A061142 -id:A061142 - cp's OEIS frontend

A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

1, 2, 3, 4, 7, 6, 15, 8, 9, 14, 56, 12, 101, 30, 21, 16, 297, 18, 490, 28, 45, 112, 1255, 24, 49, 202, 27, 60, 4565, 42, 6842, 32, 168, 594, 105, 36, 21637, 980, 303, 56, 44583, 90, 63261, 224, 63, 2510, 124754, 48, 225, 98, 891, 404, 329931, 54, 392, 120

Offset: 1

Gus Wiseman, Sep 05 2025

			The a(1) = 1 through a(7) = 15 ways:
  (1)  (2)   (3)    (2)(2)    (5)      (2)(3)     (7)
       (11)  (21)   (11)(2)   (32)     (11)(3)    (43)
             (111)  (2)(11)   (41)     (2)(21)    (52)
                    (11)(11)  (221)    (11)(21)   (61)
                              (311)    (2)(111)   (322)
                              (2111)   (11)(111)  (331)
                              (11111)             (421)
                                                  (511)
                                                  (2221)
                                                  (3211)
                                                  (4111)
                                                  (22111)
                                                  (31111)
                                                  (211111)
                                                  (1111111)

For constant partitions we have A061142, for prime indices A355731.
For prime indices instead of factors we have A299200.
The version for distinct choices is A387133, zeros A387326.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387110 counts choices of distinct distinct integer partitions of each prime index.
Cf. A063834, A120383, A299201, A335433, A335448, A355741, A357977, A383706, A387115, A387120, A387136.

Table[Length[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]]],{n,30}]

A079707 In prime factorization of n replace odd primes with their prime predecessor.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 5, 8, 4, 6, 7, 8, 11, 10, 6, 16, 13, 8, 17, 12, 10, 14, 19, 16, 9, 22, 8, 20, 23, 12, 29, 32, 14, 26, 15, 16, 31, 34, 22, 24, 37, 20, 41, 28, 12, 38, 43, 32, 25, 18, 26, 44, 47, 16, 21, 40, 34, 46, 53, 24, 59, 58, 20, 64, 33, 28, 61, 52, 38, 30, 67, 32, 71, 62, 18

Offset: 1

Reinhard Zumkeller, Jan 31 2003

Result after A061395(n)-1 iterations = A061142(n).

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

Cf. A001222, A005408, A000079, A003586, A003961, A061142, A061395, A064989, A151799.

f[p_, e_] := If[p == 2, 2, NextPrime[p, -1]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, f[i,1], precprime(f[i,1]-1))^f[i,2]);} \\ Amiram Eldar, Nov 29 2022

a(n) <= n; a(n) < n iff n > 1 is odd; a(n) = n iff n = 2^k.
A001222(a(n)) = A001222(n).
For 3-smooth numbers: a(2^i * 3^j) = 2^(i+j).
Multiplicative with 2->2 and prime(k)->prime(k-1), k>1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime > 2} ((p^2-p)/(p^2 - prevprime(p))) = 0.3310558934..., where prevprime is A151799. - Amiram Eldar, Nov 29 2022

A089693 Numbers n such that phi(n) = 2^bigomega(n).

Original entry on oeis.org

1, 3, 10, 20, 30, 40, 60, 80, 120, 160, 240, 320, 480, 640, 960, 1280, 1920, 2560, 3840, 5120, 7680, 10240, 15360, 20480, 30720, 40960, 61440, 81920, 122880, 163840, 245760, 327680, 491520, 655360, 983040

Offset: 1

Benoit Cloitre, Jan 06 2004

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

Cf. A000010, A061142, A001222, A070004.

Select[Range[100000],EulerPhi[#]==2^PrimeOmega[#]&] (* or *) Join[{1,3,10},LinearRecurrence[{0,2},{20,30},20]] (* Harvey P. Dale, Mar 01 2012 *)

a(n) = 2*a(n-2) for n>5. [Harvey P. Dale, Mar 01 2012]
For n > 2, A001222(a(n)) = 1 + floor(n/2). - Enrique Pérez Herrero, Mar 28 2012
For n > 1, a(2n) = 5*2^n and a(2n+1) = 15*2^(n-1). - Enrique Pérez Herrero, Mar 28 2012

More terms from Harvey P. Dale, Mar 01 2012

A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 12, 8, 16, 16, 20, 16, 24, 24, 32, 16, 32, 32, 36, 32, 48, 40, 44, 32, 64, 48, 64, 48, 56, 64, 60, 32, 80, 64, 96, 64, 72, 72, 96, 64, 80, 96, 84, 80, 128, 88, 92, 64, 144, 128

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.

Cf. A001222, A001620, A003958, A061142, A069205.

f:= proc(n) local f;
  mul((2*(f[1]-1))^f[2], f = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, May 19 2016

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*2^(PrimeOmega[m]), {m, 1, 100}](* G. C. Greubel, May 19 2016, based on A003958 *)
f[p_, e_] := (2*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*(p-1)*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 08 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) * A003958(n) = 2^bigomega(n) * A003958(n) = 2^A001222(n) * A003958(n).
Dirichlet g.f.: Product_{p prime} 1/(1 - 2*(p-1)*p^(-s)). - Robert Israel, May 19 2016
From Vaclav Kotesovec, Mar 08 2023: (Start)
Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - (2 - p^(2-s))/(p^s-2*p+2)).
Let f(s) = Product_{p prime} (1 - (2 - p^(2-s)) / (p^s - 2*p + 2)).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - 2/(p^2 - 2*p + 2)) = 0.353804459718477500968617797456682002952375753701841967763205003892191...,
f'(2) = f(2) * Sum_{p prime} 2*log(p) / ((p-1) * (p^2 - 2*p + 2)) = 0.350193097012820163529213089258238034020398107720137317340667886409682...
 and gamma is the Euler-Mascheroni constant A001620. (End)

