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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A343300 a(n) is p1^1 + p2^2 + ... + pk^k where {p1,p2,...,pk} are the distinct prime factors in ascending order in the prime factorization of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 2, 3, 27, 11, 11, 13, 51, 28, 2, 17, 11, 19, 27, 52, 123, 23, 11, 5, 171, 3, 51, 29, 136, 31, 2, 124, 291, 54, 11, 37, 363, 172, 27, 41, 354, 43, 123, 28, 531, 47, 11, 7, 27, 292, 171, 53, 11, 126, 51, 364, 843, 59, 136, 61, 963, 52, 2, 174, 1342, 67, 291, 532, 370, 71, 11, 73
Offset: 1

Views

Author

Giorgos Kalogeropoulos, Apr 11 2021

Keywords

Comments

From Bernard Schott, May 07 2021: (Start)
a(n) depends only on prime factors of n (see formulas).
Primes are fixed points of this sequence.
Terms are in increasing order in A344023. (End)

Examples

			a(60) = 136 because the distinct prime factors of 60 are {2, 3, 5} and 2^1 + 3^2 + 5^3 = 136.

Links

Crossrefs

Cf. A027748, A344023 (terms ordered).
See also A033845-A033851, A288162.

Programs

  • Maple
    a:= n-> (l-> add(l[i]^i, i=1..nops(l)))(sort(map(i-> i[1], ifactors(n)[2]))):
seq(a(n), n=1..73);  # Alois P. Heinz, Sep 19 2024
  • Mathematica
    {0}~Join~Table[Total[(a=First/@FactorInteger[k])^Range@Length@a],{k, 2, 100}]
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^k); \\ Michel Marcus, Apr 11 2021

Formula

a(p^k) = p for p prime and k>=1.
From Bernard Schott, May 07 2021: (Start)
a(A033845(n)) = 11;
a(A033846(n)) = 27;
a(A033847(n)) = 51;
a(A033848(n)) = 123;
a(A033849(n)) = 28;
a(A033850(n)) = 52;
a(A033851(n)) = 54;
a(A288162(n)) = 171. (End)

A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

Original entry on oeis.org

4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241
Offset: 1

Views

Author

Michel Marcus, Dec 17 2020

Keywords

Comments

Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - David A. Corneth, Dec 19 2020
From Bernard Schott, Jan 19 2021: (Start)
Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)

Examples

			   n  a(n) prime factor set
   1    4  [2]           A000079
   2    9  [3]           A000244
   3   25  [5]           A000351
   4   18  [2, 3]        A033845
   5   49  [7]           A000420
   6   80  [2, 5]        A033846
   7  121  [11]          A001020
   8  169  [13]          A001022
   9  112  [2, 7]        A033847
  10  135  [3, 5]        A033849
  11  289  [17]          A001026
  12  361  [19]          A001029
  13  441  [3, 7]        A033850
  14  352  [2, 11]       A033848
  15  529  [23]          A009967
  16  416  [2, 13]       A288162
  17  841  [29]          A009973
  18  360  [2, 3, 5]     A143207

Links

Crossrefs

Cf. A000203 (sigma), A007947 (rad).
Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
Subsequence: A001248 (squares of primes).

Programs

  • PARI
    u(n) = {my(fn=factor(n)[,1]); for (k = n, n^2, my(fk = factor(k)); if (fk[,1] == fn, if (factorback(fk[,1])^2 < sigma(fk), return (k));););}
lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", ");););}

Formula

a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020

A370266 Numbers k that are not prime powers, such that k/rad(k) >= rad(k), where rad(k) = A007947(k).

Original entry on oeis.org

36, 48, 54, 72, 96, 100, 108, 144, 160, 162, 192, 196, 200, 216, 224, 225, 250, 288, 320, 324, 375, 384, 392, 400, 405, 432, 441, 448, 484, 486, 500, 567, 576, 640, 648, 675, 676, 686, 704, 768, 784, 800, 832, 864, 896, 900, 960, 968, 972, 1000, 1029, 1080, 1089
Offset: 1

Views

Author

Michael De Vlieger, Feb 18 2024

Keywords

Comments

Numbers k = m * s, where s is composite and squarefree, rad(m) | s, and m >= s.
A177492 is a proper subset.

Examples

			For s = 6, this sequence contains {36, 48, 54, 72, 96, ...}, i.e., A033845(n) for n >= A010846(6).
For s = 10, this sequence contains {100, 160, 200, 250, 320, ...}, i.e., A033846(n) for n >= A010846(10).
For s = 14, this sequence contains {196, 224, 392, 448, 686, ...}, i.e., A033847(n) for n >= A010846(14).
For s = 15, this sequence contains {225, 375, 405, 675, 1125, ...}, i.e., A033849(n) for n >= A010846(15), etc.

Links

Crossrefs

Cf. A007947, A010846, A024619, A120944, A126706, A177492, A246547, A286708, A341645, A366250.

Programs

  • Mathematica
    Select[Select[Range[2, 1100], Not@*PrimePowerQ], #1/#2 >= #2 & @@ {#, Times @@ FactorInteger[#][[All, 1]]} &]

Formula

Set difference of A341645 and A246547.
Intersection of A341645 and A126706.
Union of A286708 and A366250.

A095678 Minimal sequence such that all triples of consecutive numbers have no common divisor greater than 1, but all three pairs within the triples are not coprime.

Original entry on oeis.org

6, 10, 15, 12, 20, 45, 18, 40, 75, 24, 50, 135, 36, 80, 225, 48, 100, 375, 54, 160, 405, 72, 200, 675, 96, 250, 1125, 108, 320, 1215, 144, 400, 1875, 162, 500, 2025, 192, 640, 3375, 216, 800, 3645, 288, 1000, 5625, 324, 1250, 6075, 384, 1280, 9375, 432
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 04 2004

Keywords

Links

Crossrefs

Cf. A001221, A003586, A003592, A003593, A006530, A020639, A033845, A033846, A033849.

