A225230 -id:A225230 - cp's OEIS frontend

A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Matthew Vandermast, May 22 2012

			36 = 2^2*3^2 has 2 positive exponents in its prime factorization. The maximal exponent in its prime factorization is also 2. Therefore, 36 belongs to this 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).

Includes subsequences A000040, A006939, A138534, A181555, A181825.
See also A212164, A212165, A212167, A212168.
Cf. A001221, A050326, A051903, A188654 (complement), A225230.

import Data.List (elemIndices)
a212166 n = a212166_list !! (n-1)
a212166_list = map (+ 1) $ elemIndices 0 a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] == Length[f]]; Select[Range[424], 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)) = 1. - Reinhard Zumkeller, May 03 2013

A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142, 143

Offset: 1

Matthew Vandermast, May 22 2012

nonn

A225230(a(n)) > 1; A050326(a(n)) > 1. - Reinhard Zumkeller, May 03 2013

Subsequence of A130092. - Ivan N. Ianakiev, Sep 17 2019

			10 = 2^1*5^1 has 2 distinct prime factors, hence 2 positive exponents in its prime factorization (although the 1s are often left implicit). 2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 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 A212165. See also A212164, A212166-A212167.

Subsequence of A188654.

import Data.List (findIndices)
a212168 n = a212168_list !! (n-1)
a212168_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 *)
Select[Range[200],Max[FactorInteger[#][[All,2]]]Harvey P. Dale, Nov 21 2018 *)

is(n,f=factor(n))=my(e=f[,2]); #e && vecmax(e)<#e \\ Charles R Greathouse IV, Jan 09 2022

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

Original entry on oeis.org

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

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

A188654 Numbers k such that the maximum exponent in its prime factorization does not equal the number of positive exponents (A051903(k) <> A001221(k)).

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 48, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 72, 74, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 108, 110, 111

Offset: 1

Reinhard Zumkeller, May 03 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001221, A212166 (complement), A225230.

Union of A212164 and A212168.

import Data.List (findIndices)
a188654 n = a188654_list !! (n-1)
a188654_list = map (+ 1) $ findIndices (/= 0) a225230_list

q[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Max[e] != Length[e]]; q[1] = False; Select[Range[120], q] (* Amiram Eldar, Sep 08 2024 *)

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

A051903(n) <> A001221(n);

A225230(a(n)) <> 0.

cp's OEIS Frontend

A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

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A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

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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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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)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A188654 Numbers k such that the maximum exponent in its prime factorization does not equal the number of positive exponents (A051903(k) <> A001221(k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula