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 1-5 of 5 results.

A381400 Numbers k >= 2 such that A115588(k) != A381399(k).

Original entry on oeis.org

64, 81, 256, 320, 405, 448, 512, 567, 625, 704, 729, 832, 891, 1024, 1053, 1088, 1216, 1280, 1377, 1472, 1539, 1600, 1792, 1856, 1863, 1875, 1984, 2240, 2349, 2368, 2401, 2511, 2560, 2624, 2752, 2816, 2835, 2997, 3008, 3072, 3136, 3321, 3328, 3392, 3483, 3520
Offset: 1

Views

Author

Paolo Xausa, Feb 22 2025

Keywords

Links

Crossrefs

Cf. A115588, A381399.

Programs

  • Mathematica
    A115588[n_] := Length[Union[NestWhile[DeleteCases[Flatten[FactorInteger[#]], 1] &, {n}, AnyTrue[#, CompositeQ] &]]];
A381399[n_] := Count[Union[Flatten[FactorInteger[n]]], _?PrimeQ];
A381400Q[k_] := A115588[k] != A381399[k];
Select[Range[2, 5000], A381400Q]

A339328 Integers m such that A240471(m) > A115588(m).

Original entry on oeis.org

16, 24, 27, 28, 32, 36, 44, 48, 50, 52, 54, 55, 60, 64, 65, 68, 72, 76, 77, 80, 81, 84, 85, 90, 91, 92, 95, 96, 98, 100, 105, 108, 110, 112, 115, 116, 119, 120, 124, 125, 126, 128, 130, 132, 133, 135, 136, 140, 143, 144, 145, 148, 150, 152, 154, 155, 156, 160
Offset: 1

Views

Author

Thomas Scheuerle, Nov 30 2020

Keywords

Comments

Integers m such that integer part of the harmonic mean of divisors of m is greater than the number of distinct prime numbers necessary to represent m.
For all m not in this sequence this integer part is equal to the number of distinct prime numbers necessary to represent m.
This correlation between A240471 and A115588 contains some apparently random component.
If the integer part of the harmonic mean of divisors of m equals 1 we will find an 1 in A115588(m) too, for all m. If the integer part of the harmonic mean of divisors of m equals 2 we will find 2 in A115588(m) too, with probability of ~0.9877 for m in range 2-1000.
For m until 10000 the only exceptions are 16 and 27. If the integer part of the harmonic mean of divisors of m equals 3 we will find 3 in A115588(m) too, with probability of ~0.1983 for m in range 2-1000. For integer parts greater than 3 the probability gets fast smaller.
If m is a square of a prime it is not in this sequence.
Let m be a semiprime with two distinct prime factors p1 and p2. If m >= 3(1+p1+p2) then m is in this sequence. Example: 55 > 3(1+5+11). This can be generalized for k-almostprimes if all factors are distinct: If m(2^k) >= (1+k)sigma(m) then m is in this sequence. Example: 105*8 > 4*192.
Let p be a prime greater than 2. Let o be a natural number >0 without divisor p, then if m = o*p^p, m is in this sequence. This can be generalized for a set of distinct primes >2 {p_1,p_2,...,p_n} and any permutation of this set {p_a,p_b,...,p_z}, then if m = o*p_1^p_a*p_2^p_b*...*p_n^p_z, m is in this sequence. Example: 3960 = 55*2^3*3^2.
The sequence includes all numbers whose prime factorization contains at least one composite exponent (A322448).

Crossrefs

Cf. A240471, A115588, A322448.

Programs

  • PARI
    listf(f, list) = {for (k=1, #f~, listput(list, f[k,1]); if (isprime(f[k,2]), listput(list, f[k,2]), if (f[k,2] > 1, my(vexp = Vec(listf(factor(f[k,2]), list))); for (i=1, #vexp, listput(list, vexp[i]););););); list;}
a8(n) = {my(f=factor(n), list=List()); #select(isprime, Set(Vec(listf(f, list))));}
a1(n) = n*numdiv(n)\sigma(n);
isok(m) = a1(m) > a8(m); \\ Michel Marcus, Dec 02 2020

A381399 a(n) is the number of prime elements in the set of bases and exponents in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3
Offset: 1

Views

Author

Paolo Xausa, Feb 22 2025

Keywords

Comments

Differs from A115588 at n = 64, 81, 256, 320, 405, 448, 512... = A381400.

Examples

			a(144) = 2 because the prime factorization of 144 is 2^4*3^2 and the set of these bases and exponents is {2, 3, 4}, containing 2 primes.

Links

Crossrefs

Cf. A115588, A381398, A381400, A381401.

Programs

  • Mathematica
    A381399[n_] := Count[Union[Flatten[FactorInteger[n]]], _?PrimeQ];
Array[A381399, 100]

A336965 a(n) is the product of the distinct prime numbers appearing in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 6, 14, 10, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70
Offset: 1

Views

Author

Rémy Sigrist, Aug 09 2020

Keywords

Comments

The prime tower factorization of a number is defined in A182318.
For any n > 0, a(n) is the product of the terms in n-th row of A336964.

Examples

			A001221(a(n)) = A115588(n) for any n > 1.
a(n) = A007947(A279513(n)).
a(n) = n iff n is squarefree (A005117).

Links

Crossrefs

Cf. A001221, A005117, A007947, A115588, A182318, A336964.

Programs

  • PARI
    a(n) = { my (f=factor(n), v=vecprod(f[,1]~)); for (k=1, #f~, v=lcm(v, a(f[k,2]))); v }

A336964 Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime numbers in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 3, 7, 2, 3, 2, 3, 2, 5, 11, 2, 3, 13, 2, 7, 3, 5, 2, 17, 2, 3, 19, 2, 5, 3, 7, 2, 11, 23, 2, 3, 2, 5, 2, 13, 3, 2, 7, 29, 2, 3, 5, 31, 2, 5, 3, 11, 2, 17, 5, 7, 2, 3, 37, 2, 19, 3, 13, 2, 3, 5, 41, 2, 3, 7, 43, 2, 11, 2, 3, 5, 2, 23, 47, 2, 3
Offset: 1

Views

Author

Rémy Sigrist, Aug 09 2020

Keywords

Comments

The prime tower factorization of a number is defined in A182318.
The n-th row includes the n-th row of A027748.

Examples

			Triangle begins:
     1    [1]
     2    [2]
     3    [3]
     4    [2]
     5    [5]
     6    [2, 3]
     7    [7]
     8    [2, 3]
     9    [2, 3]
    10    [2, 5]
    11    [11]
    12    [2, 3]
    13    [13]
    14    [2, 7]
    15    [3, 5]

Links

Crossrefs

Cf. A027748, A115588 (row lengths), A182318, A336965.

Programs

  • PARI
    row(n) = { my (f=factor(n), p=f[,1]~); for (k=1, #f~, if (f[k,2]>1, p=concat(p, row(f[k,2])));); if (#p==0, [1], Set(p)) }
Showing 1-5 of 5 results.

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