4, 9, 16, 1, 54, 49, 8, 9, 4, 25, 64, 1, 98, 27, 100, 1, 18, 1, 16, 75

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000265, A006530, A052126, A070003 (positions where a(n) = n for n > 1), A122111, A319988, A322813, A322819, A322826 (restricted growth sequence transform).

A322820(n) = if(isprime(n),1,if(1>=omega(n),n,my(f=factor(n), y=#f~); if(1==f[y,2], f[y,2] = 0; f[y-1,2]++); factorback(f)));

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A322820(n) = A052126(n)*A006530(A052126(n));

A000265(n) = (n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322820(n) = A122111(A000265(A122111(n)));

a(n) = A052126(n) * A006530(A052126(n)).
a(n) = A122111(A000265(A122111(n))).
A052126(a(n)) = A052126(n).
a(n) <= n.
For all n > 1, A010051(n) + A319988(a(n)) = 1.

A336315 The number of divisors in the conjugated prime factorization of n: a(n) = A000005(A122111(n)).

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 5, 2, 3, 6, 6, 4, 7, 8, 6, 2, 8, 4, 9, 6, 9, 10, 10, 4, 4, 12, 3, 8, 11, 8, 12, 2, 12, 14, 8, 4, 13, 16, 15, 6, 14, 12, 15, 10, 6, 18, 16, 4, 5, 6, 18, 12, 17, 4, 12, 8, 21, 20, 18, 8, 19, 22, 9, 2, 16, 16, 20, 14, 24, 12, 21, 4, 22, 24, 6, 16, 10, 20, 23, 6, 3, 26, 24, 12, 20, 28, 27, 10, 25, 8, 15, 18, 30, 30, 24

Offset: 1

Antti Karttunen, Jul 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A034444, A122111, A286621, A323173, A322813, A322819, A336316.

A336315(n) = if(1==n,n,my(p=apply(primepi,factor(n)[,1]~),m=1+p[1]); for(i=2, #p, m *= (1+p[i]-p[i-1])); (m));

a(n) = A000005(A122111(n)).

a(n) = A336316(n) + A034444(n).

cp's OEIS Frontend

A322819 a(n) = A000593(A122111(n)).

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A322865 a(n) = A000265(A122111(n)).

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A324117 Number of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A001227(A156552(n)) = A000005(A322993(n)).

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A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

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A322820 a(n) = A052126(n) * A006530(A052126(n)).

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A336315 The number of divisors in the conjugated prime factorization of n: a(n) = A000005(A122111(n)).

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