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.

Previous Showing 11-18 of 18 results.

A280301 Numbers with 73 divisors.

Original entry on oeis.org

4722366482869645213696, 22528399544939174411840147874772641, 211758236813575084767080625169910490512847900390625, 7031676478883553279994550741476882515263791803223057265323201, 955593817727321453093807642925081991552428315714137911219172409259950196321
Offset: 1

Views

Author

Omar E. Pol, Dec 31 2016

Keywords

Comments

Also, 72nd powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 73.

Examples

			a(1) = 2^72, a(2) = 3^72, a(3) = 5^72, a(4) = 7^72, a(5) = 11^72.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280299, A280346, A280347, A280349, A261700.

Programs

  • Mathematica
    Array[Prime[#]^72 &, {5}] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    a(n)=prime(n)^72

Formula

a(n) = A000040(n)^(73-1) = A000040(n)^72.
A000005(a(n)) = 73.

A175747 Numbers with 38 divisors.

Original entry on oeis.org

786432, 1310720, 1835008, 2883584, 3407872, 4456448, 4980736, 6029312, 7602176, 8126464, 9699328, 10747904, 11272192, 12320768, 13893632, 15466496, 15990784, 17563648, 18612224, 19136512, 20709376, 21757952, 23330816, 25427968, 26476544, 27000832, 28049408
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the forms p^37 and p^18*q^1, where p and q are distinct primes.

Links

Crossrefs

Cf. A175745, A175746, A139572.

Programs

  • Mathematica
    Select[Range[10000000],DivisorSigma[0,#]==38&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)
  • PARI
    is(n)=numdiv(n)==38 \\ Charles R Greathouse IV, Jun 19 2016
  • Python
    def A175747(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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//p**18) for p in primerange(integer_nthroot(x,18)[0]+1))+primepi(integer_nthroot(x,19)[0])-primepi(integer_nthroot(x,37)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

A000005(a(n))=38.

Extensions

Extended by T. D. Noe, May 08 2011

A259417 Even powers of the odd primes listed in increasing order.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649
Offset: 1

Views

Author

Hartmut F. W. Hoft, Jun 26 2015

Keywords

Comments

Each of the following sequences, p^(q-1) with p >= 2 and q > 2 primes, except their respective first elements, powers of 2, is a subsequence:
A001248(p) = p^2, A030514(p) = p^4, A030516(p) = p^6,
A030629(p) = p^10, A030631(p) = p^12, A030635(p) = p^16,
A030637(p) = p^18, A137486(p) = p^22, A137492(p) = p^28,
A139571(p) = p^30, A139572(p) = p^36, A139573(p) = p^40,
A139574(p) = p^42, A139575(p) = p^46, A173533(p) = p^52,
A183062(p) = p^58, A183085(p) = p^60.
See also the link to the OEIS Wiki.
The sequences A053182(n)^2, A065509(n)^4, A163268(n)^6 and A240693(n)^10 are subsequences of this sequence.
The odd numbers in A023194 are a subsequence of this sequence.

Examples

			a(11) = 5^4 = 625 is followed by a(12) = 3^6 = 729 since no even power of an odd prime falls between them.

Links

Programs

  • Mathematica
    a259417[bound_] := Module[{q, h, column = {}}, For[q = Prime[2], q^2 <= bound, q = NextPrime[q], For[h = 1, q^(2*h) <= bound, h++, AppendTo[column, q^(2*h)]]]; Prepend[Sort[column], 1]]
a259417[25000] (* data *)
With[{upto=25000},Select[Union[Flatten[Table[Prime[Range[2,Floor[ Sqrt[ upto]]]]^n,{n,0,Log[2,upto],2}]]],#<=upto&]] (* Harvey P. Dale, Nov 25 2017 *)

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (P(2*k) - 1/2^(2*k)) = 1.21835996432366585110..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

A280346 Numbers with 79 divisors.

Original entry on oeis.org

302231454903657293676544, 16423203268260658146231467800709255289, 3308722450212110699485634768279851414263248443603515625, 827269706064171159838078900184013751038269841857389464208009274449, 1692892739326831320764318961708001178036611459414853872137348292520966629744627081
Offset: 1

Views

Author

Omar E. Pol, Jan 01 2017

Keywords

Comments

Also, 78th powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 79.

Examples

			a(1) = 2^78, a(2) = 3^78, a(3) = 5^78, a(4) = 7^78, a(5) = 11^78.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280299, A280301, A280347, A280349, A261700.

Programs

  • Mathematica
    With[{p = 22}, Table[Prime[n]^(Prime@ p - 1), {n, 5}]] (* Michael De Vlieger, Jan 01 2017 *)
  • PARI
    a(n)=prime(n)^78

Formula

a(n) = A000040(n)^(79-1) = A000040(n)^78.
A000005(a(n)) = 79.

A280347 Numbers with 83 divisors.

Original entry on oeis.org

4835703278458516698824704, 1330279464729113309844748891857449678409, 2067951531382569187178521730174907133914530277252197265625, 1986274564260074954771227439341817016242885890299592103563430267952049, 24785642596484137367310393918366845247634028377292875541962916350799472426091085092921
Offset: 1

Views

Author

Omar E. Pol, Jan 01 2017

Keywords

Comments

Also, 82nd powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 83.

Examples

			a(1) = 2^82, a(2) = 3^82, a(3) = 5^82, a(4) = 7^82, a(5) = 11^82.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280299, A280301, A280346, A280349, A261700.

Programs

  • Mathematica
    With[{p = 23}, Table[Prime[n]^(Prime@ p - 1), {n, 5}]] (* Michael De Vlieger, Jan 01 2017 *)
  • PARI
    a(n)=prime(n)^82

Formula

a(n) = A000040(n)^(83-1) = A000040(n)^82.
A000005(a(n)) = 83.

A280349 Numbers with 89 divisors.

Original entry on oeis.org

309485009821345068724781056, 969773729787523602876821942164080815560161, 32311742677852643549664402033982923967414535582065582275390625, 233683216210633558353880137011125430143959282107856711392134007594290612801, 43909277783870034878569768760415886733743786946105343887995366053338664170638348798300219681
Offset: 1

Views

Author

Omar E. Pol, Jan 01 2017

Keywords

Comments

Also, 88th powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 89.

Examples

			a(1) = 2^88, a(2) = 3^88, a(3) = 5^88, a(4) = 7^88, a(5) = 11^88.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280299, A280301, A280346, A280347, A261700.

Programs

  • Mathematica
    With[{p = 24}, Table[Prime[n]^(Prime@ p - 1), {n, 5}]] (* Michael De Vlieger, Jan 01 2017 *)
  • PARI
    a(n)=prime(n)^88

Formula

a(n) = A000040(n)^(89-1) = A000040(n)^88.
A000005(a(n)) = 89.

A175748 Numbers with 39 divisors.

Original entry on oeis.org

36864, 102400, 200704, 495616, 692224, 1183744, 1478656, 2125764, 2166784, 3444736, 3936256, 5607424, 6885376, 7573504, 9048064, 11505664, 13286025, 14258176, 15241216, 18386944, 20647936, 21827584, 25563136, 26040609, 28217344, 32444416, 38539264, 41783296
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the forms p^38 and p^12*q^2, where p and q are distinct primes.

Links

Crossrefs

Cf. A000005, A139572, A175747.

Programs

Formula

A000005(a(n)) = 39.
Sum_{n>=1} 1/a(n) = P(2)*P(12) - P(14) + P(38) = 0.0000500204..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

Extensions

Extended by T. D. Noe, May 08 2011

A280350 Numbers with 97 divisors.

Original entry on oeis.org

79228162514264337593543950336, 6362685441135942358474828762538534230890216321, 12621774483536188886587657044524579674771302961744368076324462890625, 1347137238494276547832006567721872890819326613454654477690085519113574118965817601, 9412343651268540526001186511911506574868063110469548823950876000379062365652829504091329792873336961
Offset: 1

Views

Author

Omar E. Pol, Jan 02 2017

Keywords

Comments

Also, 96th powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 97.

Examples

			a(1) = 2^96, a(2) = 3^96, a(3) = 5^96, a(4) = 7^96, a(5) = 11^96.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280299, A280301, A280346, A280347, A280349, A261700.

Programs

  • Magma
    [NthPrime(n)^96: n in [1..5]]; // Vincenzo Librandi, Jan 06 2017
  • Mathematica
    With[{p = 25}, Table[Prime[n]^(Prime[p] - 1), {n, 5}]] (* Michael De Vlieger, Jan 02 2017 *)
  • PARI
    a(n)=prime(n)^96

Formula

a(n) = A000040(n)^(97-1) = A000040(n)^96.
A000005(a(n)) = 97.
Previous Showing 11-18 of 18 results.

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