A167294 Totally multiplicative sequence with a(p) = 2*(p-2) for prime p.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 10, 0, 4, 0, 18, 0, 22, 0, 12, 0, 30, 0, 34, 0, 20, 0, 42, 0, 36, 0, 8, 0, 54, 0, 58, 0, 36, 0, 60, 0, 70, 0, 44, 0, 78, 0, 82, 0, 24, 0, 90, 0, 100, 0, 60, 0, 102, 0, 108, 0, 68, 0, 114, 0, 118, 0, 40, 0, 132, 0, 130, 0, 84, 0, 138, 0, 142, 0

Offset: 1

Jaroslav Krizek, Nov 01 2009

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

Cf. A001222, A061142, A166586.

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 06 2016 *)
f[p_, e_] := (2*(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) = (2*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A061142(n) * A166586(n) = 2^bigomega(n) * A166586(n) = 2^A001222(n) * A166586(n).

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).

A167312 Totally multiplicative sequence with a(p) = 2*(p-3) for prime p.

Original entry on oeis.org

1, -2, 0, 4, 4, 0, 8, -8, 0, -8, 16, 0, 20, -16, 0, 16, 28, 0, 32, 16, 0, -32, 40, 0, 16, -40, 0, 32, 52, 0, 56, -32, 0, -56, 32, 0, 68, -64, 0, -32, 76, 0, 80, 64, 0, -80, 88, 0, 64, -32, 0, 80, 100, 0, 64, -64, 0, -104, 112, 0, 116, -112, 0, 64, 80, 0, 128, 112

Offset: 1

Jaroslav Krizek, Nov 01 2009

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

Cf. A001222, A061142, A166589.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 08 2016 *)
f[p_, e_] := (2*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Multiplicative with a(p^e) = (2*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A061142(n) * A166589(n) = 2^bigomega(n) * A166589(n) = 2^A001222(n) * A166589(n).

A167321 Totally multiplicative sequence with a(p) = 2*(p+3) for prime p.

Original entry on oeis.org

1, 10, 12, 100, 16, 120, 20, 1000, 144, 160, 28, 1200, 32, 200, 192, 10000, 40, 1440, 44, 1600, 240, 280, 52, 12000, 256, 320, 1728, 2000, 64, 1920, 68, 100000, 336, 400, 320, 14400, 80, 440, 384, 16000, 88, 2400, 92, 2800, 2304, 520, 100, 120000, 400, 2560

Offset: 1

Jaroslav Krizek, Nov 01 2009

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

Cf. A001222, A061142, A166591.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (2*(p+3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

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

a(n) = A061142(n) * A166591(n) = 2^bigomega(n) * A166591(n) = 2^A001222(n) * A166591(n).

A167330 Totally multiplicative sequence with a(p) = 2*(2p-1) = 4p-2 for prime p.

Original entry on oeis.org

1, 6, 10, 36, 18, 60, 26, 216, 100, 108, 42, 360, 50, 156, 180, 1296, 66, 600, 74, 648, 260, 252, 90, 2160, 324, 300, 1000, 936, 114, 1080, 122, 7776, 420, 396, 468, 3600, 146, 444, 500, 3888, 162, 1560, 170, 1512, 1800, 540, 186, 12960, 676, 1944

Offset: 1

Jaroslav Krizek, Nov 01 2009

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

Cf. A001222, A061142, A166651.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((2*fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (4*p-2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, f[k,1] = 4*f[k,1]-2;); factorback(f);} \\ Michel Marcus, Jun 06 2016

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

a(n) = A061142(n) * A166651(n) = 2^bigomega(n) * A166651(n) = 2^A001222(n) * A166651(n).

A167331 Totally multiplicative sequence with a(p) = 2*(3p-1) = 6p-2 for prime p.

Original entry on oeis.org

1, 10, 16, 100, 28, 160, 40, 1000, 256, 280, 64, 1600, 76, 400, 448, 10000, 100, 2560, 112, 2800, 640, 640, 136, 16000, 784, 760, 4096, 4000, 172, 4480, 184, 100000, 1024, 1000, 1120, 25600, 220, 1120, 1216, 28000, 244, 6400, 256, 6400, 7168, 1360, 280

Offset: 1

Jaroslav Krizek, Nov 01 2009

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

Cf. A001222, A061142, A166652.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((3*fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (6*p-2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, f[k,1] = 6*f[k,1]-2;); factorback(f);} \\ Michel Marcus, Jun 06 2016

Multiplicative with a(p^e) = (2*(3p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(3*p(k)-1))^e(k).

a(n) = A061142(n) * A166652(n) = 2^bigomega(n) * A166652(n) = 2^A001222(n) * A166652(n).

cp's OEIS Frontend

A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

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A079707 In prime factorization of n replace odd primes with their prime predecessor.

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A089693 Numbers n such that phi(n) = 2^bigomega(n).

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Extensions

A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p.

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A167294 Totally multiplicative sequence with a(p) = 2*(p-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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A167312 Totally multiplicative sequence with a(p) = 2*(p-3) for prime p.

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A167321 Totally multiplicative sequence with a(p) = 2*(p+3) for prime p.

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A167330 Totally multiplicative sequence with a(p) = 2*(2p-1) = 4p-2 for prime p.

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