Giuseppe Melfi - cp's OEIS frontend

A357439 Sums of squares of two odd primes.

18, 34, 50, 58, 74, 98, 130, 146, 170, 178, 194, 218, 242, 290, 298, 314, 338, 370, 386, 410, 458, 482, 530, 538, 554, 578, 650, 698, 722, 818, 850, 866, 890, 962, 970, 986, 1010, 1058, 1082, 1130, 1202, 1250, 1322, 1370, 1378, 1394, 1418, 1490, 1538, 1658, 1682

Offset: 1

Giuseppe Melfi, Oct 06 2022

nonn

Although this is twice A143850, it is important enough to warrant an entry of it own. - N. J. A. Sloane, Oct 10 2022

Cf. A045636, A103739, A143850, A227697.

A327621 Sums of distinct powers of 3 and powers of 4 (greater than 1).

Original entry on oeis.org

3, 4, 7, 9, 12, 13, 16, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32, 34, 36, 39, 40, 43, 46, 47, 50, 52, 55, 56, 59, 64, 67, 68, 71, 73, 76, 77, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 103, 104, 106, 107, 108, 109, 110, 111, 112

Offset: 1

Giuseppe Melfi, Sep 19 2019

nonn

From M. F. Hasler, Nov 16 2023: (Start)
Record gaps in this sequence are : a(2) - a(1) = 1, a(3) - a(2) = 3, a(30) - a(29) = 5, a(112) - a(111) = 39, a(9863) - a(9862) = 1084, a(34096) - a(34095) = 7682, ...
These gaps are closely related to the gaps in the set where 3^0 and 4^0 are (both) also allowed to be in the sum, in which case the first missing numbers are A367090 = (62, 63, 143, 144, 207, ...), see also Melfi's paper. It is obvious that the study of these gaps is crucial for the proof of Erdös conjecture.
The record gap a(9863) - a(9862) = 1084 explains the discontinuity seen in the graph of a(1..10^4). (End)

			40 is in the sequence because 40 = 27 + 9 + 4.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Nov 02 2023
S. A. Burr, P. Erdős, R. L. Graham, and W. Wen-Ching Li, Complete sequences of sets of integer powers, Acta Arithmetica 77(2) (1996), 133-138.
P. Erdős, Conjecture about the enumerating function A(x)
G. Melfi, An additive problem about powers of fixed integers, Rend. Circ. Mat. Palermo 50 (2001), 239-246.

Cf. A000244 (powers of 3), A000302 (powers of 4).

Cf. A005836 and A000695 (sums of distinct powers of 3 and of 4).

f[b_, m_] := Select[b Range[0, m/b], Max@ IntegerDigits[#, b] < 2 &]; mx=200; Union@ Select[Total /@ Tuples[{f[3, mx], f[4, mx]}], 0 < # < mx &] (* Giovanni Resta, Sep 19 2019 *)

A327621_upto(N, S=[0])={for(b=3,4, for(k=1, logint(N,b), my(p=b^k); S=setunion(S,[x+p|x<-S,x+p<=N])));S[^1]} \\ M. F. Hasler, Nov 02 2023

def A327621_upto(N):
    "list(x < N | x = sum(3^j, j in J) + sum(4^k, k in K); J, K subset N*)."
    S = {0} # empty sum
    for b in (3,4):
        p = b
        while p < N: S |= {k+p for k in S if k+p < N} ; p *= b
    return sorted(S) # includes a(0) = 0, so a(1,2,3,...) = 3,4,9,...
# M. F. Hasler, Nov 09 2023

For A(x) the enumerating function, Erdős conjectured that A(x) > c*x.

G. Melfi proved that A(x) > x^0.965 for sufficiently large x.

More terms from Giovanni Resta, Sep 19 2019

A325423 Numbers k such that sigma(2k+1) >= sigma(2k).

Original entry on oeis.org

1, 7, 31, 37, 67, 73, 97, 103, 127, 157, 199, 202, 229, 241, 247, 262, 277, 283, 307, 313, 331, 337, 346, 367, 379, 382, 397, 409, 427, 457, 472, 487, 499, 517, 547, 562, 577, 607, 619, 643, 661, 667, 682, 697, 727, 757, 769, 787

Offset: 1

Giuseppe Melfi, Sep 06 2019

nonn

The function A(x) enumerating the terms not exceeding x has the property that lim_{x->oo} A(x)/x exists (Hildebrand, 1990).

