A112714 -id:A112714 - cp's OEIS frontend

A112715 Primes in A112714.

3, 7, 11, 23, 31, 47, 79, 127, 191, 223, 239, 383, 479, 607, 863, 991, 1087, 1151, 1279, 1471, 1663, 2111, 2239, 2687, 2879, 3391, 3583, 3967, 5119, 5503, 6143, 6271, 6911, 7039, 8191, 8447, 8831, 9343, 10111, 11519, 11903, 12671, 12799, 13183, 13567

Offset: 1

Jose Brox (tautocrona(AT)terra.es), Dec 31 2005

			a(1)=3 because 3=1*2^2-1 and it is the first prime of this form.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080076.

N:= 10^6: # to get all terms <= N
sort(convert(select(isprime,{seq(seq(k*2^m-1,k=1..min((N+1)/2^m,2^m-1),2),m=1..ilog2(N+1))}),list)); # Robert Israel, May 23 2017

Take[Sort@ Select[Flatten@ Table[k 2^m - 1, {m, 0, 15}, {k, 1, 2^m - 1, 2}], PrimeQ], 45] (* Michael De Vlieger, May 23 2017, after Robert G. Wilson v at A112714 *)

for(n=2,8,for(k=2^(n-2)+1,2^n,M=k*2^n-1;if(isprime(M),print1(M","),0)))

A116882 A number k is included if (highest odd divisor of k)^2 <= k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 40, 48, 56, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1088, 1152, 1216, 1280, 1344, 1408

Offset: 1

Leroy Quet, Feb 24 2006

Also k is included if (and only if) the greatest power of 2 dividing k is >= the highest odd divisor of k. All terms of the sequence are even besides the 1.
Equivalently, positive integers of the form k*2^m, where odd k <= 2^m. - Thomas Ordowski, Oct 19 2014
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence consists of 1 and all numbers without a superior odd divisor. - Gus Wiseman, Feb 18 2021
Numbers k such that A006519(k) >= A000265(k), with equality only when k = 1. - Amiram Eldar, Jan 24 2023

			40 = 8 * 5, where 8 is highest power of 2 dividing 40 and 5 is the highest odd dividing 40. 8 is >= 5 (so 5^2 <= 40), so 40 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).

Cf. A000265, A006519, A080075, A112714.
The complement is A116883.
Positions of zeros (and 1) in A341675.
A051283 = numbers without a superior prime-power divisor (zeros of A341593).
A059172 = numbers without a superior squarefree divisor (zeros of A341592).
A063539 = numbers without a superior prime divisor (zeros of A341591).
A333805 counts strictly inferior odd divisors.
A341594 counts strictly superior odd divisors.
- Inferior: A033676, A038548, A063962, A066839, A069288, A161906, A217581.
- Superior: A033677, A063538, A070038, A072500, A161908, A341676.
- Strictly Inferior: A056924, A060775, A070039, A333806, A341596, A341674.
- Strictly Superior: A056924, A064052/A048098, A140271, A238535, A341642, A341643, A341673.
Cf. A000005, A000203, A001248, A006530, A020639, A026804, A027193, A340101, A340854 (zeros of A340832), A001008, A002805.
Subsequence of A082662, {1} U A363122.

f[n_] := Select[Divisors[n], OddQ[ # ] &][[ -1]]; Insert[Select[Range[2, 1500], 2^FactorInteger[ # ][[1]][[2]] > f[ # ] &], 1, 1] (* Stefan Steinerberger, Apr 10 2006 *)
q[n_] := 2^(2*IntegerExponent[n, 2]) >= n; Select[Range[1500], q] (* Amiram Eldar, Jan 24 2023 *)

isok(n) = vecmax(select(x->((x % 2)==1), divisors(n)))^2 <= n; \\ Michel Marcus, Sep 06 2016

isok(n) = 2^(valuation(n,2)*2) >= n \\ Jeppe Stig Nielsen, Feb 19 2019

from itertools import count, islice
def A116882_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n&-n)**2>=n,count(max(startvalue,1)))
A116882_list = list(islice(A116882_gen(),20)) # Chai Wah Wu, May 17 2023

a(n) = A080075(n-1)-1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010
a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014
Sum_{n>=1} 1/a(n) = 1 + (3/4) * Sum_{k>=1} H(2^k-1)/2^k = 2.3388865091..., where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 24 2023

More terms from Stefan Steinerberger, Apr 10 2006

A080075 Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.

Original entry on oeis.org

3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409

Offset: 1

Eric W. Weisstein, Jan 24 2003

nonn

A Proth number is a square iff it is of the form (2^(m-1)+-1)*2^(m+1)+1 = 4^m+-2^(m+1)+1 = (2^m+-1)^2 for m > 1. See A086341. - Thomas Ordowski, Apr 22 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Bertalan Borsos, Attila Kovács and Norbert Tihanyi, Tight upper and lower bounds for the reciprocal sum of Proth primes, The Ramanujan Journal (2022).
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).
Eric Weisstein's World of Mathematics, Proth Number.
Wikipedia, Proth number.

