A000079 -id:A000079 - cp's OEIS frontend

A258250 Primitive weird numbers (pwn) (A002975) whose abundance (A033880) is a power of 2 (A000079).

70, 836, 4030, 5830, 7192, 7912, 10792, 17272, 45356, 83312, 91388, 113072, 254012, 388076, 786208, 1713592, 4145216, 4559552, 4632896, 9928792, 11547352, 13086016, 15126992, 17999992, 29465852, 29581424, 34869056, 37111168, 38546576, 74899952, 89283592, 95327216

Offset: 1

Robert G. Wilson v, Jun 19 2015

nonn

Number of terms < 10^n: 0, 1, 2, 6, 11, 15, 20, 32, 38, 48, 65, ..., .
Of the total of 499 terms < 10^11 which are pwn, only about 13% have an abundance which are powers of two.
Least term whose abundance has an exponent, e, of two > 1: 70, 836, 7192, 83312, 786208, 4145216, 98196134272, 4559552, 37111168, 22889716736, 141145802752, ?13?, 3307637248, ?15?, 154153326592, ..., .
Least term which has k prime factors, not counting multiplicity > 2: 70, 4030, 29465852, 44257207676, ..., .
Least term which has k prime factors, counting multiplicity > 2: 70, 836, 7192, 83312, 786208, 4145216, 37111168, 270788864, 2529837568, 22889716736, 141145802752, ..., .

			70 is in the sequence since sigma(70) = 144 which yields an abundance of 4 = 2^2.

Robert G. Wilson v, Table of n, a(n) for n = 1..87
Mark Jaybee S. Biano, Bernadette S. Estoque, Lynden Aaron D. Galera, Maria Elena S. Rulloda, Daisy Ann A. Disu, On Weird Numbers, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 157-162.

Cf. A002975, A000079, A033880.

(* copy the terms from A002975 and assign them to lst and then *) f[n_] := DivisorSigma[1, n] - 2n; lst[[#]] & /@ Select[ Range@ 695, IntegerQ@ Log2@ f@ lst[[#]] &]

Corrected by Robert G. Wilson v, Dec 08 2015

A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

Original entry on oeis.org

1, 2, 3, 4, 6, 15, 44

Offset: 1

Gus Wiseman, Jan 07 2020

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
  ()  (2)  (4)    (6)    (12)     (72)
           (2*2)  (2*3)  (2*6)    (8*9)
                         (3*4)    (2*36)
                         (2*2*3)  (3*24)
                                  (4*18)
                                  (6*12)
                                  (2*4*9)
                                  (2*6*6)
                                  (3*3*8)
                                  (3*4*6)
                                  (2*2*18)
                                  (2*3*12)
                                  (2*2*2*9)
                                  (2*2*3*6)
                                  (2*3*3*4)
                                  (2*2*2*3*3)

The same for prime numbers (instead of powers of 2) is A330993,
Factorizations are A001055, with image A045782.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly 2^n factorizations is A330989.
Cf. A033833, A045778, A045783, A181821, A305936, A318283, A318284, A330972, A330973, A330976, A330998, A331022.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],IntegerQ[Log[2,Length[facs[Times@@Prime/@nrmptn[#]]]]]&]

A001055(A181821(a(n))) = 2^k for some k >= 0.

A171449 Powers of 2 (A000079) with 1 changed to -1.

Original entry on oeis.org

-1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Paul Curtz, Dec 09 2009

Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079.

Join[{-1}, 2^Range[50]] (* Paolo Xausa, Aug 27 2025 *)

From Stefano Spezia, Aug 26 2025: (Start)
G.f.: -(1 - 4*x)/(1 - 2*x).
E.g.f.: exp(2*x) - 2. (End)

Edited by N. J. A. Sloane, Dec 17 2009

A087266 a(n) = gcd(2^n, pi(2^n)) = gcd(A000079(n), A007053(n)).

Original entry on oeis.org

1, 2, 4, 2, 1, 2, 1, 2, 1, 4, 1, 4, 4, 4, 8, 2, 1, 8, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 2, 1, 1, 1, 4, 4, 2, 4, 4, 1, 1, 4, 2, 1, 1, 2, 2, 1, 8, 1, 4, 16, 1, 2, 4, 2, 4, 2, 1, 1, 8, 1, 4, 1, 1, 2, 16, 1, 1, 1, 1, 4, 1, 1, 4, 8, 1, 4, 4, 8, 4, 4, 2, 1, 1, 2, 4, 8, 16, 2

Offset: 1

Labos Elemer, Sep 16 2003

nonn

Cf. A000720, A007053.

