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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A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

Original entry on oeis.org

225, 675, 1089, 1125, 2601, 3025, 3267, 3375, 6075, 7225, 7803, 8649, 11979, 15125, 15129, 24025, 25947, 27225, 28125, 29403, 30375, 31329, 33275, 34969, 35937, 36125, 40401, 42025, 44217, 45387, 54675, 62001, 65025, 70227, 81675, 84375, 87025, 93987
Offset: 1

Views

Author

Gus Wiseman, Mar 24 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.

Examples

			The terms together with their prime indices (not factors) begin:
     225: {2,2,3,3}
     675: {2,2,2,3,3}
    1089: {2,2,5,5}
    1125: {2,2,3,3,3}
    2601: {2,2,7,7}
    3025: {3,3,5,5}
    3267: {2,2,2,5,5}
    3375: {2,2,2,3,3,3}
    6075: {2,2,2,2,2,3,3}
    7225: {3,3,7,7}
    7803: {2,2,2,7,7}
    8649: {2,2,11,11}
   11979: {2,2,5,5,5}
   15125: {3,3,3,5,5}
   15129: {2,2,13,13}
   24025: {3,3,11,11}
   25947: {2,2,2,11,11}
   27225: {2,2,3,3,5,5}
   28125: {2,2,3,3,3,3,3}
For example, 7803 = prime(1)^3 prime(4)^2.

Links

Crossrefs

These partitions are counted by A352493.
This is the restriction of A346068 to numbers that are not a prime power.
The prime-power version is A352519, counted by A230595.
A000040 lists the primes.
A000961 lists prime powers.
A001694 lists powerful numbers, counted by A007690.
A038499 counts partitions of prime length.
A053810 lists all numbers p^q for p and q prime, counted by A001221.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.

Programs

  • Mathematica
    Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]

Formula

Sum_{n>=1} 1/a(n) = (Product_{p prime-indexed prime} (1 + Sum_{q prime} 1/p^q)) - (Sum_{p prime-indexed prime} Sum_{q prime} 1/p^q) - 1 = 0.0106862606... . - Amiram Eldar, Aug 04 2024

A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

Original entry on oeis.org

9, 25, 27, 121, 125, 243, 289, 961, 1331, 1681, 2187, 3125, 3481, 4489, 4913, 6889, 11881, 16129, 24649, 29791, 32041, 36481, 44521, 58081, 68921, 76729, 78125, 80089, 109561, 124609, 134689, 160801, 161051, 177147, 185761, 205379, 212521, 259081, 299209
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2022

Keywords

Comments

Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.

Examples

			The terms together with their prime indices begin:
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    125: {3,3,3}
    243: {2,2,2,2,2}
    289: {7,7}
    961: {11,11}
   1331: {5,5,5}
   1681: {13,13}
   2187: {2,2,2,2,2,2,2}
   3125: {3,3,3,3,3}
   3481: {17,17}
   4489: {19,19}
   4913: {7,7,7}
   6889: {23,23}
  11881: {29,29}
  16129: {31,31}
  24649: {37,37}
  29791: {11,11,11}

Links

Crossrefs

Numbers of the form p^q for p and q prime are A053810, counted by A001221.
These partitions are counted by A230595.
This is the prime power case of A346068.
For numbers that are not a prime power we have A352518, counted by A352493.
A000040 lists the primes.
A000961 lists prime powers.
A001597 lists perfect powers.
A001694 lists powerful numbers, counted by A007690.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A164336 lists all possible power-towers of prime numbers.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.

Programs

  • Maple
    N:= 10^7: # for terms <= N
M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
R:= NULL:
for p in PP do
  q:= 1:
  do
    q:= nextprime(q);
    t:= p^q;
    if t > N then break fi;
    R:= R, t;
  od;
od:
sort([R]); # Robert Israel, Dec 08 2022
  • Mathematica
    Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{?(PrimeQ[PrimePi[#]]&),k?PrimeQ}}]&]
  • Python
    from sympy import primepi, integer_nthroot, primerange
def A352519(n):
    def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985
Offset: 0

Views

Author

Gus Wiseman, May 16 2022

Keywords

Examples

			The a(13) = 2 through a(16) = 9 compositions:
  (22333)  (77)       (555)     (3355)
  (33322)  (2255)     (33333)   (5533)
           (5522)     (222333)  (22255)
           (223322)   (333222)  (55222)
           (2222222)            (332233)
                                (2222233)
                                (2223322)
                                (2233222)
                                (3322222)

Links

Crossrefs

The first condition only is A023360, partitions A000607.
For partitions we have A351982, only run-lens A100405, only parts A008483.
The second condition only is A353401, partitions A055923.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A052284 counts compositions into nonprimes, partitions A002095.
A106356 counts compositions by number of adjacent equal parts.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A329738 counts uniform compositions, partitions A047966.
Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.

Programs

  • Maple
    b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
      add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 18 2022
  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,15}]

Extensions

a(26)-a(56) from Alois P. Heinz, May 18 2022

A352493 Number of non-constant integer partitions of n into prime parts with prime multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 3, 0, 1, 4, 5, 3, 1, 3, 5, 7, 3, 5, 6, 8, 8, 11, 7, 6, 8, 15, 14, 14, 10, 15, 17, 21, 18, 23, 20, 28, 25, 31, 27, 35, 32, 33, 37, 46, 41, 50, 45, 58, 56, 63, 59, 78, 69, 76, 81, 85, 80, 103, 107, 111, 114, 127
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2022

Keywords

Examples

			The a(n) partitions for selected n (B = 11):
n = 10    16       19        20         25          28
   ---------------------------------------------------------------
    3322  5533     55333     7733       77722       BB33
          55222    55522     77222      5533333     BB222
          3322222  3333322   553322     5553322     775522
                   33322222  5522222    55333222    55533322
                             332222222  55522222    772222222
                                        333333322   3322222222222
                                        3333322222

Crossrefs

Constant partitions are counted by A001221, ranked by A000961.
Non-constant partitions are counted by A144300, ranked A024619.
The constant version is A230595, ranked by A352519.
This is the non-constant case of A351982, ranked by A346068.
These partitions are ranked by A352518.
A000040 lists the primes.
A000607 counts partitions into primes, ranked by A076610.
A001597 lists perfect powers, complement A007916.
A038499 counts partitions of prime length.
A053810 lists primes to primes.
A055923 counts partitions with prime multiplicities, ranked by A056166.
A257994 counts prime indices that are themselves prime.
A339218 counts powerful partitions into prime parts, ranked by A352492.
Cf. A000005, A007690, A031368, A035444, A052485, A056239, A066208, A089723, A114639, A320628, A330945.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&And@@PrimeQ/@#&& And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 6, 2, 0, 6, 3, 2, 9, 2, 5, 11, 3, 5, 18, 6, 4, 20, 13, 8, 26, 10, 17, 37, 14, 16, 51, 23, 24, 58, 38, 32, 75, 44, 52, 100, 52, 59, 143, 75, 77, 159, 114, 112, 203, 132, 154, 266, 175
Offset: 0

Views

Author

Gus Wiseman, May 16 2022

Keywords

Examples

			The a(n) partitions for selected n (A = 10):
  n=9:   n=12:   n=21:      n=24:       n=30:
------------------------------------------------------
  (333)  (444)   (777)      (888)       (AAA)
         (3333)  (444333)   (6666)      (66666)
                 (3333333)  (444444)    (555555)
                            (555333)    (666444)
                            (4443333)   (777333)
                            (33333333)  (6663333)
                                        (55533333)
                                        (444333333)
                                        (3333333333)

Crossrefs

The version for only parts > 2 is A008483.
The version for only multiplicities > 2 is A100405.
The version for parts and multiplicities > 1 is A339222, ranked by A062739.
For prime parts and multiplicities we have A351982, compositions A353429.
The version for compositions is A353428 (partial A078012, A353400).
These partitions are ranked by A353502.
A000726 counts partitions with all mults <= 2, compositions A128695.
A004250 counts partitions with some part > 2, compositions A008466.
A137200 counts compositions with all parts and run-lengths <= 2.
Cf. A003242, A047966, A055923, A098859, A114901, A325534, A333755, A335464.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Min@@#>2&&Min@@Length/@Split[#]>2&]],{n,0,30}]
Previous Showing 11-15 of 15 results.

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