A166590 -id:A166590 - cp's OEIS frontend

A322362 a(n) = gcd(n, A166590(n)), where A166590 is completely multiplicative with a(p) = p+2 for prime p.

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 5, 16, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 10, 1, 32, 1, 2, 7, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 20, 1, 2, 9, 64, 5, 2, 1, 4, 1, 14, 1, 8, 1, 2, 5, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 10, 1, 4, 3, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 105

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001359, A006512, A037074, A066086, A166590, A322361.

a[n_] := If[n == 1, 1, GCD[n, Times@@ ((First[#]+2)^Last[#] &/@FactorInteger[n])]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018~ *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));

a(n) = gcd(n, A166590(n)).

a(A037074(n)) = A006512(n).

A322354 Greatest common divisor of product p and product (p+2), where p ranges over distinct prime divisors of n; a(n) = gcd(A007947(n), A166590(A007947(n))).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 7, 2, 1, 2, 3, 2, 1, 6, 1, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 10, 1, 2, 3, 2, 5, 2, 1, 2, 1, 14, 1, 2, 1, 2, 5, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 10, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 105

Offset: 1

Antti Karttunen, Dec 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..15015
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A007947, A166590, A322356, A322357, A322362.

Cf. also A066086.

f[n_] := If[n == 1, 1, Times @@ Power @@@ ({#[[1]] + 2, #[[2]]} & /@ FactorInteger[n])]; rad[n_] := Times @@ (First@# & /@ FactorInteger@n); a[n_] := GCD[rad[n], f[rad[n]]]; Array[a, 120] (* Amiram Eldar, Dec 16 2018 *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));
A007947(n) = factorback(factorint(n)[, 1]);
A322354(n) = A322362(A007947(n));
\\ Alternatively as:
A322354(n) = gcd(A007947(n), A166590(A007947(n)));

a(n) = A322362(A007947(n)) = gcd(A007947(n), A166590(A007947(n))).

a(n) = A322356(n) * A322357(n).

A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.

Original entry on oeis.org

1, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144, 68

Offset: 1

Marc LeBrun

Completely multiplicative.

Sum of divisors of n with multiplicity. If n = p^m, the number of ways to make p^k as a divisor of n is C(m,k); and sum(C(m,k)*p^k) = (p+1)^k. The rest follows because the function is multiplicative. - Franklin T. Adams-Watters, Jan 25 2010

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Index to divisibility sequences

Apart from initial terms, same as A064478.
Cf. A003958, A027746, A036987, A061142, A163407.
Cf. A063441, A165824, A165825, A165826, A165827.
Cf. A165828, A165829, A165831, A167350, A166590.
Cf. A166642, A166643, A166644, A166645, A166646.
Cf. A166647, A166649, A166650, A166586, A165830.
Cf. A167351, A168065, A168066.

a003959 1 = 1
a003959 n = product $ map (+ 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012

a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); a /@ Range[67] (* Jean-François Alcover, Apr 22 2011 *)

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X-p*X))[n]) /* Ralf Stephan */

Multiplicative with a(p^e) = (p+1)^e. - David W. Wilson, Aug 01 2001
Sum_{n>0} a(n)/n^s = Product_{p prime} 1/(1-p^(-s)-p^(1-s)) (conjectured). - Ralf Stephan, Jul 07 2013
This follows from the absolute convergence of the sum (compare with a(n) = n^2) and the Euler product for completely multiplicative functions. Convergence occurs for at least Re(s)>3. - Thomas Anton, Jul 15 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065488/2 = 1/(2*A005596) = 1.3370563627850107544802059152227440187511993141988459926... - Vaclav Kotesovec, Jul 17 2021
From Thomas Scheuerle, Jul 19 2021: (Start)
a(n) = gcd(A166642(n), A166643(n)).
a(n) = A166642(n)/A061142(n).
a(n) = A166643(n)/A165824(n).
a(n) = A166644(n)/A165825(n).
a(n) = A166645(n)/A165826(n).
a(n) = A166646(n)/A165827(n).
a(n) = A166647(n)/A165828(n).
a(n) = A166649(n)/A165830(n).
a(n) = A166650(n)/A165831(n).
a(n) = A167351(n)/A166590(n). (End)
Dirichlet g.f.: zeta(s-1) * Product_{primes p} (1 + 1/(p^s - p - 1)). - Vaclav Kotesovec, Aug 22 2021

Definition reedited (with formula) by Daniel Forgues, Nov 17 2009

A322357 a(n) = A322354(n) / A322356(n).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Dec 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

f[n_] := If[n == 1, 1, Times @@ Power @@@ ({#[[1]] + 2, #[[2]]} & /@ FactorInteger [n])]; rad[n_] := Times @@ (First@# & /@ FactorInteger@n); fun[p_, n_] := If[ PrimeQ[p + 2] && Divisible[n, p + 2], p + 2, 1]; a[n_] := GCD[rad[n], f[rad[n]]]/ Times @@ (fun[#, n] & /@ FactorInteger[n][[;; , 1]]); Array[a, 120] (* Amiram Eldar, Dec 16 2018 *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));
A007947(n) = factorback(factorint(n)[, 1]);
A322354(n) = A322362(A007947(n));
A322356(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(isprime(f[i,1]+2)&&!(n%(f[i,1]+2)), m *= (f[i,1]+2))); (m); };
A322357(n) = (A322354(n)/A322356(n));

a(n) = A322354(n) / A322356(n).

A167351 Totally multiplicative sequence with a(p) = (p+1)*(p+2) = p^2+3p+2 for prime p.

