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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A319613 a(n) = prime(n) * prime(2n).

Original entry on oeis.org

6, 21, 65, 133, 319, 481, 731, 1007, 1403, 2059, 2449, 3293, 4141, 4601, 5311, 6943, 8201, 9211, 10921, 12283, 13213, 15247, 16517, 19847, 22213, 24139, 25853, 28141, 29539, 31753, 37211, 40741, 43429, 46843, 52001, 54209, 58561, 62429, 66299, 70757, 75359
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2019

Keywords

Links

Crossrefs

Cf. A000040, A001358, A018819, A031215, A031368, A056239, A087897, A101417, A112798, A120641, A320340, A323092, A323093, A323094.

Programs

  • Maple
    a:= n-> (p-> p(n)*p(2*n))(ithprime):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 08 2019
  • Mathematica
    Table[Prime[n]*Prime[2*n],{n,50}]
  • PARI
    a(n) = prime(n)*prime(2*n) \\ Felix Fröhlich, Jan 09 2019

A332812 Numbers n for which A156552(n) == 2 (mod 3).

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 19, 24, 26, 27, 28, 29, 35, 37, 38, 43, 48, 52, 53, 54, 56, 58, 61, 63, 65, 70, 71, 74, 75, 76, 79, 86, 89, 95, 96, 101, 104, 106, 107, 108, 112, 113, 116, 117, 122, 126, 130, 131, 139, 140, 142, 143, 145, 147, 148, 150, 151, 152, 158, 163, 165, 171, 172, 173, 175, 178, 181, 185, 190, 192, 193, 199, 202
Offset: 1

Views

Author

Antti Karttunen, Feb 27 2020

Keywords

Comments

Contains neither squares nor twice squares.

Links

Crossrefs

Positions of 2's in A329903, of -1's in A332814.
Cf. A156552, A329604, A329609.
Cf. A031215 (a subsequence of prime terms).
Cf. also A332822.

Programs

  • PARI
    A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
isA332812(n) = (2==(A156552(n)%3));

A352141 Numbers whose prime factorization has all even indices and all even exponents.

Original entry on oeis.org

1, 9, 49, 81, 169, 361, 441, 729, 841, 1369, 1521, 1849, 2401, 2809, 3249, 3721, 3969, 5041, 6241, 6561, 7569, 7921, 8281, 10201, 11449, 12321, 12769, 13689, 16641, 17161, 17689, 19321, 21609, 22801, 25281, 26569, 28561, 29241, 29929, 32761, 33489, 35721
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 partitions with all even parts and all even multiplicities, counted by A035444.

Examples

			The terms together with their prime indices begin:
     1 = 1
     9 = prime(2)^2
    49 = prime(4)^2
    81 = prime(2)^4
   169 = prime(6)^2
   361 = prime(8)^2
   441 = prime(2)^2 prime(4)^2
   729 = prime(2)^6
   841 = prime(10)^2
  1369 = prime(12)^2
  1521 = prime(2)^2 prime(6)^2
  1849 = prime(14)^2
  2401 = prime(4)^4
  2809 = prime(16)^2
  3249 = prime(2)^2 prime(8)^2
  3721 = prime(18)^2
  3969 = prime(2)^4 prime(4)^2

Links

Crossrefs

The second condition alone (all even exponents) is A000290, counted by A035363.
The restriction to primes is A031215.
These partitions are counted by A035444.
The first condition alone is A066207, counted by A035363, squarefree A258117.
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 exponents, odd A162642.
A257991 counts odd 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.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.

Programs

  • Mathematica
    Select[Range[1000],#==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
  • Python
    from itertools import count, islice
from sympy import factorint, primepi
def A352141_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x: not (x[1]%2 or primepi(x[0])%2), factorint(k).items())),count(max(startvalue,1)))
A352141_list = list(islice(A352141_gen(),30)) # Chai Wah Wu, Mar 18 2022

Formula

Intersection of A000290 and A066207.
A257991(a(n)) = A162642(a(n)) = 0.
A257992(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - Amiram Eldar, Sep 19 2022

A372587 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is even.

Original entry on oeis.org

6, 7, 10, 11, 13, 14, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 44, 49, 50, 52, 56, 57, 58, 62, 69, 70, 72, 74, 75, 76, 77, 82, 83, 85, 86, 87, 88, 90, 92, 96, 98, 100, 102, 103, 104, 106, 107, 108, 109, 112, 117, 120, 123
Offset: 1

Views

Author

Gus Wiseman, May 14 2024

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
The odd version is A372586.

Examples

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {2,3}   6  (2,1)
          {1,2,3}   7  (4)
            {2,4}  10  (3,1)
          {1,2,4}  11  (5)
          {1,3,4}  13  (6)
          {2,3,4}  14  (4,1)
            {2,5}  18  (2,2,1)
          {1,2,5}  19  (8)
          {2,3,5}  22  (5,1)
        {1,2,3,5}  23  (9)
            {4,5}  24  (2,1,1,1)
          {1,4,5}  25  (3,3)
          {2,4,5}  26  (6,1)
        {1,2,4,5}  27  (2,2,2)
          {3,4,5}  28  (4,1,1)
        {2,3,4,5}  30  (3,2,1)
      {1,2,3,4,5}  31  (11)
            {1,6}  33  (5,2)
            {2,6}  34  (7,1)
          {1,2,6}  35  (4,3)
          {1,3,6}  37  (12)
          {2,3,6}  38  (8,1)

Crossrefs

Positions of even terms in A372428, zeros A372427.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372586.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum: A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[Total[bix[#]]+Total[prix[#]]]&]

Formula

Numbers k such that A029931(k) + A056239(k) is even.

A217622 Prime(prime(2*n)).

Original entry on oeis.org

5, 17, 41, 67, 109, 157, 191, 241, 283, 353, 401, 461, 547, 587, 617, 739, 797, 877, 967, 1031, 1087, 1171, 1217, 1409, 1447, 1499, 1597, 1669, 1741, 1823, 1913, 2063, 2099, 2269, 2351, 2417, 2549, 2647, 2719, 2803, 2909, 3019, 3109, 3229, 3299, 3407, 3517
Offset: 1

Views

Author

Vincenzo Librandi, Oct 13 2012

Keywords

Comments

Subsequence of A006450.
Using the Prime Number Theorem, prime(n) ~ n log n, the asymptotic behavior is A217622(n) ~ 2n (log 2n) log(2n log 2n) ~ 2n (log n)^2 ~ A230460(n). - M. F. Hasler, Oct 19 2013

Links

Crossrefs

Cf. A000040, A006450, A066066, A031215, A230460.

Programs

  • Magma
    [NthPrime(NthPrime(2*n)): n in [1..50] ]; //
  • Mathematica
    Table[Prime[Prime[2n]], {n, 100}]
  • PARI
    a(n)=prime(prime(2*n)) \\ Charles R Greathouse IV, Oct 20 2013

Formula

a(n) = A000040(A031215(n)). - Omar E. Pol, Oct 19 2013
a(n) = A006450(2n). - M. F. Hasler, Oct 20 2013

A246929 a(n) = prime(11*n).

Original entry on oeis.org

31, 79, 137, 193, 257, 317, 389, 457, 523, 601, 661, 743, 823, 887, 977, 1049, 1117, 1213, 1289, 1373, 1453, 1531, 1607, 1693, 1777, 1871, 1951, 2029, 2113, 2213, 2293, 2377, 2447, 2551, 2659, 2713, 2797, 2887, 2971, 3079, 3187, 3271, 3359, 3461, 3539
Offset: 1

Views

Author

Vincenzo Librandi, Sep 08 2014

Keywords

Links

Crossrefs

Cf. sequences of the type prime(k*n): A000040 (k=1), A031215 (k=2), A031336 - A031343 (k=3..10), this sequence (k=11), A246930 (k=12), A126588 (k=13), A246931 (k=14), A246932 (k=15), A246933 (k=16), A129480 (k=17), A031921 (k=100), A031922 (k=1000).

Programs

  • Magma
    [NthPrime(11*n): n in [1..50]];
  • Mathematica
    Prime[11 Range[50]]
  • PARI
    a(n)=prime(11*n) \\ Edward Jiang, Sep 08 2014
  • Sage
    [nth_prime(11*n) for n in (1..50)] # Bruno Berselli, Sep 08 2014

A338557 Products of three distinct prime numbers of even index.

Original entry on oeis.org

273, 399, 609, 741, 777, 903, 1113, 1131, 1281, 1443, 1491, 1653, 1659, 1677, 1729, 1869, 2067, 2109, 2121, 2247, 2373, 2379, 2451, 2639, 2751, 2769, 2919, 3021, 3081, 3171, 3219, 3367, 3423, 3471, 3477, 3633, 3741, 3801, 3857, 3913, 3939, 4047, 4053, 4173
Offset: 1

Views

Author

Gus Wiseman, Nov 08 2020

Keywords

Comments

All terms are odd.
Also sphenic numbers (A007304) with all even prime indices (A031215).
Also Heinz numbers of strict integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

Examples

			The sequence of terms together with their prime indices begins:
     273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}
     399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}
     609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}
     741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}
     777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}
     903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}
    1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}
    1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}
    1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}
    1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}
    1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}
    1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}
    1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}
    1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}
    1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

Crossrefs

For the following, NNS means "not necessarily strict".
A007304 allows all prime indices (not just even) (NNS: A014612).
A046389 allows all odd primes (NNS: A046316).
A258117 allows products of any length (NNS: A066207).
A307534 is the version for odds instead of evens (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338556 is the NNS version.
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers, with even case A039956.
A078374 counts 3-part relatively prime strict partitions (NNS: A023023).
A075819 lists even Heinz numbers of strict triples (NNS: A075818).
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

Programs

  • Mathematica
    Select[Range[1000],SquareFreeQ[#]&&PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]
  • PARI
    isok(m) = my(f=factor(m)); (bigomega(f)==3) && (omega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020
  • Python
    from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, nextprime, integer_nthroot
def A338557(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)-(b>>1) for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

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)).

A022457 a(n) = prime(2n) mod prime(n).

Original entry on oeis.org

1, 1, 3, 5, 7, 11, 9, 15, 15, 13, 17, 15, 19, 21, 19, 25, 21, 29, 29, 31, 35, 35, 33, 45, 35, 37, 45, 49, 53, 55, 39, 49, 43, 59, 51, 57, 59, 57, 63, 63, 63, 71, 61, 71, 69, 81, 69, 57, 67, 83, 91, 91, 95, 91, 87, 87, 81, 99, 93, 97, 107, 97, 87, 97, 107, 109, 95
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000040, A031215, A066066.

Programs

  • Magma
    [NthPrime(2*n) mod NthPrime(n): n in [1..50]]; // G. C. Greubel, Feb 28 2018
  • Maple
    A022457 := proc(n)
    modp(ithprime(2*n),ithprime(n)) ;
end proc:
seq(A022457(n),n=1..67) ; # R. J. Mathar, Sep 02 2016
  • Mathematica
    Table[Mod[Prime[2*n], Prime[n]], {n, 1, 50}] (* G. C. Greubel, Feb 28 2018 *)
  • PARI
    a(n) = prime(2*n) % prime(n); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A031215(n) modulo A000040(n). - Michel Marcus, Sep 30 2013

A230460 Prime(2*prime(n)).

Original entry on oeis.org

7, 13, 29, 43, 79, 101, 139, 163, 199, 271, 293, 373, 421, 443, 491, 577, 647, 673, 757, 821, 839, 929, 983, 1061, 1181, 1231, 1277, 1307, 1361, 1429, 1609, 1667, 1759, 1789, 1973, 1997, 2083, 2161, 2243, 2339, 2411, 2441, 2633, 2663, 2707, 2729, 2917
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2013

Keywords

Comments

A subsequence of A031378 (subsequence of A031215) and of A106349, which are both subsequences of A007821 which is the complement of A006450 in the primes A000040.

Examples

			a(3) = 29 because the third prime is 5, and 2 * 5 = 10, and then we see that the tenth prime is 29.
a(4) = 43 because the fourth prime is 7, and 2 * 7 = 14, and then we see that the fourteenth prime is 43.

Links

Crossrefs

Cf. A217622.

Programs

Formula

a(n) ~ 2n log(n) log(2n log(n)) ~ 2n (log n)^2.
a(n) = A000040(A100484(n)). - Omar E. Pol, Oct 19 2013
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