Cf. A080076, A086341, A112714, A116882, A157892, A157893.

Select[Range[3, 1500, 2], And[OddQ[#[[1]] ], #[[-1]] >= 1, 2^#[[-1]] > #[[1]] ] &@ Append[QuotientRemainder[#1, 2^#2], #2] & @@ {#, IntegerExponent[#, 2]} &[# - 1] &] (* Michael De Vlieger, Nov 04 2019 *)

is_A080075 = isproth(x)={!bittest(x--,0) && (x>>valuation(x+!x,2))^2 < x } \\ M. F. Hasler, Aug 16 2010; edited by Michel Marcus, Apr 23 2019, M. F. Hasler, Jul 07 2022

next_A080075(N)=N+2^(exponent(N)\2+1)
A080075_first(N)=vector(N,i,if(i>1,next_A080075(N),3)) \\ M. F. Hasler, Jul 07 2022

from itertools import count, islice
def A080075_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n:(n-1&-n+1)**2+1>=n,count(max(startvalue,3)))
A080075_list = list(islice(A080075_gen(),30)) # Chai Wah Wu, Oct 06 2024

a(n) = A116882(n+1)+1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010
a(n) = A157892(n)*2^A157893(n) + 1. - M. F. Hasler, Aug 16 2010
a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014
Sum_{n>=1} 1/a(n) = 1.09332245643583252894473574405304699874426408312553... (Borsos et al., 2022). - Amiram Eldar, Jan 29 2022
a(n+1) = a(n) + 2^round(L(n)/2), where L(n) is the number of binary digits of a(n); equivalently, floor(log_2(a(n))/2 + 1) in the exponent. [Lemma 2.2 in Borsos et al.] - M. F. Hasler, Jul 07 2022

A332011 Let k be the least positive number such that n AND floor(n/k) = 0 (where AND denotes the bitwise AND operator); a(n) = floor(n/k).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 4, 4, 5, 0, 3, 2, 1, 0, 8, 8, 9, 4, 10, 10, 1, 0, 6, 6, 5, 4, 3, 2, 1, 0, 16, 16, 17, 8, 18, 18, 9, 0, 20, 20, 21, 4, 3, 2, 1, 0, 12, 12, 12, 12, 10, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1, 0, 32, 32, 33, 16, 34, 34, 17, 8, 36, 36, 37, 4, 19, 2

Offset: 0

Rémy Sigrist, Feb 04 2020

			For n = 3:
- 3 AND floor(3/1) = 3,
- 3 AND floor(3/2) = 1,
- 3 AND floor(3/3) = 1,
- 3 AND floor(3/4) = 0,
- hence a(3) = floor(3/4) = 0.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A112714, A331985.

a(n) = for (k=1, oo, if (bitand(n, n\k)==0, return (n\k)))

a(n) = floor(n/A331985(n)).

Apparently, a(n) = 0 iff n = 0 or n belongs to A112714.

A112717 9-digit numbers n such that phi(n) = d_1 d_2 d_3d_4 d_5 d_6 d_7 d_8 d_9 where d_1 d_2 ... d_9 is the decimal expansion of n.

Original entry on oeis.org

102873384, 104754444, 104840625, 104963320, 106600792, 108512770, 108860625, 108864585, 110640784, 110756648, 116660400, 116672500, 117480648, 120297912, 120876448, 120916400, 121864290, 124704384, 125792500, 126528640, 128333700

Offset: 1

Farideh Firoozbakht, Sep 16 2005

			phi(128972250)=128*972*250, so 128972250 is in the sequence.

Cf. A058627, A112714, A112715.

Do[h=IntegerDigits[n]; k=Length[h]; If[EulerPhi[n]==Product[ 100*h[[3j-2]]+10h[[3j-1]]+h[[3j]], {j, k/3}], Print[n]], {n, 10^8, 128400000}]

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A112715 Primes in A112714.

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A116882 A number k is included if (highest odd divisor of k)^2 <= k.

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A080075 Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.

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A332011 Let k be the least positive number such that n AND floor(n/k) = 0 (where AND denotes the bitwise AND operator); a(n) = floor(n/k).

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A112717 9-digit numbers n such that phi(n) = d_1 d_2 d_3*d_4 d_5 d_6* d_7 d_8 d_9 where d_1 d_2 ... d_9 is the decimal expansion of n.

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A112717 9-digit numbers n such that phi(n) = d_1 d_2 d_3d_4 d_5 d_6 d_7 d_8 d_9 where d_1 d_2 ... d_9 is the decimal expansion of n.