A336699 -id:A336699 - cp's OEIS frontend

A366894 a(n) = A336699(A163511(n)).

1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 7, 1, 1, 1, 1, 1, 1, 61, 3, 5, 7, 1, 1, 29, 1, 5, 1, 1, 1, 1, 1, 1, 1, 23, 61, 391, 3, 5, 5, 13, 7, 101, 1, 43, 1, 29, 29, 67, 1, 1, 5, 1, 1, 1, 1, 1, 1, 7, 1, 5, 1, 1, 1, 1, 1, 547, 23, 977, 61, 391, 391, 1401, 3, 127, 5, 19, 5, 13, 13, 23, 7, 39, 101, 221, 1, 43, 43, 67, 1, 371, 29, 25, 29

Offset: 0

Antti Karttunen, Jan 03 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A000265, A000593, A163511, A324186, A336699, A351565, A366895 (rgs-transform).

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A000265(n) = (n>>valuation(n,2));
A336699(n) = A000265(1+A000265(sigma(A000265(n))));
A366894(n) = A336699(A163511(n));

a(n) = A351565(A324186(n)).

A336698 a(n) = A000265(1+A000265(sigma(n))), where A000265(k) gives the odd part of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 1, 1, 1, 1, 1, 1, 5, 5, 3, 11, 1, 5, 1, 1, 1, 11, 3, 1, 1, 5, 1, 1, 1, 7, 1, 23, 5, 1, 1, 23, 11, 1, 3, 11, 5, 5, 1, 1, 29, 47, 5, 25, 7, 1, 5, 1, 3, 23, 1, 11, 1, 1, 7, 1, 11, 5, 9, 1, 1, 5, 5, 49, 19, 29, 1, 9, 1, 11, 3, 47, 61, 1, 11, 1, 7, 17, 1, 23, 23, 59, 1, 11, 1, 5, 1, 1, 25

Offset: 1

Antti Karttunen, Aug 02 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 sigma(n)

Cf. A000203, A000265, A161942, A336699, A336701.

A000265(n) = (n>>valuation(n,2));
A336698(n) = A000265(1+A000265(sigma(n)));

a(n) = A000265(1+A000265(A000203(n))) = A000265(1+A161942(n)).

a(A000265(n)) = A336699(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));

A336844 a(n) = A336698(A003961(n)).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 61, 3, 1, 1, 7, 5, 1, 1, 1, 29, 5, 5, 5, 1, 1, 5, 23, 11, 3, 1, 101, 11, 1, 7, 3, 3, 5, 1, 23, 1, 1, 7, 91, 67, 29, 1, 59, 1, 5, 1, 1, 5, 1, 1, 5, 9, 5, 47, 547, 5, 11, 5, 33, 23, 1, 19, 39, 3, 11, 43, 5, 11, 7, 11, 61, 391, 3, 23, 59, 3, 1, 1, 9, 25, 1, 7, 49, 29, 7, 1, 137

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000265, A000203, A003961, A003973, A336698, A336699, A336842.

A000265(n) = (n>>valuation(n,2));
A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A336844(n) = A000265(1+A000265(A003973(n)));

a(n) = A000265(1+A000265(A003973(n))).

a(n) = A336698(A003961(n)) = A336699(A003961(n)).

A351565 Odd part of Kimberling's paraphrases: a(n) = A000265(A003602(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 5, 3, 3, 1, 7, 1, 1, 1, 9, 5, 5, 3, 11, 3, 3, 1, 13, 7, 7, 1, 15, 1, 1, 1, 17, 9, 9, 5, 19, 5, 5, 3, 21, 11, 11, 3, 23, 3, 3, 1, 25, 13, 13, 7, 27, 7, 7, 1, 29, 15, 15, 1, 31, 1, 1, 1, 33, 17, 17, 9, 35, 9, 9, 5, 37, 19, 19, 5, 39, 5, 5, 3, 41, 21, 21, 11, 43, 11, 11, 3, 45, 23, 23, 3, 47

Offset: 1

Antti Karttunen, Mar 27 2022

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

Cf. A000265, A003602, A023758 (gives the positions of 1's after its initial zero-term).

Cf. also A336698, A336699.

A000265(n) = (n>>valuation(n,2));
A351565(n) = A000265(1+A000265(n));

def A351565(n): return (m:=1+(n>>(~n&n-1).bit_length()))>>(~m&m-1).bit_length()  # Chai Wah Wu, Jan 03 2024

a(n) = A000265(A003602(n)) = A000265(1+A000265(n)).

A366895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366894(i) = A366894(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 3, 5, 2, 1, 1, 6, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 4, 8, 3, 5, 5, 9, 2, 10, 1, 11, 1, 6, 6, 12, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 13, 7, 14, 4, 8, 8, 15, 3, 16, 5, 17, 5, 9, 9, 7, 2, 18, 10, 19, 1, 11, 11, 12, 1, 20, 6, 21, 6, 12, 12, 7, 1, 22, 1, 5, 5

Offset: 0

Antti Karttunen, Jan 03 2024

Restricted growth sequence transform of A366894.
For all i, j >= 0:
  A366881(i) = A366881(j) => A366806(i) = A366806(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000265, A000593, A163511, A324186, A336699, A351565, A366894.

Cf. also A366806, A366881.

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; };
A000265(n) = (n>>valuation(n,2));
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A336699(n) = A000265(1+A000265(sigma(A000265(n))));
A366894(n) = A336699(A163511(n));
v366895 = rgs_transform(vector(1+up_to,n,A366894(n-1)));
A366895(n) = v366895[1+n];

cp's OEIS Frontend

A366894 a(n) = A336699(A163511(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A336698 a(n) = A000265(1+A000265(sigma(n))), where A000265(k) gives the odd part of k.

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

A336844 a(n) = A336698(A003961(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A351565 Odd part of Kimberling's paraphrases: a(n) = A000265(A003602(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A366895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366894(i) = A366894(j) for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI