cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76
Offset: 1

Views

Author

Amiram Eldar, Jul 06 2020

Keywords

Comments

Numbers k such that gcd(A008833(k), k/A008833(k)) = 1.
Numbers whose prime factorization contains exponents that are either 1 or even.
Numbers whose powerful part (A057521) is a square.
First differs from A220218 at n = 227: a(227) = 256 is not a term of A220218.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465).
Complement of A295661. - Vaclav Kotesovec, Jul 07 2020
Differs from A096432 in having or not having 1, 256, 432, 648, 768, 1280, 1728, 1792, 2000, 2160, 2304,... - R. J. Mathar, Jul 22 2020
Equivalently, numbers k whose squarefree part (A007913) is a unitary divisor, or gcd(A007913(k), A008833(k)) = 1. - Amiram Eldar, Oct 09 2022

Examples

			12 is a term since the largest square dividing 12 is 4, and 4 and 12/4 = 3 are coprime.

Links

Crossrefs

A000290, A138302 and A220218 are subsequences.
Cf. A007913, A008833, A057521, A065465, A295661.

Programs

  • Mathematica
    seqQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 1 || EvenQ[#] &];  Select[Range[100], seqQ]
  • PARI
    isok(k) = my(d=k/core(k)); gcd(d, k/d) == 1; \\ Michel Marcus, Jul 07 2020

A295662 Number of odd exponents larger than one in the canonical prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 28 2017

Keywords

Examples

			For n = 24 = 2^3 * 3^1 there are two odd exponents, but only the other is larger than 1, thus a(24) = 1.
For n = 216 = 2^3 * 3^3 there are two odd exponents larger than 1, thus a(216) = 2.

Links

Crossrefs

Cf. A056169, A162642, A295659, A295663, A295664.
Cf. A295661 (positions of nonzero terms).

Programs

Formula

Additive with a(p) = 0, a(p^e) = A000035(e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A000035(A067029(n)) + a(A028234(n)).
a(n) = A162642(n) - A056169(n).
a(n) <= A295659(n).
a(n) = 0 iff A295663(n) = 0, and when A295663(n) > 0, a(n) <= A295663(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2*(p+1)) = 0.122017493776862257491... . - Amiram Eldar, Sep 28 2023

A295663 a(n) = A295664(n) - A056169(n); 2-adic valuation of tau(n) minus the number of unitary prime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 28 2017

Keywords

Links

Crossrefs

Cf. A000005, A007814, A056169, A295662, A295664.
Cf. A295661 (positions of nonzero terms).

Programs

  • Mathematica
    Table[IntegerExponent[DivisorSigma[0, n], 2] - DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)
  • PARI
    a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Formula

Additive with a(p) = 0, a(p^e) = A007814(1+e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A007814(1+A067029(n)) + a(A028234(n)).
a(n) = A295664(n) - A056169(n).
a(n) = 0 iff A295662(n) = 0, and when A295662(n) > 0, a(n) >= A295662(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.22852676306472099280..., where f(x) = -1 + (1-x)*(-x + Sum_{k>=0} x^(2^k-1)/(1-x^(2^k))). - Amiram Eldar, Sep 28 2023

A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 500
Offset: 1

Views

Author

Amiram Eldar, Sep 11 2024

Keywords

Comments

Subsequence of A060476 and differs from it by not having the terms 1, 256, 768, 1280, 1792, 2304, ... .
Subsequence of A295661 and first differs from it at n = 51: A295661(51) = 432 is not a term of this sequence.
First differs from A325990 at n = 30: A325990(30) = 256 is not a term of this sequence.
Nonsquarefree numbers k such that A051903(k) is odd, or equivalently, numbers k such that A051903(k) is an odd number that is larger than 1.
The asymptotic density of this sequence is Sum_{k>=3} (-1)^(k+1) * (1 - 1/zeta(k)) = 0.11615617754774636364... .

Links

Crossrefs

Complement of A368714 within A013929.
Cf. A051903, A060476, A295661, A325990.

Programs

  • Mathematica
    q[n_] := n > 1 && OddQ[n]; Select[Range[500], q[Max[FactorInteger[#][[;; , 2]]]] &]
  • PARI
    is(k) = k > 1 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));

A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

216, 864, 1000, 1080, 1512, 1944, 2376, 2744, 2808, 3000, 3375, 3456, 3672, 4000, 4104, 4320, 4968, 5400, 6048, 6264, 6696, 6750, 7000, 7560, 7776, 7992, 8232, 8856, 9000, 9261, 9288, 9504, 9720, 10152, 10584, 10648, 10976, 11000, 11232, 11448, 11880, 12000, 12744, 13000
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) * (1 + Sum_{p prime} (1/(p^3+p^2-1))) = 0.0035024748296318122535... .

Links

Crossrefs

Complement of the union of A335275 and A377844.
Subsequence of A295661.
Subsequences: A162142, A179671, A190011.
Cf. A065465.

Programs

  • Mathematica
    q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] > 1; Select[Range[13000], q]
  • PARI
    is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) > 1;

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

First differs from A295661, A325990 and A376142 at n = 24: A295661(24) = A325990(24) = A376142(24) = 216 = 2^3 * 3^3 is not a term of this sequence.
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... .
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .

Links

Crossrefs

Subsequence of A295661.
Subsequences: A065036, A143610, A163569.
Cf. A060476, A065465, A325990, A376142, A377845.

Programs

  • Mathematica
    q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]
  • PARI
    is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;

A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536
Offset: 1

Views

Author

Amiram Eldar, Jun 01 2025

Keywords

Comments

Subsequence of A301517 and A374459 and first differs from them at n = 85: A374459(85) = A374459(85) = 864 = 2^5 * 3^3 is not a term of this sequence.
First differs from its subsequence A381312 at n = 21: a(21) = 216 = 2^3 * 3^3 is not a term of A381312.
Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is odd.
Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an odd power (A384518).
The asymptotic density of this sequence is Sum_{k>=1} (d(2*k+1)-1)/zeta(2) = 0.095609588748823080455..., where d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i).

Links

Crossrefs

Intersection of A268335 and A375142.
Intersection of A295661 and A375142.
Intersection of A376142 and A375142.
Equals A375142 \ A384519.
Subsequence of A301517 and A374459.
Subsequences: A381312, A384518.
Cf. A005117, A013661, A057521.

Programs

  • Mathematica
    q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && OddQ[u[[1]]]]; Select[Range[250], q]
  • PARI
    isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && e[1] % 2;}
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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