A061142 -id:A061142 - cp's OEIS frontend

A350066 Symmetric square array A(n,k) = A122111(A122111(n) * A122111(k)), n >= 1, k >= 1, read by antidiagonals.

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 7, 10, 10, 7, 6, 7, 10, 11, 9, 11, 10, 7, 8, 11, 14, 14, 14, 14, 11, 8, 9, 12, 13, 15, 13, 15, 13, 12, 9, 10, 15, 20, 22, 22, 22, 22, 20, 15, 10, 11, 14, 21, 18, 17, 21, 17, 18, 21, 14, 11, 12, 13, 22, 25, 28, 26, 26, 28, 25, 22, 13, 12, 13, 20, 17, 21, 33, 30, 19, 30, 33, 21, 17, 20, 13

Offset: 1

Antti Karttunen, Dec 13 2021

A122111 is a self-inverse permutation, so this array represents a binary operation A(.,.) over the positive integers that is isomorphic to multiplication. Its primes are the positive powers of 2 (as defined by standard multiplication): 2, 4, 8, 16, 32, ... . The positive powers of 2, as defined by A(.,.), are the prime numbers as we usually understand them: 2, 3, 5, 7, 11, ... . - Peter Munn, Aug 04 2022

			The top left 15 X 15 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  12, 13, 14,  15,
   2,  3,  5,  6,  7, 10, 11, 12, 15, 14, 13,  20, 17, 22,  21,
   3,  5,  7, 10, 11, 14, 13, 20, 21, 22, 17,  28, 19, 26,  33,
   4,  6, 10,  9, 14, 15, 22, 18, 25, 21, 26,  30, 34, 33,  35,
   5,  7, 11, 14, 13, 22, 17, 28, 33, 26, 19,  44, 23, 34,  39,
   6, 10, 14, 15, 22, 21, 26, 30, 35, 33, 34,  42, 38, 39,  55,
   7, 11, 13, 22, 17, 26, 19, 44, 39, 34, 23,  52, 29, 38,  51,
   8, 12, 20, 18, 28, 30, 44, 27, 50, 42, 52,  45, 68, 66,  70,
   9, 15, 21, 25, 33, 35, 39, 50, 49, 55, 51,  70, 57, 65,  77,
  10, 14, 22, 21, 26, 33, 34, 42, 55, 39, 38,  66, 46, 51,  65,
  11, 13, 17, 26, 19, 34, 23, 52, 51, 38, 29,  68, 31, 46,  57,
  12, 20, 28, 30, 44, 42, 52, 45, 70, 66, 68,  63, 76, 78, 110,
  13, 17, 19, 34, 23, 38, 29, 68, 57, 46, 31,  76, 37, 58,  69,
  14, 22, 26, 33, 34, 39, 38, 66, 65, 51, 46,  78, 58, 57,  85,
  15, 21, 33, 35, 39, 55, 51, 70, 77, 65, 57, 110, 69, 85,  91,

Index entries for sequences computed from indices in prime factorization

Cf. A122111, A297002 (main diagonal), A253550 (after its initial term, gives row 2 / column 2 from the second term onward).
See the formula section for the relationships with A003961, A061142.
Cf. also A003991, A129595, A331590.

up_to = 105;
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A350066sq(n,k) = A122111(A122111(n)*A122111(k));
A350066list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A350066sq(col,(a-(col-1))))); (v); };
v350066 = A350066list(up_to);
A350066(n) = v350066[n]; \\ Antti Karttunen, Dec 13 2021

A(n, A061142(n)) = A003961(n). - Peter Munn, Aug 04 2022

A248692 Fully multiplicative with a(prime(i)) = 2^i; If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime A000040(k) and c_k >= 0 then a(n) = Product_{k >= 1} 2^(k*c_k).

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 16, 8, 16, 16, 32, 16, 64, 32, 32, 16, 128, 32, 256, 32, 64, 64, 512, 32, 64, 128, 64, 64, 1024, 64, 2048, 32, 128, 256, 128, 64, 4096, 512, 256, 64, 8192, 128, 16384, 128, 128, 1024, 32768, 64, 256, 128, 512, 256, 65536, 128, 256, 128, 1024, 2048, 131072, 128, 262144, 4096, 256, 64

Offset: 1

Antti Karttunen, Oct 11 2014

Equally, if n = p_i * p_j * ... * p_k, where p_i, p_j, ..., p_k are the primes A000040(i), A000040(j), ..., A000040(k) in the prime factorization of n (indices i, j, ..., k not necessarily distinct), then a(n) = 2^i * 2^j * 2^k.
a(1) = 1 (empty product).
Fully multiplicative with a(prime(i)) = 2^i.

