A077436 -id:A077436 - cp's OEIS frontend

A236514 Primes with a binary weight greater than or equal to the binary weight of their squares.

2, 3, 7, 23, 31, 47, 79, 127, 157, 191, 223, 317, 367, 379, 383, 479, 727, 751, 887, 1087, 1151, 1277, 1279, 1451, 1471, 1531, 1663, 1783, 1789, 1951, 2297, 2557, 2927, 3067, 3259, 3319, 3581, 3583, 3967, 4253, 4349, 5119, 5231, 5503, 5807, 5821, 6079, 6143, 6271, 6653, 6871, 6911, 7039, 7103, 7151

Offset: 1

Irina Gerasimova, Jan 27 2014

Primes p such that A000120(p) = A000120(p^2): 2, 3, 7, 31, 79, 127, 157, 317, 379, 751, 1087, 1151, 1277, 1279,...

			2 is in this sequence because 2 is 10 in binary representation, and it has as many 1s as its square 4, which is 100 in binary.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A077436, A094694.

bc[n_] := DigitCount[n, 2][[1]]; Select[Range[7151], PrimeQ[#] && bc[#] >= bc[#^2] &] (* Giovanni Resta, Jan 28 2014 *)
Select[Prime[Range[1000]], DigitCount[#, 2, 1] >= DigitCount[#^2, 2, 1] &] (* Alonso del Arte, Jan 28 2014 *)

is(n)=hammingweight(n^2)<=hammingweight(n) && isprime(n) \\ Charles R Greathouse IV, Mar 18 2014

Primes p such that A000120(p) >= A000120(p^2).

A352086 a(n) is the smallest positive integer k such that wt(k^2) / wt(k) = n where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

1, 21, 2697, 4736533, 14244123157, 4804953862344753

Offset: 1

Bernard Schott, Mar 06 2022

Theorem (proofs in Diophante link):
For any n and any base b, there exists m such that sod_b(m^2) / sod_b(m) = n, where sod_b(m) = sum of digits of m in base b (A280012 for base 10).
a(n) is odd. Proof: a(n) exists. Furthermore, if a(n) is even then wt(a(n)) = wt(a(n)/2) and wt(a(n)^2) = wt((a(n)/2)^2) so then a(n)/2 so that a(n)/2 is a lesser candidate, a contradiction. - David A. Corneth, Mar 06 2022

			We have 21_10 = 10101_2, so wt(21) = 3 ones; then 21^2 = 441_10 = 110111001_2, so wt(21^2) = 6 ones; as 6/3 = 2 and 21 is the smallest integer k such that wt(k^2) / wt(k) = 2, hence a(2) = 21.

Diophante, A1730 - Des chiffres à sommer pour un entier (in French).
Wojciech Muła, Nathan Kurz and Daniel Lemire, Faster Population Counts using AVX2 Instructions, arXiv:1611.07612 [cs.DS], Sep 05 2018.

Cf. A000120, A077436, A083567, A159918, A280012, A352084, A352085.

r[n_] := Total[IntegerDigits[n^2, 2]]/Total[IntegerDigits[n, 2]]; seq[max_, nmax_] := Module[{s = Table[0, {max}], c = 0, n = 1, i}, While[c < max && n < nmax, i = r[n]; If[IntegerQ[i] && s[[i]] == 0, c++; s[[i]] = n]; n+=2]; TakeWhile[s, # > 0 &]]; seq[4, 5*10^6] (* Amiram Eldar, Mar 06 2022 *)

from gmpy2 import popcount
aDict=dict()
for k in range(1, 10**11, 2):
    if popcount(k*k)%popcount(k)==0:
        n=popcount(k*k)//popcount(k)
        if not n in aDict:
            print(n, k); aDict[n]=k # Martin Ehrenstein, Mar 16 2022

a(n) > 2^(n^2/2) for n > 1. - Charles R Greathouse IV, Mar 16 2022

a(3)-a(5) from David A. Corneth, Mar 06 2022

a(6) -- using the Muła et al. Faster Population Counts algorithm -- from Martin Ehrenstein, Mar 15 2022

A261593 Odd numbers n such that the sum of the binary digits of n and n^2 both equal 12.

Original entry on oeis.org

4095, 10239, 11263, 12159, 12223, 12255, 12271, 12279, 12283, 14333, 15351, 15355, 15743, 15807, 18431, 19455, 19967, 20351, 20477, 22015, 22495, 22511, 24031, 24303, 24431, 24445, 25599, 26615, 26621, 27519, 27631, 27639, 28095, 28411, 28413, 28511, 28541, 28575

Offset: 1

Charles R Greathouse IV, Aug 25 2015

Hare, Laishram, & Stoll show that this sequence is infinite.
12 is the least constant for which the associated sequence is known to be infinite. 1 through 8 make finite sequences, the sequences with 9 and 10 are conjectured finite, and the sequence with 11 is unknown. (See Hare-Laishram-Stoll Theorem 1.3 and 1.4 plus section 5.)
Odd numbers n such that the sum of the binary digits of n-1 and n^2-1 both equal 11. - Chai Wah Wu, Aug 26 2015

			4095 = 111111111111_2 and 4095^2 = 111111111110000000000001_2 both have 12 1s in binary.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1329
K. G. Hare, S. Laishram, and T. Stoll, The sum of digits of n and n^2, International Journal of Number Theory 7:7 (2011), pp. 1737-1752.

Subsequence of A261586 and hence of A077436.

Select[2 Range@ 15000 - 1, Total@ IntegerDigits[#, 2] == 12 && Total@ IntegerDigits[#^2, 2] == 12 &] (* Michael De Vlieger, Aug 27 2015 *)

is(n)=n%2 && hammingweight(n)==12 && hammingweight(n^2)==12

\\ List the elements below 2^(N+1).
go(N)=my(v=List(),n); forvec(u=vector(11,i,[1,N-11+i]), n=sum(i=1,11,2^u[i])+1; if(hammingweight(n^2)==12, listput(v,n)), 2); Set(v)

from itertools import combinations
A261593_list = []
for c in combinations((2**x for x in range(15)),11):
    n = sum(c)
    if sum(int(d) for d in format(n*(n+1),'b')) == 11:
        A261593_list.append(2*n+1)
A261593_list = sorted(A261593_list) # Chai Wah Wu, Aug 26 2015

A261640 Numbers n such that the digital sum of n is the same as the digital sum of n^2 in both base 2 and base 10.

Original entry on oeis.org

0, 1, 351, 379, 496, 558, 639, 1495, 1792, 3259, 4600, 5950, 6399, 6588, 8568, 10494, 10495, 12799, 17380, 17919, 26479, 38872, 38880, 44991, 44992, 46585, 48888, 56952, 59247, 60895, 64639, 89839, 89848, 89856, 92799, 105390, 142848, 168895, 174078, 179596

Offset: 1

Tom Edgar, Aug 27 2015

Intersection of A077436 and A058369.

Numbers such that A007953(n) = A007953(n^2) and A000120(n) = A000120(n^2).

			Consider the number n = 351 so n^2 = 123201. The base-10 digit sums of 351 and 123201 are both 9. Moreover, 351 has binary representation 101011111 and 123201 has binary representation 11110000101000001 and both have base-2 digit sum = 7. Thus 351 is a term in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A077436, A058369, A007953, A000120.

Select[Range[0,180000],Total[IntegerDigits[#]]==Total[IntegerDigits[#^2]]&&Total[ IntegerDigits[ #,2]]==Total[IntegerDigits[#^2,2]]&] (* Harvey P. Dale, May 29 2023 *)

[n for n in [0..200000] if sum((n).digits(2))==sum((n^2).digits(2)) and sum((n).digits())==sum((n^2).digits())]

Name (definition) and Example edited by Harvey P. Dale, May 29 2023

A265113 Primes p such that p and p^2 have the same number of 1's in their binary representations.

Original entry on oeis.org

2, 3, 7, 31, 79, 127, 157, 317, 379, 751, 1087, 1151, 1277, 1279, 1531, 1789, 1951, 2297, 2557, 2927, 3067, 3259, 3319, 3581, 4253, 4349, 5119, 5231, 5503, 5807, 5821, 6271, 6653, 6871, 8191, 8447, 8689, 9209, 10079, 10837, 11597, 11903, 12799, 13309, 13591

Offset: 1

Robert Israel, Dec 01 2015

Primes p such that p^2 is in A089042.
Primes p such that A000120(p) = A000120(p^2).
Contains all terms > 43 in A079361.
Subset of A077436.

			7 is in the sequence because 7 and 7^2 = 49 have binary representations 111 and 110001 which both have three 1's.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A000120, A077436, A079361, A089042.

