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 1-6 of 6 results.

A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.

Original entry on oeis.org

1, 4, 27, 48, 72, 108, 162, 320, 800, 1792, 2000, 3125, 3840, 5000, 5760, 6272, 8640, 9600, 10935, 12500, 12960, 14400, 18225, 19440, 21504, 21600, 21952, 24000, 29160, 30375, 31250, 32256, 32400, 36000, 43740, 45056, 48384, 48600, 50625, 54000, 60000, 65610
Offset: 1

Views

Author

Leroy Quet, May 09 2000

Keywords

Comments

Numbers for which the sum of distinct prime factors equals the sum of exponents in the prime factorization, A008472(n)=A001222(n). - Reinhard Zumkeller, Mar 08 2002

Examples

			320 is included because 320 = 2^6 * 5^1 and 2+5 = 6+1.

Links

Crossrefs

Cf. A054412, A068935, A068936, A068937, A068938.

Programs

  • Mathematica
    f[n_]:=Plus@@First/@FactorInteger[n]==Plus@@Last/@FactorInteger[n]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,0,3*8!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
max = 10^12; Sort@Reap[Sow@1; Do[p = Select[IntegerPartitions[se, All, Prime@ Range@ PrimePi@ se], Sort[#] == Union[#] &]; Do[ np = Length[f]; va = IntegerPartitions[se, {np}, Range[se]]; Do[pe = Permutations[v]; Do[z = Times @@ (f^e); If[z <= max, Sow@z], {e, pe}], {v, va}], {f, p}], {se, 2, Log2[max]}]][[2, 1]] (* Giovanni Resta, May 07 2016 *)
  • PARI
    for(n=1,10^6,if(bigomega(n)==sumdiv(n,d,isprime(d)*d),print1(n,",")))
  • PARI
    is(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]-f[i,2])==0 \\ Charles R Greathouse IV, Sep 08 2016
  • Sage
    def d(n):
    v=factor(n)[:]; L=len(v); s0=sum(v[j][0] for j in range(L)); s1=sum(v[j][1] for j in range(L))
    return s0-s1
[k for k in (1..100000) if d(k)==0] # Giuseppe Coppoletta, May 07 2016

A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 48, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 320, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 800, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1792, 1944, 2000, 2048, 2187, 2304, 2560, 2592, 2916
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 08 2002

Keywords

Comments

The product of any two terms is also a term. - Amiram Eldar, May 14 2025

Examples

			a(5) = 27 = 3^3, 3 = 3.
a(10) = 81 = 3^4, 3 < 4.
a(100) = 16000 = 2^7 * 5^3,  2+5 < 7+3.
a(1000) = 10321920 = 2^15 * 3^2 * 5 * 7, 2+3+5+7 < 15+2+1+1.

Links

Crossrefs

Cf. A001222, A008472, A068935, A068937, A068938.
Subsequences: A054411, A100716, A257278.

Programs

  • Haskell
    a068936 n = a068936_list !! (n-1)
a068936_list = [x | x <- [1..], a008472 x <= a001222 x]
-- Reinhard Zumkeller, Nov 10 2013
  • Mathematica
    fQ[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f >= Plus @@ First /@ f]; Select[ Range@3000, fQ@ # &] (* Robert G. Wilson v, Jan 16 2006 *)
Select[Range@ 3000, First@ Differences@ Map[Total, Transpose@ FactorInteger@ #] >= 0 &] (* Michael De Vlieger, Dec 08 2016 *)
  • PARI
    isok(k) = {my(f = factor(k)); vecsum(f[,1]) <= bigomega(f);} \\ Amiram Eldar, May 14 2025

Extensions

More terms from Robert G. Wilson v, Jan 16 2006

A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74
Offset: 1

Views

Author

Gus Wiseman, Mar 16 2023

Keywords

Comments

Differs from A068938 in having 1 and 4 and lacking 80.
Includes all squarefree numbers.

Examples

			The prime indices of 80 are {1,1,1,1,3}, with 5 parts and 2 distinct parts, and 2*2 < 5, so 80 is not in the sequence.

Crossrefs

Complement of A360558.
Positions of nonnegative terms in A361205.
These partitions are counted by A361394.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
less: A360254, ranks A360558
equal: A239959, ranks A067801
greater: A237365, ranks A361393
less or equal: A237363, ranks A361204
greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324521, A324560, A324562.
Cf. A360248, A360249, A360454.

Programs

  • Mathematica
    Select[Range[100],2*PrimeNu[#]>=PrimeOmega[#]&]

Formula

A001222(a(n)) <= 2*A001221(a(n)).

A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

Original entry on oeis.org

8, 16, 32, 64, 81, 96, 128, 144, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2048, 2187, 2304, 2560, 2592, 2916, 3072, 3200, 3456, 3584, 3888, 4000, 4096, 4374, 4608
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 08 2002

Keywords

Examples

			144 is included because 144 = 2^4 * 3^2 and 2+3 < 4+2.

Links

Crossrefs

Cf. A068936, A054411, A068937, A068938.

Programs

  • Mathematica
    okQ[n_]:=Module[{tfi=Transpose[FactorInteger[n]]},Total[First[tfi]]Harvey P. Dale, Jan 17 2011 *)
  • PARI
    isok(n) = vecsum(factor(n)[,1]) < bigomega(n); \\ Michel Marcus, Apr 25 2016

Extensions

More terms from Michel Marcus, Apr 25 2016

A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 08 2002

Keywords

Links

Crossrefs

Cf. A068935, A068936, A054411, A068938.

Programs

  • Mathematica
    Select[Range[76], Plus @@ (f = FactorInteger[#])[[;;,1]] >= Plus @@ f[[;;,2]] &] (* Amiram Eldar, Nov 14 2019 *)

Extensions

More terms from David Wasserman, Jun 17 2002

A356433 Numbers k such that, in the prime factorization of k, the least common multiple of the exponents equals the least common multiple of the prime factors.

Original entry on oeis.org

1, 4, 27, 72, 108, 192, 576, 800, 1458, 1728, 2916, 3125, 5120, 5832, 6272, 12500, 21600, 25600, 30375, 36000, 46656, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 388800, 395136, 583200, 600000, 653184, 691200, 750141, 802816, 823543, 857304, 979776
Offset: 1

Views

Author

Jean-Marc Rebert, Aug 07 2022

Keywords

Comments

Numbers k such that A072411(k) = A007947(k). - Michel Marcus, Aug 29 2022
Terms p^p, p prime, form the subsequence A051674. - Bernard Schott, Sep 21 2022
Terms p^q * q^p with distinct primes p and q form the subsequence A082949. - Bernard Schott, Feb 01 2023

Examples

			576 = 2^6 * 3^2, lcm(2,3) = 6 = lcm(6,2), hence 576 is a term.

Crossrefs

Cf. A054411, A054412, A068935, A068936, A068937, A068938.
Cf. A072411, A007947, A051674, A082949.

Programs

  • Mathematica
    Select[Range[10^6], Equal @@ LCM @@ FactorInteger[#] &] (* Amiram Eldar, Aug 07 2022 *)
  • PARI
    isok(k) = my(f=factor(k)); lcm(f[,1]) == lcm(f[,2]); \\ Michel Marcus, Aug 07 2022
  • Python
    from math import lcm
from sympy import factorint
def ok(n): f = factorint(n); return lcm(*f.keys()) == lcm(*f.values())
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Aug 07 2022
Showing 1-6 of 6 results.

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

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