			7 is in the sequence because sigma(14) = 1+2+7+14 = 24 <= sigma(15) = 1+3+5+15 = 24;
31 is in the sequence because sigma(62) = 1+2+31+62 = 96 <= sigma(63) = 1+3+7+9+21+63 = 104.

M. Laub, Advanced Problems: 6555. The American Mathematical Monthly, 94(8), 800 (1987). doi:10.2307/2323430.

Giuseppe Melfi, Table of n, a(n) for n = 1..2763
Mits Kobayashi, Tim Trudgian, On integers n for which sigma(2n+1)>=sigma(2n), arXiv:1904.10064 [math.NT], 2019.
M. Laub & L. Mattics, Problem 6555: Odd Integers with Relatively Large Divisor Sum, The American Mathematical Monthly, 97(4), 351-353 (1990). doi:10.2307/2324532.

Cf. A082957.

Position[Partition[DivisorSigma[1,Range[2,1601]],2],?(#[[2]] >= #[[1]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Jan 10 2022 *)

isok(n) = sigma(2*n+1) >= sigma(2*n); \\ Michel Marcus, Sep 09 2019

a(n) ~ c*n with 18.2 < c < 18.6 (claimed by Kobayashi and Trudgian).

A083567 Let B(k) be the number of binary digits in k equal to 1. This is the sequence of positive integers k such that 2B(k)=B(k^2).

Original entry on oeis.org

21, 37, 42, 45, 53, 69, 73, 74, 81, 83, 84, 90, 106, 133, 137, 138, 141, 146, 148, 155, 161, 162, 165, 166, 168, 177, 180, 211, 212, 261, 265, 266, 269, 273, 274, 276, 281, 282, 289, 291, 292, 295, 296, 299, 310, 321, 322, 324, 330, 332, 336, 354, 359, 360

Offset: 1

Giuseppe Melfi, Jun 13 2003

This includes all k > 1 such that the average of ones in the binary expansion of k is the same of the average of ones in binary expansion of k^2; these are the values in the sequence with sqrt(2)*2^j < a(k) < 2^(j+1) for some j. - Corrected by Franklin T. Adams-Watters, Aug 23 2012

Conjecture: The counting function p(n) satisfies p(n)=c n/log n + o(n/log n).

			21 is in the sequence because 21=10101_2 (3 1's) and 441=110111001_2 (6 1's).

Robert Israel, Table of n, a(n) for n = 1..10000
Giuseppe Melfi, Su alcune successioni di interi, 2° Incontro Italiano di Teoria dei Numeri, Parma, 13-15 novembre 2003.
Giuseppe Melfi, On certain positive integer sequences, arXiv:math/0404555 [math.NT], 30 Apr 2004.

Cf. A000120, A077436, A159918.

select(t -> 2*convert(convert(t,base,2),`+`) = convert(convert(t^2,base,2),`+`), [$1..1000]); # Robert Israel, Aug 27 2015

f[n_] := Total@ IntegerDigits[n, 2]; Select[Range@ 360, 2 f@ # == f[#^2] &] (* Michael De Vlieger, Aug 27 2015 *)

isok(n) =  norml2(binary(n^2)) == 2*norml2(binary(n)) \\ Michel Marcus, Jun 20 2013

A077436 Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 79, 91, 96, 112, 120, 124, 126, 127, 128, 157, 158, 159, 182, 183, 187, 192, 224, 240, 248, 252, 254, 255, 256, 279, 287, 314, 316, 317, 318, 319, 351, 364, 365, 366, 374, 375, 379, 384

Offset: 1

Giuseppe Melfi, Nov 30 2002

Superset of A023758.

Hare, Laishram, & Stoll show that this sequence contains infinitely many odd numbers. In particular for each k in {12, 13, 16, 17, 18, 19, 20, ...} there are infinitely many terms in this sequence with binary digit sum k. - Charles R Greathouse IV, Aug 25 2015

			The element 79 belongs to the sequence because 79=(1001111) and 79^2=(1100001100001), so B(79)=B(79^2)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10476, all terms <= 2^20
Karam Aloui, Damien Jamet, Hajime Kaneko, Steffen Kopecki, Pierre Popoli, and Thomas Stoll, On the binary digits of n and n^2, arXiv:2203.05451 [math.NT], 2022.
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.
Giuseppe Melfi, On simultaneous binary expansions of n and n^2, arXiv:math/0402458 [math.NT], 2004.
Giuseppe Melfi, Su alcune successioni di interi (English with an Italian title)

