A161792 -id:A161792 - cp's OEIS frontend

A386249 a(n) is the Hamming weight of A161792(n).

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 6, 1, 1, 2, 1, 1, 2, 1, 1, 2, 6, 1, 1, 2, 1, 1, 2, 1, 1, 2, 6, 1, 14, 1, 2, 15, 1, 1, 2, 1, 1, 2, 6, 1, 1, 2, 1, 1, 2, 1, 1, 2, 6, 1, 1, 2, 1, 14, 1, 2, 1, 15, 1, 2, 6, 1, 1, 2, 1, 1, 2, 1, 19, 1, 2, 6, 1, 1, 2, 1, 1, 2

Offset: 1

Rémy Sigrist, Jul 16 2025

All terms appear infinitely many times: for any n > 0, A161792(n)*2^a(n) also belongs to A161792, say it equals A161792(m) for some m > n, so a(n) = a(m), and we can find as many other occurrences of a(n) as we want.

			a(35) = A000120(2985984) = 6.

Michael S. Branicky, Table of n, a(n) for n = 1..275 (using b-file at A161792)

Cf. A000120, A161792, A325454, A386248.

{ for (n = 1, oo, if (ispower(n, h = hammingweight(n)), print1 (h", "));); }

a(n) = A000120(A161792(n)).

a(65) and beyond from Michael S. Branicky, Jul 21 2025 using A161792

A386248 a(n) is the unique integer k such that A161792(n) = k^A000120(A161792(n)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 32, 6, 64, 128, 12, 256, 512, 24, 3, 1024, 2048, 48, 4096, 8192, 96, 16384, 32768, 192, 6, 65536, 131072, 384, 262144, 524288, 768, 1048576, 2097152, 1536, 12, 4194304, 3, 8388608, 3072, 3, 16777216, 33554432, 6144, 67108864, 134217728

Offset: 1

Rémy Sigrist, Jul 16 2025

			For n = 11: A161792(11) = 144 = 12^2 = 12^A000120(144), so a(11) = 12.

Cf. A000120, A113315, A161792, A386249.

{ for (n = 1, 2^27, if (ispower(n, hammingweight(n), &r), print1 (r", "););); }

A256590 Base-2 Reacher numbers: numbers that are powers of the sum of their base-2 digits.

Original entry on oeis.org

0, 1, 81, 625, 7776, 16807, 46656, 59049, 1679616, 1475789056, 6975757441, 137858491849, 576650390625, 41426511213649, 2384185791015625, 150094635296999121, 10260628712958602189, 32768000000000000000, 243569224216081305397, 655360000000000000000

Offset: 1

Jeffrey Shallit, Apr 03 2015

Named for fictional character Jack Reacher in the series of novels by Lee Child.

There are 2709 terms with 10,000 or fewer digits; a(2709) = 15402^2388. - Charles R Greathouse IV, Nov 26 2016

Lee Child, Bad Luck and Trouble, Delacorte Press, 2007. In this book, the main character, Jack Reacher, likes the number 81 because it is the square of the sum of its base-10 digits.

Hiroaki Yamanouchi and Charles R Greathouse IV, Table of n, a(n) for n = 1..388 (first 141 terms from Hiroaki Yamanouchi)
J. Shallit, Mathematics in a Jack Reacher Novel, blog post, Sep 08 2007.

Cf. A023106, A161792.

fQ[n_] := Block[{wt = DigitCount[n, 2, 1]},
Which[n <= 1, True, wt <= 1, False, True, IntegerQ@ Log[wt, n]]]; Select[Range[10^5], fQ] (* Michael De Vlieger, Apr 04 2015 *)

is(n)= n<=1 || (ispower(n,,&r) && (r==hammingweight(n) || (r^ispower(n=hammingweight(n))==n && n>1))) \\ Michel Marcus and M. F. Hasler, Apr 04 2015

list(lim)=my(v=List([0,1]),H,t); for(e=3,logint(lim\=1,3), for(b=2,min(solve(x=e,lim,x-e*log(x)/log(2)-1),sqrtnint(lim,e)), H=hammingweight(t=b^e); if(H>1 && b^valuation(H,b)==H, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Nov 26 2016

a(1) prepended and a(14)-a(20) added by Hiroaki Yamanouchi, Apr 03 2015

A161793 If b(n) is the number of 0's in the binary representation of n, then the positive integer n is included if n = k^b(n), for some k = integer.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 23, 25, 27, 29, 30, 32, 47, 55, 59, 61, 62, 64, 81, 95, 111, 119, 121, 123, 125, 126, 128, 191, 223, 239, 247, 251, 253, 254, 256, 343, 383, 447, 479, 495, 503, 507, 509, 510, 512, 767, 895, 959, 991, 1007, 1015, 1019, 1021

Offset: 1

Leroy Quet, Jun 19 2009

			25 in binary is 11001, which has two 0's. Since 25 = 5^2, with an exponent of 2, then 25 is included in this sequence.

