A202274 -id:A202274 - 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]

A332455 Starting from sigma(n)+1, number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 6, 0, 5, 6, 2, 5, 5, 7, 7, 0, 6, 1, 1, 9, 8, 6, 7, 5, 0, 9, 40, 10, 39, 42, 8, 0, 7, 41, 7, 4, 11, 5, 10, 33, 9, 43, 4, 1, 11, 42, 7, 39, 5, 38, 42, 7, 41, 34, 42, 34, 6, 33, 5, 16, 39, 43, 12, 0, 1, 42, 3, 15, 43, 42, 42, 7, 3, 10, 39, 3, 43, 16, 6, 14, 5, 15, 1, 17, 41, 8, 34, 4, 33, 46, 2, 16, 44, 42, 34, 39

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, A006667, A088580, A202274, A332453, A332454.

Cf. also A324294, A332459.

A006667(n) = { my(t=0); while(n>1, if(n%2, t++; n=3*n+1, n>>=1)); (t); };
A332455(n) = A006667(1+sigma(n));

a(n) = A006667(A088580(n)) = A006667(1+sigma(n)).

a(2^n) = 0 for all n >= 0. [Zero occurs at least also at a(25). See A202274]

A378983 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-(1+sigma(k))), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 11, 15, 16, 17, 25, 26, 29, 32, 33, 35, 39, 41, 57, 59, 64, 71, 93, 101, 107, 125, 128, 137, 149, 161, 179, 191, 197, 227, 239, 256, 269, 281, 311, 347, 419, 431, 461, 512, 521, 569, 599, 617, 641, 659, 782, 809, 821, 827, 857, 881, 1019, 1024, 1030, 1031, 1034, 1049, 1054, 1061, 1091, 1151

Offset: 1

Antti Karttunen, Dec 13 2024

nonn

Conjecture: A202274 gives all terms of A028982 that occur in this sequence.

			For k=16 we have A003961(16) = 81, A003961(k)-2*k = 49, and 49 divides (A003961(k)-(1+sigma(k))) = 81-32 = 49, therefore 16 is included in this sequence.
For k=25 we have A003961(25) = 49, A003961(k)-2*k = -1, and -1 divides (A003961(k)-(1+sigma(k))) regardless of what the latter is, therefore 25 is included.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's in A378982.
Subsequences: A048674, A348514, A202274.
Cf. also A378980.

PARI
```
isA378983(n) = !A378982(n);
```

A202273 Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 3m.

Original entry on oeis.org

1, 2, 5, 8, 14, 15, 20, 23, 24, 26, 30, 38, 40, 41, 45, 46, 51, 54, 55, 56, 58, 59, 60, 71, 74, 78, 87, 88, 89, 90, 92, 95, 106, 113, 118, 123, 136, 137, 143, 145, 146, 153, 167, 173, 178, 179, 215, 233, 263, 269, 303, 317, 335, 353

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 54 terms.

			m=1, 3m=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 3m=6, sigma(h)=6 for h=5; number 5 is in sequence.
m=5, 3m=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 3m=24, sigma(h)=24 for h=14,15,23; numbers 14,15,23 are in sequence.

Cf. A000203, A002191, A202274.

cp's OEIS Frontend

A332459 Odd part of 1+sigma(n).

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A332455 Starting from sigma(n)+1, number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

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A378983 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-(1+sigma(k))), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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PARI

A202273 Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 3m.

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