Table[GCD[2^n, PrimePi[2^n]], {n, 40}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = gcd(2^n, primepi(2^n)); \\ Michel Marcus, Mar 26 2017

a(53)-a(92) from Amiram Eldar, Jun 09 2024

A152537 Convolution sequence: this sequence convolved with A000041 gives powers of 2, (A000079).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 18, 37, 74, 148, 296, 592, 1183, 2366, 4732, 9463, 18926, 37852, 75704, 151408, 302816, 605632, 1211265, 2422530, 4845060, 9690120, 19380241, 38760482, 77520964, 155041928, 310083856, 620167712, 1240335424, 2480670848, 4961341696, 9922683391

Offset: 0

Gary W. Adamson, Dec 06 2008

nonn

Terms are very similar to those of A178841. - Georg Fischer, Mar 23 2019

			a(5) = 9 = 32 - 23 = (32 - ((7,5,3,2,1) dot (1,1,1,2,4)))
(1,1,2,3) convolved with (1,1,1,2) = 8, where (1,1,2,3...) = the first four partition numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A010815, A000041, A000079, A152538, A178841.

a:= proc(n) option remember;
      2^n-add(combinat[numbpart](j)*a(n-j), j=1..n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2025

nmax = 40; CoefficientList[Series[Product[1-x^k, {k, 1, nmax}] / (1-2*x), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 02 2018 *)

/* computation by definition (division of power series) */
N=55;
A000079=vector(N,n,2^(n-1));
S000079=Ser(A000079);
A000041=vector(N,n,numbpart(n-1));
S000041=Ser(A000041);
S152537=S000079/S000041;
A152537=Vec(S152537) /* show terms */  /* Joerg Arndt, Feb 06 2011 */

/* computation using power series eta(x) and 1/(1-2*x) */
x='x+O('x^55);  S152537=eta(x)/(1-2*x);
A152537=Vec(S152537) /* show terms */  /* Joerg Arndt, Feb 06 2011 */

Construct an array of rows such that n-th row = partial sums of (n-1)-th row of A010815: (1, -1, -1, 0, 0, 1, 0, 1,...).
A152537 = sums of antidiagonal terms of the array.
The sequence may be obtained directly from the following set of operations:
Our given sequence = A000041: (1, 1, 2, 3, 5, 7, 11,...). Delete the first "1" then consider (1, 2, 3, 5, 7, 11,...) as an operator Q which we write in reverse with 1,2,3,...terms for each operation. Letting R = the target sequence (1,2,4,8,...); we begin a(0) = 1, a(1) = 1, then perform successive operations of: "next term in (1,2,4,...) - dot product of Q*R" where Q is written right to left and R (the ongoing result) written left to right).
Examples: Given 4 terms Q, R, we have: (5,3,2,1) dot (1,1,1,2) = (5+3+2+2) = 12, which we subtract from 16, = 4.
Given 5 terms of Q,R and A152537, we have (7,5,3,2,1) dot (1,1,1,2,4) = 23 which is subtracted from 32 giving 9. Continue with analogous operations to generate the series.
a(n) = Sum_{j=0..n} A010815(j)*2^(n-j). G.f.: A000079(x)/A000041(x) = A010815(x)/(1-2x), where A......(x) denotes the g.f. of the associated sequence. - R. J. Mathar, Dec 09 2008
a(n) ~ c * 2^n, where c = A048651 = 0.28878809508660242127889972192923078... - Vaclav Kotesovec, Jun 02 2018
a(n) = 2^n - Sum_{j=1..n} A000041(j)*a(n-j). - Alois P. Heinz, Feb 02 2025

A264980 Base-3 reversal of 2^n: a(n) = A030102(A000079(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 64, 32, 184, 352, 704, 1408, 1880, 2824, 14032, 10328, 56128, 100576, 145784, 189472, 370304, 731752, 4388248, 2924096, 11175712, 15965704, 31930448, 63861880, 383165344, 255439712, 1021772344, 510875648, 2550188248, 5619691648, 9689861048, 17830350904, 79068724264, 34109913224, 192259976368, 133338241880

Offset: 0

Antti Karttunen, Dec 05 2015

			2^5 = 32 in base 3 = "1012" (= A007089(32)) as 1*27 + 1*3 + 2*1 = 32. 2^6 = 64 in base 3 = "2101" as 2*27 + 1*9 + 1*1 = 64. "1012" reversed is "2101" and vice versa, thus a(5) = 64 and a(6) = 32.

Antti Karttunen, Table of n, a(n) for n = 0..512

Leftmost column of A265345.
Cf. A000079, A007089, A030102, A263273, A265210, A265342.
Cf. also A036215.

base(n) = {my(a=[n%3]); while(0Altug Alkan, Dec 29 2015

a(n) = A030102(A000079(n)) = A263273(A000079(n)).

a(0) = 1, for n >= 1, a(n) = A265342(a(n-1)).

A266186 a(n) = A266196(A000079(n)); indices of powers of 2 in A266195.

Original entry on oeis.org

1, 2, 4, 7, 12, 16, 25, 42, 50, 82, 104, 116, 201, 227, 243, 455, 477, 517, 1035, 1093, 1155, 1217, 2599, 2695, 4377, 4491, 4773, 4947, 5137, 13409, 13537, 14125, 14299, 14631, 15123, 34005, 34447, 34781, 36017, 36867, 37289, 37913, 155106, 155700, 157254

Offset: 0

Antti Karttunen, Dec 26 2015

Very likely also the positions of records in A264982.

Rémy Sigrist, C++ program

Cf. A000079, A264982, A266195, A266196, A266187.

a(n) = A266196(A000079(n)).

More terms from Rémy Sigrist, Oct 04 2022

A269373 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 11, 8, 9, 10, 7, 16, 21, 32, 15, 22, 17, 12, 19, 64, 13, 18, 31, 128, 41, 24, 63, 14, 29, 256, 43, 512, 33, 42, 23, 1024, 37, 20, 127, 30, 25, 2048, 35, 48, 61, 34, 255, 4096, 81, 8192, 47, 38, 125, 96, 27, 40, 57, 26, 511, 44, 85, 16384, 1023, 62, 65, 32768, 83, 65536, 45, 82, 2047, 131072, 73, 262144, 39

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269374.
Cf. A000035, A000079, A260438, A260439.
Cf. also A249813, A269383.

a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A133806 Alternate terms of A131708 and A000079.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 8, 5, 16, 10, 32, 21, 64, 43, 128, 86, 256, 171, 512, 341, 1024, 682, 2048, 1365, 4096, 2731, 8192, 5462, 16384, 10923, 32768, 21845, 65536, 43690, 131072, 87381, 262144, 174763, 524288, 349526, 1048576, 699051, 2097152

Offset: 0

Paul Curtz, Jan 06 2008

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,2).

Cf. A000079.

G.f.: -x*(1+x-x^2-x^3+x^4) / ( (2*x^2-1)*(x^4-x^2+1) ). - R. J. Mathar, Jul 16 2015

A153237 a(n) = A000079(n) - A153130(n).

Original entry on oeis.org

0, 0, 0, 0, 9, 27, 63, 126, 252, 504, 1017, 2043, 4095, 8190, 16380, 32760, 65529, 131067, 262143, 524286, 1048572, 2097144, 4194297, 8388603, 16777215, 33554430, 67108860, 134217720, 268435449, 536870907, 1073741823, 2147483646

Offset: 0

Paul Curtz, Dec 21 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,3,-2).

LinearRecurrence[{3,-2,-1,3,-2},{0,0,0,0,9},40] (* Harvey P. Dale, Dec 26 2021 *)

a(n) = 9 *A153234(n). G.f. 9*x^4 / ( (x-1)*(2*x-1)*(1+x)*(x^2-x+1) ). - R. J. Mathar, Dec 17 2012

Definition corrected by Omar E. Pol, Dec 24 2008

Edited by N. J. A. Sloane, Dec 31 2008

cp's OEIS Frontend

A258250 Primitive weird numbers (pwn) (A002975) whose abundance (A033880) is a power of 2 (A000079).

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A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

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A171449 Powers of 2 (A000079) with 1 changed to -1.

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A087266 a(n) = gcd(2^n, pi(2^n)) = gcd(A000079(n), A007053(n)).

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A152537 Convolution sequence: this sequence convolved with A000041 gives powers of 2, (A000079).

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A264980 Base-3 reversal of 2^n: a(n) = A030102(A000079(n)).

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A266186 a(n) = A266196(A000079(n)); indices of powers of 2 in A266195.

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A269373 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).

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