Cf. A058369, A000120, A000290, A083567.
Cf. A211676 (number of n-bit numbers in this sequence).
A261586 is a subsequence. Subsequence of A352084.

import Data.List (elemIndices)
import Data.Function (on)
a077436 n = a077436_list !! (n-1)
a077436_list = elemIndices 0
   $ zipWith ((-) `on` a000120) [0..] a000290_list
-- Reinhard Zumkeller, Apr 12 2011

[n: n in [0..400] | &+Intseq(n, 2) eq &+Intseq(n^2, 2)]; // Vincenzo Librandi, Aug 30 2015

select(t -> convert(convert(t,base,2),`+`) = convert(convert(t^2,base,2),`+`), [$0..1000]); # Robert Israel, Aug 27 2015

t={}; Do[If[DigitCount[n, 2, 1] == DigitCount[n^2, 2, 1], AppendTo[t, n]], {n, 0, 364}]; t
f[n_] := Total@ IntegerDigits[n, 2]; Select[Range[0, 384], f@ # == f[#^2] &] (* Michael De Vlieger, Aug 27 2015 *)

is(n)=hammingweight(n)==hammingweight(n^2) \\ Charles R Greathouse IV, Aug 25 2015

def ok(n): return bin(n).count('1') == bin(n**2).count('1')
print([m for m in range(400) if ok(m)]) # Michael S. Branicky, Mar 11 2022

A159918(a(n)) = A000120(a(n)). - Reinhard Zumkeller, Apr 25 2009

Initial 0 added by Reinhard Zumkeller, Apr 28 2012, Apr 12 2011

A052287 Start with 3; the general rule is "if x is present then so is x*y for every y <= x".

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 27, 30, 36, 45, 48, 54, 60, 63, 72, 81, 84, 90, 96, 108, 120, 126, 132, 135, 144, 150, 162, 168, 180, 189, 192, 198, 210, 216, 225, 234, 240, 243, 252, 264, 270, 288, 297, 300, 306, 312, 315, 324, 330, 336, 351, 360, 378, 384, 390, 396

Offset: 1

Giuseppe Melfi, Feb 08 2000

			63 is an element because 63 = 3*3*7 and 3 <= 3 and 7 <= 3*3.

T. D. Noe, Table of n, a(n) for n = 1..1000

If instead we start with 2, we obtain the "Nullwertzahlen sequence" A047836.

Cf. A196149.

import Data.List.Ordered (union)
a052287 n = a052287_list !! (n-1)
a052287_list = f [3] where
   f (x:xs) = x : f (xs `union` map (x *) [2..x])
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

N:= 1000: # get all terms <= N
S:= {3}:
New:= {3}:
while New <> {} do
  x:= New[1];
  New:= subsop(1=NULL,New);
  R:= {seq(k*x, k=1..min(x,N/x))} minus S;
  S:= S union R;
  New:= New union R;
od:
sort(convert(S,list));  # Robert Israel, Aug 27 2015

3 Select[Range[132], Max[#[[2]]/#[[1]] & /@ Partition[Divisors[#], 2, 1]] <= 3 &] (* Michael De Vlieger, Aug 27 2015, after Harvey P. Dale at A196149 *)

x is a term if and only if x = 3*p1*p2*...*pk with primes 2 <= p1 <= p2 <= ... <= pk and 3*p1*p2*...*pi >= p(i+1) for all i < k.
a(n) = 3 * A196149(n). - Reinhard Zumkeller, Sep 28 2011
The number of terms <= x is c*x/log(x) + O(x/(log(x))^2), where c = 0.68514..., and a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 1.45954... This follows from the formula just above. - Andreas Weingartner, Jun 30 2021

More terms from Reinhard Zumkeller, Jun 22 2003

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User: Giuseppe Melfi

A357439 Sums of squares of two odd primes.

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A327621 Sums of distinct powers of 3 and powers of 4 (greater than 1).

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A325423 Numbers k such that sigma(2*k+1) >= sigma(2*k).

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A083567 Let B(k) be the number of binary digits in k equal to 1. This is the sequence of positive integers k such that 2B(k)=B(k^2).

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A077436 Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2).

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A052287 Start with 3; the general rule is "if x is present then so is x*y for every y <= x".

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A325423 Numbers k such that sigma(2k+1) >= sigma(2k).