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 1-10 of 13 results. Next

A348076 Number k such that k and k+1 both have an equal number of even and odd exponents in their prime factorization (A187039).

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 175, 207, 244, 332, 368, 387, 404, 507, 548, 603, 604, 656, 724, 800, 832, 844, 847, 891, 908, 931, 963, 1052, 1075, 1083, 1124, 1250, 1251, 1323, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 1791, 2096, 2224, 2312, 2348, 2367, 2511, 2523
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2021

Keywords

Comments

First differs from A049103 and A074172 at n=7.

Examples

			44 is a term since 44 = 2^2 * 11 and 44 + 1 = 45 = 3^2 * 5 both have one even and one odd exponent in their prime factorization.

Links

Crossrefs

Subsequence of A187039.
A074172 is a subsequence.
Cf. A049103.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; Select[Range[2500], q[#] && q[# + 1] &]
  • Python
    from sympy import factorint
def aupto(limit):
    alst, cond = [], False
    for nxtk in range(3, limit+2):
        evenodd = [0, 0]
        for e in factorint(nxtk).values():
            evenodd[e%2] += 1
        nxtcond = (evenodd[0] == evenodd[1])
        if cond and nxtcond:
            alst.append(nxtk-1)
        cond = nxtcond
    return alst
print(aupto(2523)) # Michael S. Branicky, Sep 27 2021

A348077 Starts of runs of 3 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

Original entry on oeis.org

603, 1250, 1323, 2523, 4203, 4923, 4948, 7442, 10467, 12591, 18027, 20402, 21123, 23823, 31507, 31850, 36162, 40327, 54475, 54511, 55323, 58923, 63747, 64386, 71523, 73204, 79011, 83151, 85291, 88047, 97675, 103923, 104211, 118323, 120787, 122571, 124891, 126927
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2021

Keywords

Examples

			603 is a term since 603 = 3^2 * 67, 603 + 1 = 604 = 2^2 * 151 and 603 + 2 = 605 = 5 * 11^2 all have one even and one odd exponent in their prime factorization.

Links

Crossrefs

Subsequence of A187039 and A348076.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; v = q /@ Range[3]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 2]], {k, 4, 130000}]; seq
  • Python
    from sympy import factorint
def aupto(limit):
    alst, condvec = [], [False, False, False]
    for kp2 in range(4, limit+3):
        evenodd = [0, 0]
        for e in factorint(kp2).values():
            evenodd[e%2] += 1
        condvec = condvec[1:] + [evenodd[0] == evenodd[1]]
        if all(condvec):
            alst.append(kp2-2)
    return alst
print(aupto(126927)) # Michael S. Branicky, Sep 27 2021

A348078 Starts of runs of 4 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

Original entry on oeis.org

906596, 1141550, 1243275, 12133673, 13852924, 19293209, 20738672, 22997761, 23542001, 26587348, 30731822, 31237450, 39987773, 41419024, 43627148, 54040975, 54652148, 56487148, 70289225, 75855625, 77449300, 79677772, 80665072, 82126448, 91420721, 93883850, 95162849
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2021

Keywords

Examples

			906596 is a term since 906596 = 2^2 * 226649, 906596 + 1 = 906597 = 3^2 * 100733, 906596 + 2 = 906598 = 2 * 7^2 * 11 * 29^2 and 906596 + 3 = 906599 = 71 * 113^2 all have an equal number of even and odd exponents in their prime factorization.

Links

Crossrefs

Subsequence of A187039, A348076 and A348077.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; v = q /@ Range[4]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 3]], {k, 5, 2*10^7}]; seq
  • Python
    from sympy import factorint
def cond(n):
    evenodd = [0, 0]
    for e in factorint(n).values():
        evenodd[e%2] += 1
    return evenodd[0] == evenodd[1]
def afind(limit, startk=5):
    condvec = [cond(startk+i) for i in range(4)]
    for kp3 in range(startk+3, limit+4):
        condvec = condvec[1:] + [cond(kp3)]
        if all(condvec):
            print(kp3-3, end=", ")
afind(125*10**4) # Michael S. Branicky, Sep 27 2021

A356415 a(n) is the least start of exactly n consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039), or -1 if no such run of consecutive numbers exists.