[NthPrime(n): n in [1..2000] | Multiplicity({* z: z in Intseq(NthPrime(n)^2, 2) *}, 1) eq &+Intseq(NthPrime(n), 2)]; // Vincenzo Librandi, Dec 02 2015

f:= proc(n) isprime(n) and (convert(convert(n,base,2),`+`) = convert(convert(n^2,base,2),`+`)) end proc:
select(f, [2,seq(i,i=3..10^5,2)]);

Select[ Prime@ Range@ 1700, DigitCount[n, 2, 1] == DigitCount[n^2, 2, 1],  &] (* Robert G. Wilson v, Dec 01 2015 *)

c(k, d, b) = {my(c=0, f); while (k>b-1, f=k-b*(k\b); if (f==d, c++); k\=b); if (k==d, c++); return(c)}
forprime(p=2, 1e5, if(c(p, 1, 2) == c(p^2, 1, 2), print1(p, ", "))) \\ Altug Alkan, Dec 02 2015

A340100 Fixed points of A340069.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 730, 1449, 4803, 8506, 10837, 17012, 18321, 21674, 43348, 86696, 103622, 103628, 153696, 173392, 185395, 207244, 227393, 370780, 454786, 693568, 936337, 989572, 1172353, 1284865, 1387136, 1413399, 1820929, 1872674, 2344706, 2569730, 2774272, 2826798

Offset: 1

Thomas Scheuerle, Dec 28 2020

A subset of A077436 because A000120(a(n)) = A000120(a(n)^2).

Cf. A340069, A077436, A000120.

a(n) = A340069(a(n)).

Incorrect terms removed and a(16)-a(33) added by Chai Wah Wu, Jan 08 2021

a(34)-a(38) from Chai Wah Wu, Jan 14 2021

A091819 Numbers k that have, in binary, the same number of ones as are in k^2 and which have no nonincreasing binary representation.

Original entry on oeis.org

79, 91, 157, 158, 159, 182, 183, 187, 279, 287, 314, 316, 317, 318, 319, 351, 364, 365, 366, 374, 375, 379, 445, 558, 573, 574, 575, 628, 632, 634, 636, 637, 638, 639, 702, 703, 728, 730, 732, 735, 748, 750, 751, 758, 759, 763, 815, 890, 893, 975, 1071

Offset: 1

Ralf Stephan, Mar 08 2004

nonn

In A077436 but not in A023758.

G. Melfi, On simultaneous binary expansions of n and n^2, arXiv:math/0402458 [math.NT], 2004.

A232245 Sum of the number of ones in binary representation of n and n^2.

Original entry on oeis.org

0, 2, 2, 4, 2, 5, 4, 6, 2, 5, 5, 8, 4, 7, 6, 8, 2, 5, 5, 8, 5, 9, 8, 7, 4, 8, 7, 10, 6, 9, 8, 10, 2, 5, 5, 8, 5, 9, 8, 11, 5, 8, 9, 11, 8, 12, 7, 9, 4, 8, 8, 9, 7, 12, 10, 12, 6, 10, 9, 12, 8, 11, 10, 12, 2, 5, 5, 8, 5, 9, 8, 11, 5, 9, 9, 13, 8, 11, 11, 10, 5, 9

Offset: 0

Jon Perry, Nov 20 2013

The sequence is never 1 or 3, but seems to take on all other values. The fact it is never 3 can be used to prove if n^2 has exactly 4 1's then it must have an even number of 0's (A231898).

			5 is 101 and 25 is 11001, so a(5) = 2 + 3 = 5.

Cf. A000120, A159918, A077436, A094694, A231898, A232243.

function bitCount(n) {
var i,c,s;
c=0;
s=n.toString(2);
for (i=0;i

a(n) = A159918(n) + A000120(n).

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A236514 Primes with a binary weight greater than or equal to the binary weight of their squares.

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A352086 a(n) is the smallest positive integer k such that wt(k^2) / wt(k) = n where wt(k) = A000120(k) is the binary weight of k.

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A261593 Odd numbers n such that the sum of the binary digits of n and n^2 both equal 12.

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A261640 Numbers n such that the digital sum of n is the same as the digital sum of n^2 in both base 2 and base 10.

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A265113 Primes p such that p and p^2 have the same number of 1's in their binary representations.

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A340100 Fixed points of A340069.

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A091819 Numbers k that have, in binary, the same number of ones as are in k^2 and which have no nonincreasing binary representation.

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A232245 Sum of the number of ones in binary representation of n and n^2.

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