A332455 -id:A332455 - cp's OEIS frontend

A332459 Odd part of 1+sigma(n).

1, 1, 5, 1, 7, 13, 9, 1, 7, 19, 13, 29, 15, 25, 25, 1, 19, 5, 21, 43, 33, 37, 25, 61, 1, 43, 41, 57, 31, 73, 33, 1, 49, 55, 49, 23, 39, 61, 57, 91, 43, 97, 45, 85, 79, 73, 49, 125, 29, 47, 73, 99, 55, 121, 73, 121, 81, 91, 61, 169, 63, 97, 105, 1, 85, 145, 69, 127, 97, 145, 73, 49, 75, 115, 125, 141, 97, 169, 81, 187, 61, 127

Offset: 1

Antti Karttunen, Feb 16 2020

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 related to sigma(n)

Cf. A000203, A000265, A002487, A088580, A161942, A202274, A324294, A332454, A332455.

A332459(n) = { my(s=1+sigma(n)); (s>>valuation(s,2)); };

a(n) = A000265(A088580(n)) = A000265(1+sigma(n)).
A002487(a(n)) = A324294(n).
a(2^n) = 0 for all n >= 0. [Zero occurs at least also at a(25). See A202274]

A332453 Starting from sigma(n), number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

Original entry on oeis.org

0, 2, 0, 5, 2, 2, 0, 5, 2, 6, 2, 5, 5, 2, 2, 39, 6, 11, 1, 1, 0, 6, 2, 5, 39, 1, 1, 5, 5, 6, 0, 39, 2, 41, 2, 33, 6, 5, 5, 4, 1, 2, 4, 1, 11, 6, 2, 39, 10, 4, 6, 7, 41, 5, 6, 5, 1, 4, 5, 1, 39, 2, 2, 15, 1, 6, 3, 39, 2, 6, 6, 43, 6, 10, 39, 3, 2, 1, 1, 4, 34, 39, 1, 5, 41, 8, 5, 4, 4, 5, 5, 1, 0, 6, 5, 39, 7, 45, 11, 7, 7, 41, 2, 12, 2

Offset: 1

Antti Karttunen, Feb 16 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences related to sigma(n)

Cf. A000203, A006667, A332209, A332452, A332455.

A006667(n) = { my(t=0); while(n>1, if(n%2, t++; n=3*n+1, n>>=1)); (t); };
A332453(n) = A006667(sigma(n));

a(n) = A006667(A000203(n)).

a(n) = A332209(n) - A332452(n).

A332454 Starting from sigma(n)+1, number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

Original entry on oeis.org

1, 2, 4, 3, 11, 7, 13, 4, 12, 14, 7, 13, 12, 16, 16, 5, 14, 7, 6, 20, 18, 15, 16, 14, 5, 20, 69, 22, 67, 73, 18, 6, 17, 71, 17, 13, 23, 14, 22, 59, 20, 75, 12, 8, 24, 73, 17, 69, 14, 67, 73, 18, 71, 61, 73, 61, 16, 59, 14, 33, 68, 75, 26, 7, 8, 74, 11, 31, 75, 74, 73, 19, 11, 23, 69, 12, 75, 33, 16, 30, 15, 31, 8, 35, 72, 20

Offset: 1

Antti Karttunen, Feb 16 2020

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 3x+1 (or Collatz) problem
Index entries for sequences related to sigma(n)

Cf. A000203, A006666, A088580, A332452, A332455.

Cf. also A324294.

A006666(n) = { my(t=0); while(n>1, if(n%2, n=3*n+1, n>>=1; t++)); (t); }; \\ From A006666
A332454(n) = A006666(1+sigma(n));

a(n) = A006666(A088580(n)) = A006666(1+sigma(n)).

A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 8, 9, 10, 11, 12, 13, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 4, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 41, 46, 47, 48, 41, 49, 50, 51, 52, 53, 54, 55, 56, 4, 57, 58, 59, 60, 55, 58, 61, 62, 63, 64, 65, 66, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 76

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A048250(i) = A048250(j),
  a(i) = a(j) => A332455(i) = A332455(j),
  a(i) = a(j) => A332459(i) = A332459(j).

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

Cf. A003557, A046523, A048250, A295300, A332455, A332459.

Cf. also A326199.

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; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A209229(n) = (n && !bitand(n,n-1));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux332230(n) = if((n>2)&&A209229(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v332230 = rgs_transform(vector(up_to,n,Aux332230(n)));
A332230(n) = v332230[n];

cp's OEIS Frontend

A332459 Odd part of 1+sigma(n).

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A332453 Starting from sigma(n), number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

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A332454 Starting from sigma(n)+1, number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

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A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

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