A180094 -id:A180094 - cp's OEIS frontend

A007013 Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.

2, 3, 7, 127, 170141183460469231731687303715884105727

Offset: 0

N. J. A. Sloane, Nik Lygeros (webmaster(AT)lygeros.org)

nonn

The next term is too large to include.
Orbit of 2 under iteration of the "Mersenne operator" M: n -> 2^n-1 (0 and 1 are fixed points of M). - M. F. Hasler, Nov 15 2006
Also called the Catalan sequence. - Artur Jasinski, Nov 25 2007
a(n) divides a(n+1)-1 for every n. - Thomas Ordowski, Apr 03 2016
Proof: if 2^a == 2 (mod a), then 2^a = 2 + ka for some k, and 2^(2^a-1) = 2^(1 + ka) = 2*(2^a)^k == 2 (mod 2^a-1). Given that a(1) = 3 satisfies 2^a == 2 (mod a), that gives you all 2^a(n) == 2 (mod a(n)), and since a(n+1) - 1 = 2^a(n) - 2 that says a(n) | a(n+1) - 1. - Robert Israel, Apr 05 2016
All terms shown are primes, the status of the next term is currently unknown. - Joerg Arndt, Apr 03 2016
The next term is a prime or a Fermat pseudoprime to base 2 (i.e., a member of A001567). If it is a pseudoprime, then all succeeding terms are pseudoprimes. - Thomas Ordowski, Apr 04 2016
a(n) is the least positive integer that requires n+1 steps to reach 1 under iteration of the binary weight function A000120. - David Radcliffe, Jun 25 2018
If the next term were prime, it would be a counterexample to the New Mersenne conjecture. It is known that (2^a(4) + 1) / 3 is composite, with factor 886407410000361345663448535540258622490179142922169401 = 5209834514912200*a(4)+1. - William Hu, Jul 30 2024
a(n) is the smallest number of additive persistence n+1 in base 2.  (Similar to A006050 but for binary instead of decimal.) - J. Beach, Nov 17 2024

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 81.
W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 91.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chris K. Caldwell, Mersenne Primes.
Alex Kritov, Explicit Values for Gravitational and Hubble Constants from Cosmological Entropy Bound and Alpha-Quantization of Particle Masses, 2021, see p. 8.
Double Mersennes Prime Search Status of M(M(p)) where M(p) is a Mersenne prime [outdated link of Will Edgington replaced by _Georg Fischer_, Jan 18 2019].
Carlos Rivera, Conjecture 15. The New Mersenne Conjecture, The Prime Puzzles & Problems Connection.
W. Sierpiński, A Selection of Problems in the Theory of Numbers, Macmillan, NY, 1964, p. 91-92. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Catalan-Mersenne Number
Eric Weisstein's World of Mathematics, Double Mersenne Number.

Cf. A000668, A001567, A014221, A006050, A180094.

M:=n->2^n-1; '(M@@i)(2)'$i=0..4; # M. F. Hasler, Nov 15 2006

NestList[2^#-1&,2,4] (* Harvey P. Dale, Jul 18 2011 *)

a(n)=if(n,2^a(n-1)-1,2) \\ Charles R Greathouse IV, Sep 07 2016

a(n) = M(a(n-1)) = M^n(2) with M: n-> 2^n-1. - M. F. Hasler, Nov 15 2006

A180094(a(n)) = n + 1.

Edited by Henry Bottomley, Nov 07 2002

Amended title name by Marc Morgenegg, Apr 14 2016

A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 2, 1, 8, 1, 9, 2, 1, 10, 2, 1, 11, 3, 2, 1, 12, 2, 1, 13, 3, 2, 1, 14, 3, 2, 1, 15, 4, 1, 16, 1, 17, 2, 1, 18, 2, 1, 19, 3, 2, 1, 20, 2, 1, 21, 3, 2, 1, 22, 3, 2, 1, 23, 4, 1, 24, 2, 1, 25, 3, 2, 1, 26, 3, 2, 1

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0  1  2  3
  ----------------
   0 |  0;
   1 |  1;
   2 |  2, 1;
   3 |  3, 2, 1;
   4 |  4, 1;
   5 |  5, 2, 1;
   6 |  6, 2, 1;
   7 |  7, 3, 2, 1;
   8 |  8, 1;
   9 |  9, 2, 1;
  10 | 10, 2, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11724 (rows 0..3000 of triangle, flattened).

Cf. A000120, A078627 (row lengths), A078677 (row sums), A180094.

Cf. A381963, A381965.

A381962row[n_] := NestWhileList[DigitSum[#, 2] &, n, # > 1 &];
Array[A381962row, 30, 0]

def row(n):
    out = [n] if n > 1 else []
    while (n:=n.bit_count()) > 1:
        out += [n]
    return out + [n]
print([e for n in range(27) for e in row(n)]) # Michael S. Branicky, Mar 12 2025

T(n,0) = n and, for k = 1..A180094(n), T(n,k) = A000120(T(n,k-1)).

A078627 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = 1 + number of steps required.

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 2, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 4, 2, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 4, 3, 4, 4, 3, 4, 3, 3, 4, 4, 3

Offset: 1

Frank Schwellinger (nummer_eins(AT)web.de), Dec 12 2002

The terms a(n) are unbounded. The smallest n with a(n) = m, n_min(m), however may be exorbitantly large, even for small m. It can be calculated by the following recurrence: n_min(1) = 1; n_min(2) = 2; n_min(m) = 2^n_min(m-1) - 1 {if m > 2};

			a(13) = 4 because 13 = (1101) -> (1+1+0+1 = 11) -> (1+1 = 10) -> (1+0 = 1) = 1. (Three iterations were required to reach 1.)

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A000120.

One more than A180094. Row lengths of A381962.

for n from 1 to 500 do h := n:a[n] := 1:while(h>1) do a[n] := a[n]+1: b := convert(h,base,2):h := sum(b[j],j=1..nops(b)):od:od:seq(a[j],j=1..500);

Table[Length[NestWhileList[Total[IntegerDigits[#,2]]&,n,#>1&]],{n,110}] (* Harvey P. Dale, Oct 10 2011 *)

A078627(n) = { my(k=1); while(n>1, n = hammingweight(n); k += 1); (k); }; \\ Antti Karttunen, Jul 09 2017

def a(n):
    c = 1 if n > 1 else 0
    while (n:=n.bit_count()) > 1:
        c += 1
    return c + 1
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Mar 12 2025

a(1) = 1; for n > 1, a(n) = 1 + a(A000120(n)), where A000120 gives the number of occurrences of digit 1 in binary representation of n.

a(n) = 1 + A180094(n). - Antti Karttunen, Jul 09 2017

Description corrected by Antti Karttunen, Jul 09 2017

A078677 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = sum of these sums (including '1' and n itself).

Original entry on oeis.org

1, 3, 6, 5, 8, 9, 13, 9, 12, 13, 17, 15, 19, 20, 20, 17, 20, 21, 25, 23, 27, 28, 28, 27, 31, 32, 32, 34, 34, 35, 39, 33, 36, 37, 41, 39, 43, 44, 44, 43, 47, 48, 48, 50, 50, 51, 55, 51, 55, 56, 56, 58, 58, 59, 63, 62, 62, 63, 67, 65, 69, 70, 72, 65, 68, 69, 73, 71, 75, 76, 76, 75

Offset: 1

Frank Schwellinger (nummer_eins(AT)web.de), Dec 17 2002

			a(13) = 19 because 13 = (1101) -> (1+1+0+1 = 11) -> (1+1 = 10) -> (1+0 = 1) = 1 and 1101+11+10+1 (binary) = 19 (decimal).

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000120, A078627, A180094.

Row sums of A381962.

A078677[n_] := Total[NestWhileList[DigitSum[#, 2] &, n, # > 1 &]];
Array[A078677, 100] (* Paolo Xausa, Mar 11 2025 *)

def a(n):
    s = n if n > 1 else 0
    while (n:=n.bit_count()) > 1:
        s += n
    return s + 1
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 12 2025

a(1) = 1; for n > 1, a(n) = n + a(A000120(n)).

a(n) = Sum_{k = 0..A180094(n)} A381962(n,k). - Paolo Xausa, Mar 12 2025

cp's OEIS Frontend

A007013 Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.

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A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

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A078627 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = 1 + number of steps required.

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Maple

Mathematica

PARI

Python

Formula

Extensions

A078677 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = sum of these sums (including '1' and n itself).

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Author

Keywords

Examples

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Programs

Mathematica

Python

Formula