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.

A030516 Numbers with 7 divisors. 6th powers of primes.

Original entry on oeis.org

64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889, 594823321, 887503681, 2565726409, 4750104241, 6321363049, 10779215329, 22164361129, 42180533641, 51520374361, 90458382169, 128100283921
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

These are the numbers p^6 with p prime. - Jorge Coveiro, Apr 13 2004
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

Links

Crossrefs

Subsequence of A046306.
Cf. A013664, A050997, A085966, A131993, A000005, A000040, A001248, A030514, A258603, A269404.

Programs

Formula

a(n) = A000040(n)^(7-1) = A000040(n)^6. - Omar E. Pol, May 06 2008
A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(6) = 0.0170700868... (A085966). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(6)/zeta(12) = 675675/(691*Pi^6) (A269404).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(6) = 945/Pi^6 = 1/A013664. (End)

A258600 a(n) is the index m such that A036966(m) = prime(n)^3.

Original entry on oeis.org

2, 4, 8, 13, 23, 29, 39, 45, 57, 75, 81, 99, 110, 117, 130, 149, 169, 176, 197, 209, 212, 236, 250, 270, 295, 309, 317, 328, 337, 354, 399, 414, 436, 445, 477, 483, 506, 529, 541, 563, 585, 591, 631, 635, 654, 657, 697, 747, 758, 765, 781, 803, 809, 845, 864
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

Examples

			.   n |  p |  a(n) | A036966(a(n)) = A030078(n) = p^3
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             8
.   2 |  3 |     4 |            27
.   3 |  5 |     8 |           125
.   4 |  7 |    13 |           343
.   5 | 11 |    23 |          1331
.   6 | 13 |    29 |          2197
.   7 | 17 |    39 |          4913
.   8 | 19 |    45 |          6859
.   9 | 23 |    57 |         12167
.  10 | 29 |    75 |         24389
.  11 | 31 |    81 |         29791
.  12 | 37 |    99 |         50653
.  13 | 41 |   110 |         68921
.  14 | 43 |   117 |         79507
.  15 | 47 |   130 |        103823
.  16 | 53 |   149 |        148877
.  17 | 59 |   169 |        205379
.  18 | 61 |   176 |        226981
.  19 | 67 |   197 |        300763
.  20 | 71 |   209 |        357911
.  21 | 73 |   212 |        389017
.  22 | 79 |   236 |        493039
.  23 | 83 |   250 |        571787
.  24 | 89 |   270 |        704969
.  25 | 97 |   295 |        912673  .

Links

Crossrefs

Cf. A258568, A000040, A030078, A036966, A258599, A258601, A258602, A258603.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a258600 = (+ 1) . fromJust . (`elemIndex` a258568_list) . a000040
  • Mathematica
    With[{m = 60}, c = Select[Range[Prime[m]^3], Min[FactorInteger[#][[;; , 2]]] > 2 &]; 1 + Flatten[FirstPosition[c, #] & /@ (Prime[Range[m]]^3)]] (* Amiram Eldar, Feb 07 2023 *)
  • Python
    from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258600(n):
    c, m = 0, prime(n)**3
    for w in range(1,integer_nthroot(m,5)[0]+1):
        if all(d<=1 for d in factorint(w).values()):
            for y in range(1,integer_nthroot(z:=m//w**5,4)[0]+1):
                if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                    c += integer_nthroot(z//y**4,3)[0]
    return c # Chai Wah Wu, Sep 10 2024

Formula

A036966(a(n)) = A030078(n) = prime(n)^3.
A036966(m) mod prime(n) > 0 for m < a(n).
Also smallest number m such that A258568(m) = prime(n):
A258568(a(n)) = A000040(n) and A258568(m) != A000040(n) for m < a(n).

Extensions

a(11)-a(55) and example corrected by Amiram Eldar, Feb 07 2023

A258599 a(n) is the index m such that A001694(m) = prime(n)^2.

Original entry on oeis.org

2, 4, 6, 10, 16, 20, 28, 31, 39, 48, 51, 65, 71, 75, 84, 94, 107, 110, 120, 129, 133, 145, 152, 163, 180, 187, 191, 199, 202, 212, 238, 246, 258, 261, 282, 286, 297, 309, 319, 330, 342, 344, 366, 372, 377, 382, 407, 431, 440, 443, 450, 463, 468, 487, 498
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

Examples

			.   n |  p |  a(n) | A001694(a(n)) = A001248(n) = p^2
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             4
.   2 |  3 |     4 |             9
.   3 |  5 |     6 |            25
.   4 |  7 |    10 |            49
.   5 | 11 |    16 |           121
.   6 | 13 |    20 |           169
.   7 | 17 |    28 |           289
.   8 | 19 |    31 |           361
.   9 | 23 |    39 |           529
.  10 | 29 |    48 |           841
.  11 | 31 |    51 |           961
.  12 | 37 |    65 |          1369
.  13 | 41 |    71 |          1681
.  14 | 43 |    75 |          1849
.  15 | 47 |    84 |          2209
.  16 | 53 |    94 |          2809
.  17 | 59 |   107 |          3481
.  18 | 61 |   110 |          3721
.  19 | 67 |   120 |          4489
.  20 | 71 |   129 |          5041
.  21 | 73 |   133 |          5329
.  22 | 79 |   145 |          6241
.  23 | 83 |   152 |          6889
.  24 | 89 |   163 |          7921
.  25 | 97 |   180 |          9409  .

Links

Crossrefs

Cf. A258567, A000040, A001248, A001694, A258600, A258601, A258602, A258603.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a258599 = (+ 1) . fromJust . (`elemIndex` a258567_list) . a000040
  • Mathematica
    With[{m = 60}, c = Select[Range[Prime[m]^2], Min[FactorInteger[#][[;; , 2]]] > 1 &]; 1 + Flatten[FirstPosition[c, #] & /@ (Prime[Range[m]]^2)]] (* Amiram Eldar, Feb 07 2023 *)
  • Python
    from math import isqrt
from sympy import prime, integer_nthroot, factorint
def A258599(n):
    m = prime(n)**2
    return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # Chai Wah Wu, Sep 10 2024

Formula

A001694(a(n)) = A001248(n) = prime(n)^2.
A001694(m) mod prime(n) > 0 for m < a(n).
Also smallest number m such that A258567(m) = prime(n):
A258567(a(n)) = A000040(n) and A258567(m) != A000040(n) for m < a(n).

A258601 a(n) is the index m such that A036967(m) = prime(n)^4.

Original entry on oeis.org

2, 5, 10, 16, 28, 37, 55, 61, 80, 105, 113, 142, 163, 170, 190, 219, 249, 260, 286, 310, 318, 352, 374, 407, 448, 472, 482, 505, 511, 536, 614, 634, 672, 682, 740, 754, 783, 821, 842, 878, 916, 924, 984, 996, 1015, 1032, 1103, 1171, 1201, 1213, 1233, 1270, 1286, 1343, 1379
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

Comments

A036967(a(n)) = A030514(n) = prime(n)^4;
A036967(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258569(m) = prime(n):
A258569(a(n)) = A000040(n) and A258569(m) != A000040(n) for m < a(n).

Examples

			.   n |  p |  a(n) | A036967(a(n)) = A030514(n) = p^4
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            16
.   2 |  3 |     5 |            81
.   3 |  5 |    10 |           625
.   4 |  7 |    16 |          2401
.   5 | 11 |    28 |         14641
.   6 | 13 |    37 |         28561
.   7 | 17 |    55 |         83521
.   8 | 19 |    61 |        130321
.   9 | 23 |    80 |        279841
.  10 | 29 |   105 |        707281
.  11 | 31 |   113 |        923521
.  12 | 37 |   142 |       1874161
.  13 | 41 |   163 |       2825761
.  14 | 43 |   170 |       3418801
.  15 | 47 |   190 |       4879681
.  16 | 53 |   219 |       7890481
.  17 | 59 |   249 |      12117361
.  18 | 61 |   260 |      13845841
.  19 | 67 |   286 |      20151121
.  20 | 71 |   310 |      25411681
.  21 | 73 |   318 |      28398241
.  22 | 79 |   352 |      38950081
.  23 | 83 |   374 |      47458321
.  24 | 89 |   407 |      62742241
.  25 | 97 |   448 |      88529281

Links

Crossrefs

Cf. A258569, A000040, A030514, A036967, A258599, A258600, A258602, A258603.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a258601 = (+ 1) . fromJust . (`elemIndex` a258569_list) . a000040
  • PARI
    \\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^4,4) \\ Andrew Howroyd, Sep 10 2024
  • Python
    from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258601(n):
    c, m = 0, prime(n)**4
    for u in range(1,integer_nthroot(m,7)[0]+1):
        if all(d<=1 for d in factorint(u).values()):
            for w in range(1,integer_nthroot(a:=m//u**7,6)[0]+1):
                if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                    for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                        if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                            c += integer_nthroot(z//y**5,4)[0]
    return c # Chai Wah Wu, Sep 10 2024

Extensions

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024

A258602 a(n) is the index m such that A069492(m) = prime(n)^5.

Original entry on oeis.org

2, 5, 12, 20, 37, 45, 68, 82, 106, 142, 154, 196, 219, 234, 260, 305, 342, 360, 407, 434, 451, 496, 528, 573, 635, 668, 681, 720, 737, 770, 885, 919, 966, 984, 1065, 1087, 1139, 1193, 1228, 1283, 1331, 1348, 1440, 1455, 1484, 1509, 1624, 1731, 1767, 1789
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

Comments

A069492(a(n)) = A050997(n) = prime(n)^5;
A069492(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258570(m) = prime(n):
A258570(a(n)) = A000040(n) and A258570(m) != A000040(n) for m < a(n).

Examples

			.   n |  p |  a(n) | A069492(a(n)) = A050997(n) = p^5
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            32
.   2 |  3 |     5 |           243
.   3 |  5 |    12 |          3125
.   4 |  7 |    20 |         16807
.   5 | 11 |    37 |        161051
.   6 | 13 |    45 |        371293
.   7 | 17 |    68 |       1419857
.   8 | 19 |    82 |       2476099
.   9 | 23 |   106 |       6436343
.  10 | 29 |   142 |      20511149
.  11 | 31 |   154 |      28629151
.  12 | 37 |   196 |      69343957
.  13 | 41 |   219 |     115856201
.  14 | 43 |   234 |     147008443
.  15 | 47 |   260 |     229345007
.  16 | 53 |   305 |     418195493
.  17 | 59 |   342 |     714924299
.  18 | 61 |   360 |     844596301
.  19 | 67 |   407 |    1350125107
.  20 | 71 |   434 |    1804229351
.  21 | 73 |   451 |    2073071593
.  22 | 79 |   496 |    3077056399
.  23 | 83 |   528 |    3939040643
.  24 | 89 |   573 |    5584059449
.  25 | 97 |   635 |    8587340257  .

Links

Crossrefs

Cf. A258570, A000040, A050997, A069492, A258599, A258600, A258601, A258603.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a258602 = (+ 1) . fromJust . (`elemIndex` a258570_list) . a000040
  • PARI
    \\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^5,5) \\ Andrew Howroyd, Sep 10 2024
  • Python
    from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258602(n):
    c, m = 0, prime(n)**5
    for t in range(1,integer_nthroot(m,9)[0]+1):
        if all(d<=1 for d in factorint(t).values()):
            for u in range(1,integer_nthroot(s:=m//t**9,8)[0]+1):
                if gcd(t,u)==1 and all(d<=1 for d in factorint(u).values()):
                    for w in range(1,integer_nthroot(a:=s//u**8,7)[0]+1):
                        if gcd(u,w)==1 and gcd(t,w)==1 and all(d<=1 for d in factorint(w).values()):
                            for y in range(1,integer_nthroot(z:=a//w**7,6)[0]+1):
                                if gcd(w,y)==1 and gcd(u,y)==1 and gcd(t,y)==1 and all(d<=1 for d in factorint(y).values()):
                                    c += integer_nthroot(z//y**6,5)[0]
    return c # Chai Wah Wu, Sep 10 2024

Extensions

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024

A069493 6-full numbers: if p divides n then so does p^6.

Original entry on oeis.org

1, 64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 46656, 59049, 65536, 78125, 93312, 117649, 131072, 139968, 177147, 186624, 262144, 279936, 373248, 390625, 419904, 524288, 531441, 559872, 746496, 823543, 839808
Offset: 1

Views

Author

Benoit Cloitre, Apr 15 2002

Keywords

Comments

a(m) mod prime(n) > 0 for m < A258603(n); a(A258600(n)) = A030516(n) = prime(n)^6. - Reinhard Zumkeller, Jun 06 2015

Examples

			2^7*3^6 = 93312 is a member (although not of A076470).

Links

Crossrefs

Cf. A036967, A036966, A001694. Different from A076470.
Cf. A030516, A258603.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, fromList, union)
a069493 n = a069493_list !! (n-1)
a069493_list = 1 : f (singleton z) [1, z] zs where
   f s q6s p6s'@(p6:p6s)
     | m < p6 = m : f (union (fromList $ map (* m) ps) s') q6s p6s'
     | otherwise = f (union (fromList $ map (* p6) q6s) s) (p6:q6s) p6s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030516_list
-- Reinhard Zumkeller, Jun 03 2015
  • Mathematica
    Join[{1},Select[Range[900000],Min[FactorInteger[#][[All,2]]]>5&]] (* Harvey P. Dale, Mar 03 2018 *)
  • PARI
    for(n=1,560000,if(n*sumdiv(n,d,isprime(d)/d^6)==floor(n*sumdiv(n,d,isprime(d)/d^6)),print1(n,",")))
  • Python
    from math import gcd
from sympy import factorint, integer_nthroot
def A069493(n):
    def f(x):
        c = n+x
        for y1 in range(1,integer_nthroot(x,11)[0]+1):
            if all(d<=1 for d in factorint(y1).values()):
                for y2 in range(1,integer_nthroot(z2:=x//y1**11,10)[0]+1):
                    if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
                        for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
                            if gcd(y3,y1)==1 and gcd(y3,y2)==1 and all(d<=1 for d in factorint(y3).values()):
                                for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
                                    if gcd(y4,y1)==1 and gcd(y4,y2)==1 and gcd(y4,y3)==1 and all(d<=1 for d in factorint(y4).values()):
                                        for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
                                            if gcd(y5,y1)==1 and gcd(y5,y2)==1 and gcd(y5,y3)==1 and gcd(y5,y4)==1 and all(d<=1 for d in factorint(y5).values()):
                                                c -= integer_nthroot(z5//y5**7,6)[0]
        return c
    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 10 2024

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^5*(p-1))) = 1.0334657852594050612296726462481884631303137561267151463866539131591664... - Amiram Eldar, Jul 09 2020

A258571 Smallest prime factors of 6-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 3, 2, 5, 2, 7, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 3, 2, 2, 7, 2, 2, 2, 2, 2, 2, 3, 2, 11, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 13, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 11, 2, 2, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

Links

Crossrefs

Cf. A069493, A020639, A258603, A258567, A258568, A258569, A258570.

Programs

  • Haskell
    a258571 = a020639 . a069493
  • PARI
    lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 5, print1(f[1, 1], ", "))); } \\ Amiram Eldar, Sep 12 2024

Formula

a(n) = A020639(A069493(n)).
a(A258603(n)) = A000040(n) and a(m) != A000040(n) for m < A258603(n).
Showing 1-7 of 7 results.

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