A001221 -id:A001221 - cp's OEIS frontend

A212164 Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k)).

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Matthew Vandermast, May 22 2012

			40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (namely, 3 and 1, although the 1 is often left implicit).   2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 belongs to the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212167.
See also A212165, A212166, A212168.
Cf. A001221, A050326, A051903, A225230.
Subsequence of A188654.

import Data.List (elemIndices)
a212164 n = a212164_list !! (n-1)
a212164_list = map (+ 1) $ findIndices (< 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] > Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); #e && vecmax(e) > #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) < 0; A050326(a(n)) = 0. - Reinhard Zumkeller, May 03 2013

A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Matthew Vandermast, May 22 2012

nonn

Union of A212166 and A212168. Includes numerous subsequences that are subsequences of neither A212166 nor A212168.

			40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although the 1 is often left implicit).  2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 does not belong to the sequence. But 10 = 2^1*5^1 and 20 = 2^2*5^1 belong, since the maximal exponents in their prime factorizations are 1 and 2 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212164. See also A212165.
Subsequences (none of which are subsequences of A212166 or A212168) include A002110, A051451, A129912, A179983, A181826, A181827, A182862, A182863. Includes all members of A003418.
Cf. A001221, A050326, A051903, A188654, A225230.

import Data.List (findIndices)
a212167 n = a212167_list !! (n-1)
a212167_list = map (+ 1) $ findIndices (>= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

isA212167 := proc(n)
    simplify(A051903(n) <= A001221(n)) ;
end proc:
for n from 1 to 1000 do
    if isA212167(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 06 2021

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] <= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) <= #e; } \\ Amiram Eldar, Sep 09 2024

A225230(a(n)) >= 0; A050326(a(n)) > 0. - Reinhard Zumkeller, May 03 2013

A323171 Numerator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

Original entry on oeis.org

2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 4, 2, 17, 5, 19, 7, 5, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 7, 19, 6, 5, 37, 21, 8, 7, 41, 4, 43, 13, 4, 25, 47, 5, 7, 7, 10, 15, 53, 5, 8, 9, 11, 31, 59, 10, 61, 33, 5, 2, 9, 16, 67, 19, 13, 14, 71, 5, 73, 39, 4, 21, 9, 6, 79, 7, 3, 43, 83, 4, 11, 45, 16, 13, 89, 10, 10, 25, 17, 49, 12, 5

Offset: 2

Antti Karttunen, Jan 05 2019

			Fractions begins with 2, 3, 2, 5, 5/2, 7, 2, 3, 7/2, 11, 5/2, 13, ...

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A323172 (denominators).

Cf. A001221, A008472, A123528.

a[n_] := Numerator[Mean[FactorInteger[n][[;; , 1]]]]; Array[a, 100, 2] (* Amiram Eldar, Sep 17 2024 *)

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A323171(n) = (numerator(A008472(n)/omega(n)));

A323172 Denominator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2

Offset: 2

Antti Karttunen, Jan 05 2019

Antti Karttunen, Table of n, a(n) for n = 2..65539

Cf. A323171 (numerators).

Cf. A001221, A008472, A123529, A322591.

a[n_] := Denominator[Mean[FactorInteger[n][[;; , 1]]]]; Array[a, 100, 2] (* Amiram Eldar, Sep 17 2024 *)

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A323172(n) = (denominator(A008472(n)/omega(n)));

A159560 Minimal recursive sequence beginning with 3 such that A001221(a(n)) = A001221(n).

Original entry on oeis.org

3, 4, 5, 7, 10, 11, 13, 16, 18, 19, 20, 23, 24, 26, 27, 29, 33, 37, 38, 39, 40, 41, 44, 47, 48, 49, 50, 53, 60, 61, 64, 65, 68, 69, 72, 73, 74, 75, 76, 79, 84, 89, 91, 92, 93, 97, 98, 101, 104, 106, 108, 109, 111, 112, 115, 116, 117, 121, 126, 127, 129, 133, 137, 141, 150

Offset: 2

Vladimir Shevelev, Apr 15 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000
V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Cf. A001221, A159559.

A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:
A159560 := proc(n) option remember; if n = 2 then 3; else for a from procname(n-1)+1 do if A001221(a) = A001221(n) then return a; end if; end do: end if; end proc:
seq(A159560(n),n=2..70) ; # R. J. Mathar, Oct 30 2010

a[2] = 3; a[n_] := a[n] = For[an = a[n - 1] + 1, True, an++, If[PrimeNu[an] == PrimeNu[n], Return[an]]];
Table[a[n], {n, 2, 66}] (* Jean-François Alcover, Nov 14 2017 *)

a(n+1) = min{m > a(n): A001221(m) = A001221(n+1)}.