A161792

More terms from Sean A. Irvine, Mar 14 2010

A285438 Perfect powers that are also the sum of two powers of a prime p.

Original entry on oeis.org

4, 8, 9, 16, 32, 36, 64, 128, 144, 256, 324, 512, 576, 1024, 2048, 2304, 2744, 2916, 4096, 8192, 9216, 16384, 26244, 32768, 36864, 65536, 131072, 147456, 236196, 262144, 524288, 589824, 941192, 1048576, 2097152, 2125764, 2359296, 4194304, 8388608, 9437184

Offset: 1

Michael Josephy, Apr 18 2017

nonn

Integers n such that there exist integers i, j, k, m, p with i, j >= 0, m, k >= 2 and p prime, such that n = m^k = p^i + p^j.
These are numbers of the form 2^r = 2^(r-1) + 2^(r-1) when r >= 2, numbers of the form (3*2^r)^2 = 2^(2*r) + 2^(2*r+3) and numbers of the form (2*p^r)^k = p^(r*k) + p^(r*k+1) when p = 2^k - 1 is a Mersenne prime. [Edited by Jinyuan Wang, Nov 30 2019]
If n = p^i + p^j is a term with exactly two sets of integer solutions (p, i, j), where i <= j, then n must be 36 = 6^2 = 2^2 + 2^5 = 3^2 + 3^3 or of the form 2^k = 2^(k-1) + 2^(k-1) = p^0 + p^1 where p = 2^k - 1 is a Mersenne prime. There is no n = p^i + p^j in this sequence with at least three sets of integer solutions (p, i, j), where i <= j. - Jinyuan Wang, Nov 30 2019

			324 = 18^2 = 3^4 + 3^5.

Robert Israel, Table of n, a(n) for n = 1..6655
W. Weakley, Problem 11936, Amer. Math. Monthly, 123 (2016), 941.

Cf. A000668, A048645, A072103, A161792, A282550.

N:= 10^9: # to get all terms <= N
R1:= {seq(2^i,i=2..ilog2(N))}:
R2:= {seq(9*2^(2*r), r=0..ilog2(floor(N/9))/2)}:
R3:= {seq(seq(2^k*(2^k-1)^(r*k),r=1..floor(log[2^k-1](N/2^k)/k)),k=select(t -> isprime(2^t-1),[$2..ilog2(N)]))}:
sort(convert(R1 union R2 union R3, list)); # Robert Israel, Apr 25 2017

upto(nn) = {my(v=List([]), k=1); for(r=2, logint(nn, 2), listput(v, 2^r)); for(r=0, logint(nn\9, 4), listput(v, 9*4^r)); while((2*2^k-2)^kJinyuan Wang, Nov 30 2019

a(19)-a(40) from Robert Israel, Apr 25 2017

A386372 Odd numbers of the form k^m that have m binary digits 1.

Original entry on oeis.org

1, 9, 729, 4782969, 14348907, 19073486328125, 363797880709171295166015625

Offset: 1

David A. Corneth, Jul 19 2025

5^89 is a term.

			9 = 3^2 = 1001_2 is in the sequence since it is a second power and it has two binary digits 1.
729 = 3^6 = 1011011001_2 is in the sequence since it is a sixth power and it has six binary digits 1.

David A. Corneth, PARI program

Cf. A000120, A001597, A075109, A161792.

With[{nn = 2^30}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[OddQ[#], 1 < #2 == DigitCount[#1, 2, 1]] & @@ {#, GCD @@ FactorInteger[#][[;; , -1]]} &] ] (* Michael De Vlieger, Jul 21 2025 *)

PARI
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\\ See Corneth link
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cp's OEIS Frontend

A386249 a(n) is the Hamming weight of A161792(n).

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A386248 a(n) is the unique integer k such that A161792(n) = k^A000120(A161792(n)).

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A256590 Base-2 Reacher numbers: numbers that are powers of the sum of their base-2 digits.

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Mathematica

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A161793 If b(n) is the number of 0's in the binary representation of n, then the positive integer n is included if n = k^b(n), for some k = integer.

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A285438 Perfect powers that are also the sum of two powers of a prime p.

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Maple

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A386372 Odd numbers of the form k^m that have m binary digits 1.

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Mathematica

PARI