Original entry on oeis.org

1, 12, 20, 144, 42, 240, 72, 1728, 400, 504, 156, 2880, 210, 864, 840, 20736, 342, 4800, 420, 6048, 1440, 1872, 600, 34560, 1764, 2520, 8000, 10368, 930, 10080, 1056, 248832, 3120, 4104, 3024, 57600, 1482, 5040, 4200, 72576, 1806, 17280, 1980, 22464

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)

Multiplicative with a(p^e) = ((p+1)*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+1)*(p(k)+2))^e(k). a(n) = A003959(n) * A166590(n).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 3*p + 1)) = 1.224476389903759550811745481197762941643093896189832037452375111814242433... - Vaclav Kotesovec, Sep 20 2020

A167303 Totally multiplicative sequence with a(p) = 2*(p+2) for prime p.

Original entry on oeis.org

1, 8, 10, 64, 14, 80, 18, 512, 100, 112, 26, 640, 30, 144, 140, 4096, 38, 800, 42, 896, 180, 208, 50, 5120, 196, 240, 1000, 1152, 62, 1120, 66, 32768, 260, 304, 252, 6400, 78, 336, 300, 7168, 86, 1440, 90, 1664, 1400, 400, 98, 40960, 324, 1568

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A061142, A166590.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (2*(p+2))^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+2))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)+2))^e(k).

a(n) = A061142(n) * A166590(n) = 2^bigomega(n) * A166590(n) = 2^A001222(n) * A166590(n).

A167304 Totally multiplicative sequence with a(p) = 3*(p+2) for prime p.

Original entry on oeis.org

1, 12, 15, 144, 21, 180, 27, 1728, 225, 252, 39, 2160, 45, 324, 315, 20736, 57, 2700, 63, 3024, 405, 468, 75, 25920, 441, 540, 3375, 3888, 93, 3780, 99, 248832, 585, 684, 567, 32400, 117, 756, 675, 36288, 129, 4860, 135, 5616, 4725, 900, 147, 311040, 729, 5292

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A165824, A166590.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (3*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Multiplicative with a(p^e) = (3*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)+2))^e(k).

a(n) = A165824(n) * A166590(n) = 3^bigomega(n) * A166590(n) = 3^A001222(n) * A166590(n).

A167305 Totally multiplicative sequence with a(p) = 4*(p+2) for prime p.

Original entry on oeis.org

1, 16, 20, 256, 28, 320, 36, 4096, 400, 448, 52, 5120, 60, 576, 560, 65536, 76, 6400, 84, 7168, 720, 832, 100, 81920, 784, 960, 8000, 9216, 124, 8960, 132, 1048576, 1040, 1216, 1008, 102400, 156, 1344, 1200, 114688, 172, 11520, 180, 13312, 11200, 1600

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A165825, A166590.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*4^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (4*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Multiplicative with a(p^e) = (4*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)+2))^e(k).

a(n) = A165825(n) * A166590(n) = 4^bigomega(n) * A166590(n) = 4^A001222(n) * A166590(n).

A167306 Totally multiplicative sequence with a(p) = 5*(p+2) for prime p.

Original entry on oeis.org

1, 20, 25, 400, 35, 500, 45, 8000, 625, 700, 65, 10000, 75, 900, 875, 160000, 95, 12500, 105, 14000, 1125, 1300, 125, 200000, 1225, 1500, 15625, 18000, 155, 17500, 165, 3200000, 1625, 1900, 1575, 250000, 195, 2100, 1875, 280000, 215, 22500, 225, 26000

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A165826, A166590.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*5^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (5*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Multiplicative with a(p^e) = (5*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (5*(p(k)+2))^e(k).

a(n) = A165826(n) * A166590(n) = 5^bigomega(n) * A166590(n) = 5^A001222(n) * A166590(n).

A167307 Totally multiplicative sequence with a(p) = 6*(p+2) for prime p.

Original entry on oeis.org

1, 24, 30, 576, 42, 720, 54, 13824, 900, 1008, 78, 17280, 90, 1296, 1260, 331776, 114, 21600, 126, 24192, 1620, 1872, 150, 414720, 1764, 2160, 27000, 31104, 186, 30240, 198, 7962624, 2340, 2736, 2268, 518400, 234, 3024, 2700, 580608, 258, 38880, 270

Offset: 1

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A165827, A166590.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*6^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (6*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Multiplicative with a(p^e) = (6*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (6*(p(k)+2))^e(k).

a(n) = A165827(n) * A166590(n) = 6^bigomega(n) * A166590(n) = 6^A001222(n) * A166590(n).

cp's OEIS Frontend

A322362 a(n) = gcd(n, A166590(n)), where A166590 is completely multiplicative with a(p) = p+2 for prime p.

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A322354 Greatest common divisor of product p and product (p+2), where p ranges over distinct prime divisors of n; a(n) = gcd(A007947(n), A166590(A007947(n))).

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A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.

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A322357 a(n) = A322354(n) / A322356(n).

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A167351 Totally multiplicative sequence with a(p) = (p+1)*(p+2) = p^2+3p+2 for prime p.

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A167303 Totally multiplicative sequence with a(p) = 2*(p+2) for prime p.

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A167304 Totally multiplicative sequence with a(p) = 3*(p+2) for prime p.

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A167305 Totally multiplicative sequence with a(p) = 4*(p+2) for prime p.

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A167306 Totally multiplicative sequence with a(p) = 5*(p+2) for prime p.