Corrected (4 inserted, 8 removed) and extended beyond 53 by R. J. Mathar, Oct 30 2010

A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104

Offset: 1

Matthew Vandermast, May 22 2012

Union of A212164 and A212166.  Includes numerous subsequences that are subsequences of neither A212164 nor A212166.
Includes all factorials except A000142(3) = 6.
Observation: all terms in DATA section are also the first 65 numbers n whose difference between the arithmetic derivative of n and the sum of the divisors of n is nonnegative. - Omar E. Pol, Dec 19 2012

			10 = 2^1*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although 1s are often left implicit).  2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 does not belong to the sequence. But 20 = 2^2*5^1 and 40 = 2^3*5^1 belong, since the largest exponents in their prime factorizations are 2 and 3 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212168.
See also A212167.
Subsequences (none of which are subsequences of A212164 or A212166) include A000079, A001021, A066120, A087980, A130091, A141586, A166475, A181818, A181823, A181824, A182763, A212169. Also includes all terms in A181813 and A181814.
Cf. A001221, A051903, A188654, A225230.

import Data.List (findIndices)
a212165 n = a212165_list !! (n-1)
a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) >= #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) <= 0. - Reinhard Zumkeller, May 03 2013

A280195 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 8, 11, 19, 28, 47, 72, 116, 182, 289, 460, 724, 1153, 1820, 2891, 4572, 7249, 11482, 18190, 28821, 45651, 72338, 114582, 181549, 287596, 455647, 721847, 1143588, 1811748, 2870239, 4547232, 7203907, 11412882, 18080833, 28644680, 45380392, 71894054, 113898439, 180443915, 285869028, 452888824, 717490903, 1136687237

Offset: 0

Ilya Gutkovskiy, Dec 28 2016

nonn

Number of compositions (ordered partitions) into prime powers (1 excluded).

			a(6) = 4 because we have [4, 2], [3, 3], [2, 4] and [2, 2, 2].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A001221, A023360, A023894, A246655.

nmax = 48; CoefficientList[Series[1/(1 - Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} floor(1/omega(k))*x^k).

A306071 Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).

Original entry on oeis.org

8, 0, 7, 3, 3, 0, 8, 2, 1, 6, 3, 6, 2, 0, 5, 0, 3, 9, 1, 4, 8, 6, 5, 4, 2, 7, 9, 9, 3, 0, 0, 3, 1, 1, 3, 4, 0, 2, 5, 8, 4, 5, 8, 2, 5, 0, 8, 1, 5, 5, 6, 6, 4, 4, 0, 1, 8, 0, 0, 5, 2, 0, 7, 7, 0, 4, 4, 1, 3, 8, 1, 4, 8, 4, 9, 3, 7, 5, 1, 8, 6, 4, 9, 6, 9, 5, 6, 0, 9, 3, 5, 0, 9, 6, 2, 9, 4, 8, 3, 7, 6, 5, 0, 1, 1, 8

Offset: 0

Amiram Eldar, Jun 19 2018

The constant A that appears in the asymptotic formulae for the sums of the bi-unitary divisor function (A306069) and the bi-unitary totient function (A306070).
The product in Suryanarayana's 1972 paper has a error that was corrected in his 1975 paper.
The probability that 2 randomly selected numbers will be unitary coprime (i.e. their largest common unitary divisor is 1). - Amiram Eldar, Aug 27 2019

			0.80733082163620503914...

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 508), preprint, arXiv:1310.7053 [math.NT] (2014) (see p. 21).

Cf. A000010, A001221, A286324, A306069, A306070.

cc = CoefficientList[Series[Log[1 - (p - 1)/(p^2*(p + 1))] /. p -> 1/x, {x, 0, 36}], x]; f = FindSequenceFunction[cc]; digits = 20; A = Exp[NSum[f[n + 1 // Floor]*(PrimeZetaP[n]), {n, 2, Infinity}, NSumTerms -> 16 digits, WorkingPrecision -> 16 digits]]; RealDigits[A, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
$MaxExtraPrecision = 1000; Do[Print[Zeta[2] * Exp[-N[Sum[q = Expand[(2*p^2 - 2*p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}], 120]]], {t, 300, 1000, 100}] (* Vaclav Kotesovec, May 29 2020 *)

prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ Amiram Eldar, Mar 18 2021

Equals Product_{p prime} (1 - (p-1)/(p^2 * (p+1))).