Original entry on oeis.org

1, 44, 603, 906596, 792007675
Offset: 1

Views

Author

Amiram Eldar, Aug 06 2022

Keywords

Comments

a(6) > 6.5*10^10, if it exist.

Examples

			a(2) = 44 since 44 = 2^2 * 11 and 45 = 3^2 * 5 both have one even and one odd exponent in their prime factorization, 43 and 46 have no even exponent, and 44 is the least number with this property.

Crossrefs

Cf. A187039, A348076, A348077, A348078.

Programs

  • Mathematica
    f[p_, e_] := (-1)^e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; seq[len_, nmax_] := Module[{s = Table[0, {len}], v = {1}, n = 2, c = 0, m}, While[c <= len && n <= nmax, If[q[n], v = Join[v, {n}], m = Length[v]; v = {}; If[0 <= m <= len && s[[m]] == 0, c++; s[[m]] = n - m]]; n++]; s]; seq[4, 10^6]

A348079 Starts of runs of 5 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

Original entry on oeis.org

792007675, 2513546971, 2820448771, 3201296272, 4742326672, 4894282924, 5462510272, 5664816448, 6947006272, 7814337424, 8784450448, 9085360624, 10147712524, 10246365547, 11537724975, 11861786572, 11907710548, 12456672496, 13338112048, 13510075471, 13931933948
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2021

Keywords

Examples

			792007675 is a term since 792007675 = 2^2 * 31680307, 792007675 + 1 = 792007676 = 2^2 * 198001919, 792007675 + 2 = 792007677 = 3^2 * 88000853, 792007675 + 3 = 792007678 = 2 * 7^2 * 11^2 * 66791 and 792007675 + 4 = 792007679 = 17^2 * 2740511 all have an equal number of even and odd exponents in their prime factorization.

Crossrefs

Subsequence of A187039, A348076, A348077 and A348078.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; v = q /@ Range[5]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 4]], {k, 6, 3*10^9}]; seq
  • Python
    from sympy import factorint
def cond(n):
    evenodd = [0, 0]
    for e in factorint(n).values():
        evenodd[e%2] += 1
    return evenodd[0] == evenodd[1]
def afind(limit, startk=6):
    condvec = [cond(startk+i) for i in range(5)]
    for kp4 in range(startk+4, limit+5):
        condvec = condvec[1:] + [cond(kp4)]
        if all(condvec):
            print(kp4-4, end=", ")
afind(10**9) # Michael S. Branicky, Sep 27 2021

A316523 Number of odd multiplicities minus number of even multiplicities in the canonical prime factorization of n.

Original entry on oeis.org

0, 1, 1, -1, 1, 2, 1, 1, -1, 2, 1, 0, 1, 2, 2, -1, 1, 0, 1, 0, 2, 2, 1, 2, -1, 2, 1, 0, 1, 3, 1, 1, 2, 2, 2, -2, 1, 2, 2, 2, 1, 3, 1, 0, 0, 2, 1, 0, -1, 0, 2, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 0, -1, 2, 3, 1, 0, 2, 3, 1, 0, 1, 2, 0, 0, 2, 3, 1, 0, -1, 2, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2018

Keywords

Links

Crossrefs

Cf. A000040, A000607, A071321, A100118, A112798.
Cf. A187039 (where a(n)=0). - Michel Marcus, Jul 08 2018
Cf. A077761, A162641, A162642, A179119.

Programs

  • Maple
    f:= proc(n) local F;
  F:= map(t -> t[2],ifactors(n)[2]);
  2*nops(select(type,F,odd))-nops(F);
end proc:
map(f, [$1..100]); # Robert Israel, Aug 27 2018
  • Mathematica
    Table[Total[-(-1)^If[n==1,{},FactorInteger[n][[All,2]]]],{n,100}]
  • PARI
    a(n) = my(f=factor(n)); -sum(k=1, #f~, (-1)^(f[k,2])); \\ Michel Marcus, Jul 08 2018; corrected Jun 13 2022

Formula

If i and j are coprime, a(i*j) = a(i)+a(j). - Robert Israel, Aug 27 2018
From Amiram Eldar, Oct 05 2023: (Start)
Additive with a(p^e) = (-1)^(e+1).
a(n) = A162642(n) - A162641(n).
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 - 2*A179119 = -0.398962... . (End)

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

First differs from A187039 in having 2520 and lacking 1 and 12600.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The unordered prime signature of 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.