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A000040, A000079, A003961, A003965, A048675, A056239, A061142, A065446, A122111, A297109.

a:= n-> mul((2^numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..64);  # Alois P. Heinz, Jan 14 2021

a[n_] := Product[{p, e} = pe; (2^PrimePi[p])^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Jan 03 2022 *)

A248692(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = 2^primepi(f[i,1])); factorback(f)); \\ Antti Karttunen, Feb 01 2021

a(n) = 2^A056239(n) = A000079(A056239(n)).
Other identities. For all n >= 1:
a(A122111(n)) = a(n).
a(A000040(n)) = A000079(n).
For all n >= 0:
a(A000079(n)) = A000079(n).
a(n) = Product_{d|n} 2^A297109(d). - Antti Karttunen, Feb 01 2021
Sum_{n>=1} 1/a(n) = A065446. - Amiram Eldar, Dec 24 2022

A165872 Totally multiplicative sequence with a(p^e) = (-2)^e.

Original entry on oeis.org

1, -2, -2, 4, -2, 4, -2, -8, 4, 4, -2, -8, -2, 4, 4, 16, -2, -8, -2, -8, 4, 4, -2, 16, 4, 4, -8, -8, -2, -8, -2, -32, 4, 4, 4, 16, -2, 4, 4, 16, -2, -8, -2, -8, -8, 4, -2, -32, 4, -8, 4, -8, -2, 16, 4, 16, 4, 4, -2, 16, -2, 4, -8, 64, 4, -8, -2, -8, 4, -8, -2, -32, -2, 4, -8, -8, 4, -8, -2, -32, 16, 4, -2, 16, 4, 4, 4, 16

Offset: 1

Jaroslav Krizek, Sep 28 2009

The Dirichlet series is Product_{prime p} 1/(1+2/p^s), related to the Feller-Tornier constants like A065474. - R. J. Mathar, Feb 07 2011

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A061142, A065474, A122803.

(-2)^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 16 2016 *)

a(n) = (-2)^bigomega(n); \\ Altug Alkan, Apr 16 2016

a(n) = A122803(A001222(n)) = (-2)^bigomega(n) = (-2)^A001222(n).

Abs(a(n)) = A061142(n).

A303809 Multiplicative with a(p^k) = 2^a(k).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 16, 2, 4, 4, 8, 2, 8, 2, 8, 8, 4, 2, 32, 4, 8, 4, 8, 2, 8, 4, 8, 4, 4, 2, 16, 2, 4, 8, 16, 4, 8, 2, 8, 4, 8, 2, 16, 2, 4, 8, 8, 4, 8, 2, 32, 16, 4, 2, 16

Offset: 1

Rémy Sigrist, Apr 30 2018

This sequence contains every power of 2; see A303819 for the corresponding least indices.

For any n > 0, a(n) only depends on the prime signature of n.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002110, A014221, A061142, A303819.

a(n) = my (f=factor(n)); prod (i=1, #f~, 2^a(f[i, 2]))

a(n) <= n with equality iff n = A014221(k) for some k >= 0.
a(A002110(k)) = 2^k for any k >= 0.
a(n) = 2 iff n is prime.

A133205 Fully multiplicative with a(p) = p*(p+1)/2 for prime p.

Original entry on oeis.org

1, 3, 6, 9, 15, 18, 28, 27, 36, 45, 66, 54, 91, 84, 90, 81, 153, 108, 190, 135, 168, 198, 276, 162, 225, 273, 216, 252, 435, 270, 496, 243, 396, 459, 420, 324, 703, 570, 546, 405, 861, 504, 946, 594, 540, 828, 1128, 486, 784, 675, 918, 819, 1431, 648, 990, 756

Offset: 1

Jonathan Vos Post, Oct 10 2007

There are analogs with the triangular numbers replaced by some other sequence, but this was chosen because of the parity coincidences of A034953.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A003958, A003959, A003960, A003961, A003964, A034953, A061142, A167338.

f[p_, e_] := (p*(p + 1)/2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Dec 24 2022 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],binomial(f[i,1]+1,2)^f[i,2]) /* Charles R Greathouse IV, Sep 09 2010 */

for(n=1, 100, print1(direuler(p=2, n, 1 + (p^2 + p) / (2/X - p^2 - p))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023

a((p_1)^(e_1)*(p_2)^(e_2)*...*(p_k)^(e_k)) = T(p_1)^(e_1)*T(p_2)^(e_2)*...*T(p_k)^(e_k), where T(i) = A000217(i). a(prime(i)) = A034953(i).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - 2/(p*(p+1)))^(-1) = 2.12007865309570462566... . - Amiram Eldar, Dec 24 2022
Dirichlet g.f.: Product_{p prime} (1 + (p^2 + p) / (2*p^s - p^2 - p)). - Vaclav Kotesovec, Apr 05 2023
a(n) = A167338(n)/A061142(n). - Vaclav Kotesovec, Jan 28 2025
Conjecture: Sum_{k=1..n} a(k) = O(n^3/log(n)). - Vaclav Kotesovec, Jan 28 2025

A166642 Totally multiplicative sequence with a(p) = 2*(p+1) for prime p.

Original entry on oeis.org

1, 6, 8, 36, 12, 48, 16, 216, 64, 72, 24, 288, 28, 96, 96, 1296, 36, 384, 40, 432, 128, 144, 48, 1728, 144, 168, 512, 576, 60, 576, 64, 7776, 192, 216, 192, 2304, 76, 240, 224, 2592, 84, 768, 88, 864, 768, 288, 96, 10368, 256, 864

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A003959, A061142.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*2^(PrimeOmega[n]), {n, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (2*(p+1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

Multiplicative with a(p^e) = (2*(p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)+1))^e(k).

a(n) = A061142(n) * A003959(n) = 2^bigomega(n) * A003959(n) = 2^A001222(n) * A003959(n).

A326376 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: for any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) o f(k)) (where o denotes function composition).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 4, 2, 2, 1, 1, 4, 3, 2, 1, 2, 2, 4, 4, 2, 1, 1, 4, 5, 4, 5, 2, 1, 8, 2, 6, 16, 4, 6, 2, 1, 1, 8, 7, 8, 11, 4, 7, 2, 1, 2, 4, 8, 256, 10, 90, 4, 8, 2, 1, 1, 4, 9, 8, 17, 12, 17, 4, 9, 2, 1, 4, 2, 10, 16, 8, 47250, 14, 512, 4, 10, 2, 1, 1, 8

Offset: 1

Rémy Sigrist, Jul 02 2019

This sequence has connections with A297845.

The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.

			Array T(n, k) begins:
  n\k|  1  2   3    4   5      6   7          8        9       10
  ---+-----------------------------------------------------------
    1|  1  1   1    1   1      1   1          1        1        1
    2|  2  2   2    2   2      2   2          2        2        2
    3|  1  2   3    4   5      6   7          8        9       10
    4|  4  4   4    4   4      4   4          4        4        4
    5|  1  2   5   16  11     90  17        512      625      550
    6|  2  4   6    8  10     12  14         16       18       20
    7|  1  2   7  256  17  47250  29  134217728  5764801  5656750
    8|  8  8   8    8   8      8   8          8        8        8
    9|  1  4   9   16  25     36  49         64       81      100
   10|  2  4  10   32  22    180  34       1024     1250     1100
The corresponding polynomials are:
  f(n)\f(k)| 0 1 x     2 x^2   x+1             x^3   3  2*x     x^2+1
  ---------+---------------------------------------------------------------------
          0| 0 0 0     0 0     0               0     0  0       0
          1| 1 1 1     1 1     1               1     1  1       1
          x| 0 1 x     2 x^2   x+1             x^3   3  2*x     x^2+1
          2| 2 2 2     2 2     2               2     2  2       2
        x^2| 0 1 x^2   4 x^4   x^2+2*x+1       x^6   9  4*x^2   x^4+2*x^2+1
        x+1| 1 2 x+1   3 x^2+1 x+2             x^3+1 4  2*x+1   x^2+2
        x^3| 0 1 x^3   8 x^6   x^3+3*x^2+3*x+1 x^9   27 8*x^3   x^6+3*x^4+3*x^2+1
          3| 3 3 3     3 3     3               3     3  3       3
        2*x| 0 2 2*x   4 2*x^2 2*x+2           2*x^3 6  4*x     2*x^2+2
      x^2+1| 1 2 x^2+1 5 x^4+1 x^2+2*x+2       x^6+1 10 4*x^2+1 x^4+2*x^2+2

See A326377 for the main diagonal of T.

