A337195 -id:A337195 - cp's OEIS frontend

A337194 a(n) = 1 + A000265(sigma(n)), where A000265 gives the odd part.

2, 4, 2, 8, 4, 4, 2, 16, 14, 10, 4, 8, 8, 4, 4, 32, 10, 40, 6, 22, 2, 10, 4, 16, 32, 22, 6, 8, 16, 10, 2, 64, 4, 28, 4, 92, 20, 16, 8, 46, 22, 4, 12, 22, 40, 10, 4, 32, 58, 94, 10, 50, 28, 16, 10, 16, 6, 46, 16, 22, 32, 4, 14, 128, 22, 10, 18, 64, 4, 10, 10, 196, 38, 58, 32, 36, 4, 22, 6, 94, 122, 64, 22, 8, 28, 34

Offset: 1

Antti Karttunen, Aug 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

One more than A161942.

Cf. A000203, A000265, A336698, A337195, A337196.

A000265(n) = (n>>valuation(n,2));
A337194(n) = (1+A000265(sigma(n)));

a(n) = 1+A000265(A000203(n)).
a(n) = A336698(n) * 2^A337195(n).
A007949(a(n)) = A337196(n).

A336701 Numbers k for which A000265(1+A000265(sigma(k))) is equal to A000265(1+k).

Original entry on oeis.org

1, 3, 7, 15, 31, 127, 1023, 8191, 34335, 57855, 131071, 524287, 2147483647

Offset: 1

Antti Karttunen, Aug 02 2020

Numbers k such that A336698(k) [= A000265(1+A161942(k))] is equal to A000265(1+k).
Numbers k such that A337194(k) = 2^e * A000265(1+k), for some e >= 1, where that e = A337195(k).
Any odd perfect number would trivially satisfy this condition.
Also, all hypothetical quasiperfect numbers, numbers k that satisfy sigma(k) = 2k+1, would be members.
Question: Is A066175 a subsequence of this sequence?
From Antti Karttunen, Aug 23 2020: (Start)
Numbers k such that (1+k) = 2^e * A336698(k), for some e >= 0.
Thus numbers k such that for some e >= 0, (1+k) = 2^(e-A337195(k)) * A337194(k), or equally, that A337194(k) = 2^(A337195(k)-e) * (1+k).
Conjecture: There are no even terms. This is equivalent to claim that there are no k such that A336698(k) = 1+k: If we assume that k is even, then in above equations we set e=0, and the requirement will then become that A337194(k) = 2^A337195(k)*(1+k), thus 1+k = A336698(k) = A000265(1+A000265(sigma(k))).
(End)

Paolo Cattaneo, Sui numeri quasiperfetti, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62.
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.
Eric Weisstein's World of Mathematics, Quasiperfect Number
Index entries for sequences where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Subsequence of A336700.
Cf. A000203, A000265, A066175, A161942, A336698, A336699, A336937, A337194, A337195.
Cf. A000668 (a subsequence).
See also comments in A326042, A332223.

Block[{f}, f[n_] := n/2^IntegerExponent[n, 2]; Select[Range[2^20], f[1 + f[DivisorSigma[1, #]]] == f[1 + #] &] ] (* Michael De Vlieger, Aug 22 2020 *)

A000265(n)  = (n>>valuation(n,2));
isA336701(n) = (A000265(1+A000265(sigma(n))) == A000265(1+n));

A337196 The 3-adic valuation of 1+A000265(sigma(n)), where A000265 gives the odd part.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Aug 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A000265, A007949, A074941, A161942, A336698, A336700, A337194, A337195.

Cf. A337197 (the first occurrence of each n).

Array[IntegerExponent[1 + #/2^IntegerExponent[#, 2], 3] &@ DivisorSigma[1, #] &, 105] (* Michael De Vlieger, Feb 22 2021 *)

A007949(n) = valuation(n,3);
A000265(n) = (n>>valuation(n,2));
A337196(n) = A007949(1+A000265(sigma(n)));

a(n) = A007949(A337194(n)) = A007949(1+A000265(A000203(n))).

a(n) = A007949(A336698(n)).

A378998 Number of trailing 1-bits in the binary representation of sigma(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Dec 16 2024

nonn

Cf. A000203, A007814, A088580, A028982 (positions of terms > 0), A028983 (of 0's), A072461 (of 1's), A072462 (of terms > 1), A337195, A378999 [= a(n^2)].

IntegerExponent[DivisorSigma[1, Range[100]] + 1, 2] (* Paolo Xausa, Dec 19 2024 *)

PARI
```
A378998(n) = valuation(sigma(n)+1,2);
```

a(n) = A007814(A088580(n)). [the 2-adic valuation of 1+sigma(n)]

For all n in A028982, a(n) = A337195(n).

cp's OEIS Frontend

A337194 a(n) = 1 + A000265(sigma(n)), where A000265 gives the odd part.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A336701 Numbers k for which A000265(1+A000265(sigma(k))) is equal to A000265(1+k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A337196 The 3-adic valuation of 1+A000265(sigma(n)), where A000265 gives the odd part.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378998 Number of trailing 1-bits in the binary representation of sigma(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula