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 21-27 of 27 results.

A190111 Numbers with prime factorization p*q*r*s^2*t^3 (where p, q, r, s, t are distinct primes).

Original entry on oeis.org

27720, 32760, 41580, 42840, 46200, 47880, 49140, 51480, 54600, 57960, 64260, 64680, 67320, 71400, 71820, 72072, 73080, 75240, 76440, 77220, 78120, 79560, 79800, 85800, 86940, 88920, 91080, 93240, 94248, 96600, 99960, 100980, 101640, 103320
Offset: 1

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Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Keywords

Examples

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^2)*5*7*11 = 27720;
a(2) = (2^3)*(3^2)*5*7*13 = 32760;
a(3) = (2^2)*(3^3)*5*7*11 = 41580;
a(4) = (2^3)*(3^2)*5*7*17 = 42840;
a(5) = (2^3)*3*(5^2)*7*11 = 46200.
(End)

Links

Crossrefs

Cf. A179645, A189990, A190109, A190110.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,2,3};Select[Range[150000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4); forprime(p=2,sqrtnint(lim\420, 3), t1=p^3; forprime(q=2,sqrtint(lim\(30*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(6*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\(2*t3), if(s==p || s==q || s==r, next); t4=s*t3; forprime(t=2,lim\t4, if(t==p || t==q || t==r || t==s, next); listput(v, t4*t)))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A272191 Either 8th power of a prime, or product of a square and a cube of two different primes.

Original entry on oeis.org

72, 108, 200, 256, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6561, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436, 27848, 29768
Offset: 1

Views

Author

Paolo P. Lava, Apr 22 2016

Keywords

Comments

Numbers such that the sum of the number of divisors of their aliquot parts is four times the number of their divisors.

Examples

			72 = 2^3 * 3^2;  256 = 2^8.

Links

Crossrefs

Cf. A030516, A143610, A179645, A272190.
Cf. A085548, A085541, A085965, A085968.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);
if 4*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);
  • Mathematica
    Select[Range[30000], MemberQ[{{8}, {2, 3}}, Sort[FactorInteger[#][[;; , 2]]]] &] (* Amiram Eldar, Oct 03 2023 *)
  • PARI
    isok(n) = 4*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ Michel Marcus, Apr 22 2016
  • PARI
    is(n) = {my(e = vecsort(factor(n)[, 2])~); e == [8] || e == [2, 3];} \\ Amiram Eldar, Oct 03 2023

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) + P(8) = A085548 * A085541 - A085965 + A085968 = 0.047342..., where P is the prime zeta function. - Amiram Eldar, Oct 03 2023

A375270 Numbers of the form p^Fibonacci(2*k), where p is a prime and k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256
Offset: 1

Views

Author

Amiram Eldar, Aug 09 2024

Keywords

Comments

Differs from A186285 by having the terms 1, 2^8 = 256, 3^8 = 6561, ..., and not having the terms 2^9 = 512, 3^9 = 19683, ... .
The partial products of this sequence (A375271) are the sequence of numbers with record numbers of Zeckendorf-infinitary divisors (A318465).

Examples

			The positive even-indexed Fibonacci numbers are 1, 3, 8, 21, ..., so the sequence includes 2^1 = 2, 2^3 = 8, 2^8 = 256, ..., 3^1 = 3, 3^3 = 27, 3^8 = 6561, ... .

Links

Crossrefs

Subsequence of A115975.
Subsequences: A000040, A030078, A179645.
Cf. A000045, A001906, A050376, A186285, A318465, A375271 (partial products).

Programs

  • Mathematica
    fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k += 2; f = Fibonacci[k]]; s];
seq[max_] := Module[{s = {1}, p = 2, e = 1, f = {}}, While[e > 0, e = Floor[Log[p, max]]; If[f == {}, f = fib[e], f = Select[f, # <= e &]]; s = Join[s, p^f]; p = NextPrime[p]]; Sort[s]]; seq[256]
  • PARI
    fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k += 2; f = fibonacci(k)); Vec(s);}
lista(pmax) = {my(s = [1], p = 2, e = 1, f = []); while(e > 0, e = logint(pmax, p); if(#f == 0, f = fib(e), f = select(x -> x <= e, f)); s = concat(s, apply(x -> p^x, f)); p = nextprime(p+1)); vecsort(s);}

Formula

a(n) = A375271(n)/A375271(n-1) for n >= 2.

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

Links

Crossrefs

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

Programs

  • Mathematica
    q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]
  • PARI
    isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

A236214 Sum of the eighth powers of the first n primes.

Original entry on oeis.org

256, 6817, 397442, 6162243, 220521124, 1036251845, 8012009286, 24995572327, 103306557608, 603552970569, 1456444008010, 4968923461931, 12953848691052, 24642048968653, 48453335630414, 110713026041775, 257543463646096, 449250776643377, 855318454200018
Offset: 1

Views

Author

Robert Price, Jan 20 2014

Keywords

Links

Crossrefs

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
Partial sums of A179645.

Programs

  • Mathematica
    Table[Sum[Prime[k]^8, {k, n}], {n, 1000}]
Accumulate[Prime[Range[20]]^8] (* Harvey P. Dale, Feb 25 2016 *)

Formula

a(n) = Sum_{k=1..n} prime(k)^8.

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228
Offset: 1

Views

Author

Amiram Eldar, Jan 16 2024

Keywords

Comments

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

Links

Crossrefs

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

Programs

  • Mathematica
    gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]
  • PARI
    gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

Links

Crossrefs

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

Programs

  • Mathematica
    q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]
  • PARI
    isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));
Previous Showing 21-27 of 27 results.

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