Programs

  • Mathematica
    seq1[p_, q_, lim_] := Sort[Flatten[Table[p^i * q^j, {i, 1, Log[p, lim]}, {j, 1, Log[q, lim/p^i]}]]];
seq[lim_] := Module[{s1 = seq1[2, 3, lim], s2 = seq1[2, 5, lim], s3 = seq1[3, 5, lim], ns}, ns = Length[s3]; Flatten[Transpose[{s1[[1;;ns]], s2[[1;;ns]], s3}]]]; seq[10^4] (* Amiram Eldar, Sep 29 2024 *)

Formula

gcd(a(n),a(n+1),a(n+2)) = 1, gcd(a(n),a(n+1)) > 1, gcd(a(n),a(n+2)) > 1 and gcd(a(n+1),a(n+2)) > 1.
A001221(a(n)) = 2; 2 <= A020639(a(n)) <= 3 <= A006530(a(n)) <= 5.
From Jianing Song, Jun 08 2022: (Start)
a(3n-2) = A033845(n) = 6*A003586(n);
a(3n-1) = A033846(n) = 10*A003592(n);
a(3n) = A033849(n) = 15*A003593(n). (End)
Sum_{n>=1} 1/a(n) = 7/8. - Amiram Eldar, Sep 29 2024

A370409 Numbers k = m * s, where s is composite and squarefree, rad(m) divides s, and 1 < m <= s, where rad() = A007947().

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 50, 52, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 98, 99, 100, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 196, 198, 204, 207, 208
Offset: 1

Views

Author

Michael De Vlieger, Feb 22 2024

Keywords

Comments

A177492 is a proper subset.
Proper subset of A126706.

Examples

			Let T(j,k) = row j of A162306 and let s = A120944(n), n > 1.
This sequence contains finite sequences R(s) = s * T(s, 2..A010846(s)). The cardinality of R(s) is A010846(s)-1.
For s = 6, this sequence contains {12, 18, 24, 36},
  i.e., A033845(2..A010846(6)).
For s = 10, this sequence contains {20, 40, 50, 80, 100},
  i.e., A033846(2..A010846(10)).
For s = 14, this sequence contains {28, 56, 98, 112, 196},
  i.e., A033847(2..A010846(14)).
For s = 15, this sequence contains {45, 75, 135, 225},
  i.e., A033849(2..A010846(15)), etc.

Links

Crossrefs

Cf. A005117, A007947, A120944, A126706, A162306, A177492, A370266.

A309944 Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, where p_1 < ... < p_k are primes, then for all i < k, p_i = A000720(p_{i+1}).

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 55, 60, 72, 75, 90, 96, 108, 119, 120, 135, 144, 150, 162, 165, 180, 192, 216, 225, 240, 270, 275, 288, 300, 324, 330, 341, 360, 375, 384, 405, 432, 450, 480, 486, 495, 533, 540, 576, 600, 605, 648, 660, 675, 720, 750, 768
Offset: 1

Views

Author

Michel Lagneau, Aug 24 2019

Keywords

Comments

Numbers m such that for all k, d(k) = prime(d(k-1)), where d(k) is the k-th prime factor of m.
The primitive subsequence b(k), k = 1, 2, ... begins with 6, 15, 30, 55, 110, 165, 330, 341, 533, ... because if d(i) is the i-th prime factor of b(k), so b(k)*d(i)^m is in the sequence, m = 0, 1, 2, ...
Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then for all i > 1, p_i = A000040(p_{i-1}). - Antti Karttunen, Aug 24 2019

Examples

			330 is in the sequence because the prime factors are {2, 3, 5, 11} with 3 = prime(2), 5 = prime(3) and 11 = prime(5).
1299210 is in the sequence because the prime factors are {2, 3, 5, 11, 31, 127} with 3 = prime(2), 5 = prime(3), 11 = prime(5), 31 = prime(11) and 127 = prime(31).

Links

Crossrefs

Cf. A000040, A000720, A006450, A027746, A027748
Contains A033845, A033849, A143207.

Programs

  • Magma
    sol:=[]; s:=1; for m in [2..1000] do v:=PrimeDivisors(m);  if #v ge 2 then nr:=0; for k in [2..#v] do  if v[k] eq NthPrime(v[k-1])  then nr:=nr+1;  end if; end for; if nr eq #v-1 then sol[s]:=m;s:=s+1; end if; end if; end for;  sol; // Marius A. Burtea, Aug 24 2019
  • Maple
    with(numtheory):nn:=10^3:
for n from 1 to nn do:
d:=factorset(n):n0:=nops(d):it:=0:
  if n0>1
  then
  for i from 2 to n0 do :
   if d[i]=ithprime(d[i-1])
    then
    it:=it+1:
    else fi:
   od:
    if it=n0-1
    then
    printf(`%d, `,n):
    else fi:fi:
od:
  • Mathematica
    aQ[n_] := (m = Length[(p = FactorInteger[n][[;; , 1]])]) > 1 && NestList[Prime@# &, p[[1]], m - 1] == p; Select[Range[770], aQ] (* Amiram Eldar, Aug 24 2019 *)
  • PARI
    isok(m) = {my(f=factor(m)[,1]~); if (#f < 2, return(0)); for (i=2, #f, if (f[i] != prime(f[i-1]), return (0));); return (1);} \\ Michel Marcus, Aug 25 2019

Extensions

Edited by N. J. A. Sloane, Oct 05 2019, using definition suggested by Antti Karttunen, Aug 24 2019
Previous Showing 21-26 of 26 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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