Equals zeta(2) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4).

a(1)-a(20) from Jean-François Alcover, Jun 19 2018
a(20)-a(24) from Jon E. Schoenfield, May 27 2019
More terms from Vaclav Kotesovec, May 29 2020

A328959 a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.

Original entry on oeis.org

-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 02 2019. The idea for this sequence came from Mats Granvik

sign

Conjecture: All terms are nonnegative except for a(1) = -1.

			a(72) = sigma_0(72) - 2 - (omega(72) - 1) * nu(72) = 12 - 2 - (5 - 1) * 2 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

The positions of positive terms are conjectured to be A320632.
Positions of first appearances are A328963.
omega(n) * nu(n) is A113901(n).
(omega(n) - 1) * nu(n) is A307409.
sigma_0(n) - omega(n) * nu(n) is A328958(n).
Cf. A000005, A001221, A001222, A112798, A124010, A307408, A323023, A328956, A328960, A328961, A328962, A328965.

Table[DivisorSigma[0,n]-2-(PrimeOmega[n]-1)*PrimeNu[n],{n,100}]

A307408(n) = 2+((bigomega(n)-1)*omega(n));
A328959(n) = (numdiv(n) - A307408(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A000005(n) - A307408(n). - Antti Karttunen, Nov 17 2019

A328963 Smallest k such that n = sigma_0(k) - ((bigomega(k)-1)*omega(k)), where sigma_0 = A000005, omega = A001221, bigomega = A001222.

Original entry on oeis.org

1, 2, 36, 72, 144, 180, 576, 420, 360, 864, 1296, 720, 36864, 1080, 1440, 1260, 5184, 1800, 2160, 3360, 5760, 15552, 4620, 2520, 150994944, 6480, 5400, 13440, 8640, 6300, 9663676416, 5040, 12960, 9240, 331776, 7560, 186624, 248832, 34560, 10080, 1327104, 13860

Offset: 1

Gus Wiseman, Nov 02 2019

nonn

a(n) = smallest k for which A328959(k) = n-2. a(31) > 2^28. - Antti Karttunen, Nov 17 2019

a(n) <= 2^(n-1)*3^2, with equality for n = 3, 4, 5, 7, 13, 25, 31, 43,... . - Giovanni Resta, Nov 18 2019

			The sequence of terms together with their prime signatures begins:
        1: ()
        2: (1)
       36: (2,2)
       72: (3,2)
      144: (4,2)
      180: (2,2,1)
      576: (6,2)
      420: (2,1,1,1)
      360: (3,2,1)
      864: (5,3)
     1296: (4,4)
      720: (4,2,1)
    36864: (12,2)
     1080: (3,3,1)
     1440: (5,2,1)
     1260: (2,2,1,1)
     5184: (6,4)
     1800: (3,2,2)
     2160: (4,3,1)
     3360: (5,1,1,1)
     5760: (7,2,1)
    15552: (6,5)
     4620: (2,1,1,1,1)
     2520: (3,2,1,1)
150994944: (24,2)

Positions of first appearances in A328959.
All terms are in A025487.
Cf. A000005, A001221, A001222, A113901, A124010, A307409, A320632, A323023, A328956, A328958, A328963, A328964, A328965.

dat=Table[DivisorSigma[0,n]-(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];
Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]

search_up_to = 2^28;
A307408(n) = 2+((bigomega(n)-1)*omega(n));
A328959(n) = (numdiv(n) - A307408(n));
A328963(search_up_to) = { my(m=Map(),t,lista=List([])); for(n=1,search_up_to,t =
A328959(n); if(!mapisdefined(m,t+2), mapput(m,t+2,n))); for(u=1,oo,if(!mapisdefined(m,u,&t),return(Vec(lista)), listput(lista,t))); };
v328963 = A328963(search_up_to);
A328963(n) = v328963[n]; \\ Antti Karttunen, Nov 17 2019

Definition corrected and terms a(25) - a(30) added by Antti Karttunen, Nov 17 2019

a(31)-a(42) from Giovanni Resta, Nov 18 2019

cp's OEIS Frontend

A212164 Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k)).

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A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

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A323171 Numerator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

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A323172 Denominator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

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A159560 Minimal recursive sequence beginning with 3 such that A001221(a(n)) = A001221(n).

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A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

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A280195 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

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