Cf. A006519, A061142, A297473, A297845, A319525.

g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))
f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i,2] * v^(primepi(f[i,1]) - 1))
T(n,k) = g(f(n, f(k)))

For any m, n, k > 0 and any i >= 0:
- T(1, k) = 1,
- T(2^i, k) = 2^i,
- T(3, k) = k,
- T(3^i, k) = k^i,
- T(5, k) = A297473(k),
- T(6, k) = 2*k,
- T(n, 1) = A006519(n),
- T(n, 2) = A061142(n),
- T(n, 3) = n,
- T(n, 5) = A319525(n),
- T(m*n, k) = T(m, k) * T(n, k).

A335073 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 29, 30, 32, 33, 35, 36, 37, 38, 42, 44, 45, 49, 51, 52, 53, 54, 70, 71, 72, 73, 77, 78, 79, 80, 84, 85, 86, 87, 89, 91, 92, 93, 101, 103, 105, 106, 108, 109, 113, 114, 118, 119, 120, 121, 123, 124, 125, 127

Offset: 1

Daniel Suteu, May 22 2020

nonn

Partial sums of A162510.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001221, A001222, A046660, A061142, A069205, A162510.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      2^(bigomega(n)-nops(factorset(n)))+a(n-1))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, May 22 2020

Accumulate[Table[2^(PrimeOmega[n]-PrimeNu[n]),{n,70}]] (* Harvey P. Dale, Aug 14 2020 *)

a(n) = sum(k=1, n, 2^(bigomega(k) - omega(k)));

a(n) = Sum_{k=1..n} A008683(k) * A069205(floor(n/k)).

a(n) = Sum_{k=1..n} A061142(k) * A002321(floor(n/k)).

A349921 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 2 * k^(-s)).

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 14, 6, 6, 2, 18, 2, 6, 6, 34, 2, 18, 2, 18, 6, 6, 2, 46, 6, 6, 14, 18, 2, 22, 2, 74, 6, 6, 6, 58, 2, 6, 6, 46, 2, 22, 2, 18, 18, 6, 2, 114, 6, 18, 6, 18, 2, 46, 6, 46, 6, 6, 2, 70, 2, 6, 18, 166, 6, 22, 2, 18, 6, 22, 2, 150, 2, 6, 18

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A001055, A061142, A070933, A301830, A349922.

A351521 Dirichlet g.f.: Product_{p prime} (1 + 4*p^(-s)).

Original entry on oeis.org

1, 4, 4, 0, 4, 16, 4, 0, 0, 16, 4, 0, 4, 16, 16, 0, 4, 0, 4, 0, 16, 16, 4, 0, 0, 16, 0, 0, 4, 64, 4, 0, 16, 16, 16, 0, 4, 16, 16, 0, 4, 64, 4, 0, 0, 16, 4, 0, 0, 0, 16, 0, 4, 0, 16, 0, 16, 16, 4, 0, 4, 16, 0, 0, 16, 64, 4, 0, 16, 64, 4, 0, 4, 16, 0, 0, 16, 64

Offset: 1

Vaclav Kotesovec, Feb 17 2022

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A074823, A347149.
Cf. A061142, A165824, A165825.
Cf. A008966, A035116.

Table[MoebiusMu[n]^2 * 4^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X))[n], ", "))

Dirichlet g.f.: zeta(s)^4 * Product_{prime p} (1 + (4 - 15*p^s + 20*p^(2*s) - 10*p^(3*s))/p^(5*s)).
a(n) = A008966(n) * A035116(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 4, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

cp's OEIS Frontend

A350066 Symmetric square array A(n,k) = A122111(A122111(n) * A122111(k)), n >= 1, k >= 1, read by antidiagonals.

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A248692 Fully multiplicative with a(prime(i)) = 2^i; If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime A000040(k) and c_k >= 0 then a(n) = Product_{k >= 1} 2^(k*c_k).

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A165872 Totally multiplicative sequence with a(p^e) = (-2)^e.

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A303809 Multiplicative with a(p^k) = 2^a(k).

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A133205 Fully multiplicative with a(p) = p*(p+1)/2 for prime p.

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A166642 Totally multiplicative sequence with a(p) = 2*(p+1) for prime p.

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A335073 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)).

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