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.

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A352140 Numbers whose prime factorization has all even prime indices and all odd exponents.

Original entry on oeis.org

1, 3, 7, 13, 19, 21, 27, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.
All terms are odd. - Michael S. Branicky, Mar 12 2022

Examples

			The terms together with their prime indices begin:
      1 = 1
      3 = prime(2)^1
      7 = prime(4)^1
     13 = prime(6)^1
     19 = prime(8)^1
     21 = prime(4)^1 prime(2)^1
     27 = prime(2)^3
     29 = prime(10)^1
     37 = prime(12)^1
     39 = prime(6)^1 prime(2)^1
     43 = prime(14)^1
     53 = prime(16)^1
     57 = prime(8)^1 prime(2)^1
     61 = prime(18)^1
     71 = prime(20)^1

Crossrefs

The restriction to primes is A031215.
These partitions are counted by A055922 (aerated).
The first condition alone is A066207, counted by A035363.
The squarefree case is A258117.
The second condition alone is A268335, counted by A055922.
A056166 = exponents all prime, counted by A055923.
A066208 = prime indices all odd, counted by A000009.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even prime exponents, odd A162642.
A257991 counts odd prime indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.

Programs

  • Mathematica
    Select[Range[100],And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
  • Python
    from sympy import factorint, primepi
def ok(n):
    if n%2 == 0: return False
    return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())
print([k for k in range(318) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Formula

Intersection of A066207 and A268335.
A257991(a(n)) = A162641(a(n)) = 0.
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A001222(a(n)).

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

Examples

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

Links

Crossrefs

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Programs

  • Mathematica
    Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
  • Python
    from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Formula

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

Examples

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

Crossrefs

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Formula

Intersection of A066208 and A346635.

A338469 Products of three odd prime numbers of odd index.

Original entry on oeis.org

125, 275, 425, 575, 605, 775, 935, 1025, 1175, 1265, 1331, 1445, 1475, 1675, 1705, 1825, 1955, 2057, 2075, 2255, 2425, 2575, 2585, 2635, 2645, 2725, 2783, 3175, 3179, 3245, 3425, 3485, 3565, 3685, 3725, 3751, 3925, 3995, 4015, 4175, 4301, 4475, 4565, 4715
Offset: 1

Views

Author

Gus Wiseman, Nov 08 2020

Keywords

Comments

Also Heinz numbers of integer partitions with 3 parts, all of which are odd and > 1. These partitions are counted by A001399.

Examples

			The sequence of terms together with their prime indices begins:
     125: {3,3,3}     1825: {3,3,21}    3425: {3,3,33}
     275: {3,3,5}     1955: {3,7,9}     3485: {3,7,13}
     425: {3,3,7}     2057: {5,5,7}     3565: {3,9,11}
     575: {3,3,9}     2075: {3,3,23}    3685: {3,5,19}
     605: {3,5,5}     2255: {3,5,13}    3725: {3,3,35}
     775: {3,3,11}    2425: {3,3,25}    3751: {5,5,11}
     935: {3,5,7}     2575: {3,3,27}    3925: {3,3,37}
    1025: {3,3,13}    2585: {3,5,15}    3995: {3,7,15}
    1175: {3,3,15}    2635: {3,7,11}    4015: {3,5,21}
    1265: {3,5,9}     2645: {3,9,9}     4175: {3,3,39}
    1331: {5,5,5}     2725: {3,3,29}    4301: {5,7,9}
    1445: {3,7,7}     2783: {5,5,9}     4475: {3,3,41}
    1475: {3,3,17}    3175: {3,3,31}    4565: {3,5,23}
    1675: {3,3,19}    3179: {5,7,7}     4715: {3,9,13}
    1705: {3,5,11}    3245: {3,5,17}    4775: {3,3,43}

Links

Crossrefs

A046316 allows all primes (strict: A046389).
A338471 allows all odd primes (strict: A307534).
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005408 lists odds (strict: A056911).
A008284 counts partitions by sum and length.
A014311 is a ranking of 3-part compositions (strict: A337453).
A014612 lists Heinz numbers of 3-part partitions (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A066207 lists numbers with all even prime indices (strict: A258117).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict 3-part partitions.
Cf. A001221, A001222, A002620, A005117, A037144, A056239, A112798.

Programs

  • Maple
    N:= 10000: # for terms <= N
P0:= [seq(ithprime(i),i=3..numtheory:-pi(floor(N/25)),2)]:
sort(select(`<=`,[seq(seq(seq(P0[i]*P0[j]*P0[k],k=1..j),j=1..i),i=1..nops(P0))], N)); # Robert Israel, Nov 12 2020
  • Mathematica
    Select[Range[1,1000,2],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
  • PARI
    isok(m) = my(f=factor(m)); (m%2) && (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338469(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(5,integer_nthroot(x,3)[0]+1),3)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A366839 Sum of even prime factors of 2n, counted with multiplicity.

Original entry on oeis.org

2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 12, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 14, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2023

Keywords

Examples

			The prime factors of 2*60 are {2,2,2,3,5}, of which the even factors are {2,2,2}, so a(60) = 6.

Crossrefs

A compound version is A001414, triangle A331416.
Dividing by 2 gives A001511.
Positions of 2's are A005408.
For count instead of sum we have A007814, odd version A087436.
The partition triangle for this statistic is A116598 aerated.
For indices we have A366531, halved A366533, triangle A113686/A174713.
The odd version is A366840.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A000720, A066208, A258117, A325698.

Programs

  • Mathematica
    Table[2*Length[NestWhileList[#/2&,n,EvenQ]],{n,100}]
  • PARI
    a(n) = 2 * valuation(n, 2) + 2; \\ Amiram Eldar, Sep 13 2024

Formula

a(n) = 2*A001511(n).
a(n) = A100006(n) - A366840(2n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Sep 13 2024

A366840 Sum of odd prime factors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 6, 5, 11, 3, 13, 7, 8, 0, 17, 6, 19, 5, 10, 11, 23, 3, 10, 13, 9, 7, 29, 8, 31, 0, 14, 17, 12, 6, 37, 19, 16, 5, 41, 10, 43, 11, 11, 23, 47, 3, 14, 10, 20, 13, 53, 9, 16, 7, 22, 29, 59, 8, 61, 31, 13, 0, 18, 14, 67, 17, 26, 12, 71, 6
Offset: 1

Views

Author

Gus Wiseman, Oct 27 2023

Keywords

Comments

Contains all positive integers except 1, 2, 4.

Examples

			The prime factors of 60 are {2,2,2,3,5}, of which the odd factors are {3,5}, so a(60) = 8.

Crossrefs

The compound version is A001414, triangle A331416.
For count instead of sum we have A087436, even version A007814.
Odd-indexed terms are A100005.
Positions of odd terms are A335657, even A036349.
For prime indices we have A366528, triangle A113685 (without zeros A365067)
The even version is A366839 = 2*A001511.
The partition triangle for this statistic is A366851, even version A116598.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
Cf. A000009, A066208, A113686, A174713, A258117, A325698, A366531, A366850.

Programs

  • Mathematica
    Table[Total[Times@@@DeleteCases[If[n==1,{}, FactorInteger[n]],{2,_}]],{n,100}]
  • PARI
    a(n) = my(f=factor(n), j=if(n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ Michel Marcus, Oct 30 2023

Formula

a(n) = A100006(n) - A366839(n).
a(2n) = a(n).
a(2n-1) = A001414(2n-1) = A100005(n).
Completely additive with a(2^e) = 0 and a(p^e) = e*p for an odd prime p. - Amiram Eldar, Nov 03 2023
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