Crossrefs

A subset of A030231.
For mean instead of median we have A070011.
Positions of odd terms in A360460.
The complement is A360553 (without 1), counted by A360687.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A304038, A316413, A326621, A348551, A360006, A360009, A360248, A360453.

Programs

  • Mathematica
    Select[Range[2,100],!IntegerQ[Median[Last/@FactorInteger[#]]]&]

A348097 Numbers having equally many unitary and nonunitary prime divisors.

Original entry on oeis.org

1, 12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 75, 76, 80, 88, 92, 96, 98, 99, 104, 112, 116, 117, 124, 135, 136, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 207, 208, 212, 224, 232, 236, 242, 244, 245, 248, 250, 261
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2021

Keywords

Comments

A prime divisor of k is unitary if its exponent in the prime factorization of k is 1, and is nonunitary otherwise.
Numbers k such that A056169(k) = A056170(k) = A001221(k)/2.
A345381 is a subsequence. After a(1) = 1, a(238) = 1260 is the next term that is not in A345381.

Examples

			12 = 2^2 * 3 is a term since it has one unitary prime divisor (3) and one nonunitary prime divisor (2).

Links

Crossrefs

Cf. A001221, A056169, A056170, A345381.
Subsequence of A030231.
Similar sequences: A016825 (odd and even divisors), A048109, A187039.

Programs

  • Mathematica
    q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; Select[Range[300], q]

A279456 Numbers k such that number of distinct primes dividing k is odd and number of prime divisors (counted with multiplicity) of k is even.

Original entry on oeis.org

4, 9, 16, 25, 49, 60, 64, 81, 84, 90, 121, 126, 132, 140, 150, 156, 169, 198, 204, 220, 228, 234, 240, 256, 260, 276, 289, 294, 306, 308, 315, 336, 340, 342, 348, 350, 360, 361, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 504, 516, 522, 525, 528, 529, 532, 540, 550, 558, 560, 564, 572, 580, 585, 600
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2016

Keywords

Comments

Intersection of A028260 and A030230.
Numbers k such that A000035(A001221(k)) = 1 and A000035(A001222(k)) = 0.
Numbers k such that A076479(k) = -1 and A008836(k) = 1.

Examples

			90 is in the sequence because 90 = 2*3^2*5 therefore omega(90) = 3 {2,3,5} is odd and bigomega(90) = 4 {2,3,3,5} is even.

Links

Crossrefs

Cf. A000035, A001221, A001222, A008836, A028260, A030230, A076479, A082522 (subsequence), A187039, A279457, A279458.

Programs

  • Mathematica
    Select[Range[600], Mod[PrimeNu[#1], 2] == 1 && Mod[PrimeOmega[#1], 2] == 0 & ]
  • PARI
    is(k) = {my(f = factor(k)); omega(f) % 2 && !(bigomega(f) % 2);} \\ Amiram Eldar, Sep 17 2024

A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

Original entry on oeis.org

1, 60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726, 732, 735
Offset: 1

Views

Author

Amiram Eldar, Aug 06 2022

Keywords

Comments

Numbers k such that A350386(k) = A350387(k).
A085987 is a subsequence. Terms that are not in A085987 are 1, 2160, 3024, ...

Examples

			60 is a term since A350386(60) = A350387(60) = 2.

Links

Crossrefs

Cf. A350386, A350387.
Subsequence of A028260.
Subsequences: A085987, A179698, A190109, A190110.
Similar sequences: A048109, A187039, A348097.

Programs

  • Mathematica
    f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[1000], q]
  • PARI
    isok(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};
Showing 1-10 of 13 results. Next

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