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 51-57 of 57 results.

A377044 The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).

Original entry on oeis.org

-1, 1, 4, 4, 9, 17, 18, 21, 23, 33, 47, 62, 77, 96, 98, 99, 113, 137, 159, 175, 182, 196, 207, 236, 265, 282, 297, 333, 370, 411, 433, 448, 493, 536, 579, 628, 681, 734, 791, 848, 879, 899, 962, 1028, 1094, 1159, 1192, 1220, 1293, 1364, 1437, 1514, 1559, 1591
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2024

Keywords

Comments

Perfect-powers (A001597) are numbers with a proper integer root.

Crossrefs

Including 1 with the prime-powers gives A377043.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, A093555, A376596.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A025475 lists numbers that are both a perfect-power and a prime-power.
A031218 gives the greatest prime-power <= n.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
A246655 lists the prime-powers, complement A361102, A375708.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A376560, A376561, A377051.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
per=Select[Range[1000],perpowQ];
per-NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,2,Length[per]-1]
  • Python
    from sympy import mobius, primepi, integer_nthroot
def A377044(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: 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-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def g(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024

Formula

a(n) = A001597(n) - A246655(n).

A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

Original entry on oeis.org

2, 6, 10, 14, 22, 30, 34, 38, 42, 46, 58, 61, 66, 70, 73, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 133, 138, 142, 145, 154, 158, 166, 173, 178, 182, 186, 190, 194, 202, 205, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262, 266, 273, 277, 282
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2024

Keywords

Comments

These are the positions of 2 in A378369 (difference between n and the next nonsquarefree number).
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = A065474 - A206256 = 0.19714711803343537224... . - Amiram Eldar, Dec 03 2024

Links

Crossrefs

Complement of A007675 within A007674.
The version for prime power instead of nonsquarefree is a subset of A006549.
Another variation is A073247.
The version for nonprime instead of squarefree is A179384.
Positions of 0 in A378369 are A013929.
Positions of 1 in A378369 are A373415.
Positions of 2 in A378369 are A378458 (this).
Positions of 3 in A378369 are A007675.
A000961 lists the powers of primes, differences A057820.
A120327 gives the least nonsquarefree number >= n.
A378373 counts composite numbers between nonsquarefree numbers.
Cf. A065474, A067535, A081221, A206256, A236575, A377046, A378033, A378039, A378086.

Programs

  • Mathematica
    Select[Range[100],NestWhile[#+1&,#,SquareFreeQ[#]&]==#+2&]
  • PARI
    list(lim) = my(q1 = 1, q2 = 1, q3); for(k = 3, lim, q3 = issquarefree(k); if(q1 && q2 &&!q3, print1(k-2, ", ")); q1 = q2; q2 = q3); \\ Amiram Eldar, Dec 03 2024

A180868 Numbers n such that n and n+1 are semiprime powers.

Original entry on oeis.org

9, 14, 15, 21, 25, 33, 34, 35, 38, 57, 64, 81, 85, 86, 93, 94, 118, 121, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 215, 216, 217, 218, 225, 253, 298, 301, 302, 326, 334, 361, 381, 393, 394, 445, 446, 453, 481, 484, 501
Offset: 1

Views

Author

Jonathan Vos Post, Jan 22 2011

Keywords

Comments

This is to semiprimes A001358 and powers of semiprimes A085155 as A006549 is to primes A000040 and powers of primes A000961.

Examples

			15 is in the sequence because 15 = (3*5)^1 and 15+1 = 16 = (2*2)^2 are both semiprime powers.

Links

Crossrefs

Cf. A000961, A001358, A006549, A085155.

Programs

  • Maple
    spp:= proc(n) option remember; local l;
        if n<2 or isprime(n) then false
        else l:= ifactors(n)[2];
             if nops(l)>2 then false
           elif nops(l)=2 then evalb(l[1][2]=l[2][2])
           else evalb(irem(l[1][2], 2)=0)
             fi
        fi
      end:
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 8, a(n-1))
        while not spp(k) or not spp(k+1)
      do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 22 2011
  • Mathematica
    sppQ[n_] := With[{f = FactorInteger[n][[All, 2]]}, n==1 || Length[f]==1 && EvenQ[f[[1]]] || Length[f]==2 && f[[1]]==f[[2]]];
Select[Range[1000], sppQ[#] && sppQ[#+1]&] (* Jean-François Alcover, Nov 21 2020 *)

Formula

{ n : {n,n+1} is subset of {A085155} } = { n : n = A001358(i)^j and n+1 = A001358(k)^m }.

Extensions

More terms and edited by Alois P. Heinz, Jan 22 2011

A294280 a(n) = least positive k such that omega(n+k) > max(omega(n), omega(k)), where omega(m) = A001221(m), the number of distinct primes dividing m.

Original entry on oeis.org

1, 4, 3, 2, 1, 24, 3, 2, 1, 20, 1, 18, 1, 16, 15, 2, 1, 12, 1, 10, 9, 8, 1, 6, 1, 4, 1, 2, 1, 180, 2, 1, 9, 8, 7, 6, 1, 4, 3, 2, 1, 168, 1, 16, 15, 14, 1, 12, 1, 10, 9, 8, 1, 6, 5, 4, 3, 2, 1, 150, 1, 4, 3, 1, 1, 144, 1, 2, 1, 140, 1, 6, 1, 4, 3, 2, 1, 132, 1
Offset: 1

Views

Author

Rémy Sigrist, Oct 26 2017

Keywords

Comments

For any n > 0, a(n) <= n * (A053669(n) - 1).
Apparently, a(n) = n * (A053669(n) - 1) iff n belongs to A077011.
a(n) = 1 iff omega(n) < omega(n+1).
a(p) = 1 for any prime power p not in A006549.
The scatterplot of the sequence shows segments of slope -1, corresponding to frequent values of n+a(n); these segments correspond to the strands in the plot of the ordinal transform of n+a(n) (see plots in Links section).

Examples

			For n=2:
- omega(2+1) = 1 = omega(2),
- omega(2+2) = 1 = omega(2),
- omega(2+3) = 1 = omega(2),
- omega(2+4) = 2 > max(omega(2), omega(4)) = 1,
- hence, a(2) = 4.

Links

Crossrefs

Cf. A001221, A006549, A053669, A077011, A294277.

Programs

  • PARI
    a(n) = my (on=omega(n)); for (k=1, oo, if (omega(n+k) > max(on, omega(k)), return (k)))

A376163 Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.

Original entry on oeis.org

4, 7, 8, 14, 15, 16, 18, 19, 22, 23, 26, 27, 29, 30, 31, 32, 35, 37, 39, 40, 43, 44, 45, 46, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 66, 67, 70, 71, 73, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105
Offset: 1

Views

Author

Gus Wiseman, Sep 13 2024

Keywords

Examples

			The non-prime-powers (inclusive) are 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ... which increase by 1 after positions 4, 7, 8, ...

Crossrefs

For prime-powers inclusive (A000961) we have A375734, differences A373671.
For nonprime numbers (A002808) we have A375926, differences A373403.
For prime-powers exclusive (A246655) we have A375734(n+1) + 1.
First differences are A373672.
The exclusive version is a(n) - 1 = A375713.
Positions of 1's in A375735.
For non-perfect-powers we have A375740.
Prime-powers inclusive:
- terms: A000961
- differences: A057820
- runs: A373675, A373673, A373674, A174965
- antiruns: A373576, A120430, A006549, A373671
Non-prime-powers inclusive:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- antiruns: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.

Programs

  • Mathematica
    ce=Select[Range[2,100],!PrimePowerQ[#]&];
Select[Range[Length[ce]-1],ce[[#+1]]==ce[[#]]+1&]

A269721 Integers k such that k, k+2, k+4 and k+6 are prime powers (A000961).

Original entry on oeis.org

1, 3, 5, 7, 23, 25
Offset: 1

Views

Author

Altug Alkan, Mar 04 2016

Keywords

Comments

At least one of a(n), a(n)+2, a(n)+4 and a(n)+6 must be a power of 3. See comments in A264734.

Examples

			5 is a term because 5, 7, 11 are prime numbers and 9 = 3^2.
23 is a term because 23 and 29 are prime numbers and 25 = 5^2, 27 = 3^3.
25 is a term because 25 = 5^2, 27 = 3^3, 29 and 31 are prime numbers.

Crossrefs

Cf. A000961, A006549, A120431, A264734.

Programs

  • Mathematica
    Select[Range[0, 10^5], AllTrue[Range[0, 6, 2] + #, Or[# == 1, PrimePowerQ@ #] &] &] (* Michael De Vlieger, Mar 04 2016, Version 10 *)
  • PARI
    lista(nn) = for(n=1, nn, if(n==1 || (isprimepower(n) && isprimepower(n+2) && isprimepower(n+4) && isprimepower(n+6)), print1(n, ", ")));

A325480 a(n) is the largest integer m such that the product of n consecutive integers starting at m is divisible by at most n primes.

Original entry on oeis.org

16, 24, 24, 45, 48, 49, 120, 120, 125, 189, 240, 240, 350, 350, 350, 350, 374, 494, 494, 714, 714, 714, 714, 825, 832, 1078, 1078, 1078, 1078, 1425, 1440, 1440, 1856, 2175, 2175, 2175, 2175, 2175, 2175, 2175, 2870, 2870, 2870, 2871, 2880, 2880, 2880, 3219
Offset: 3

Views

Author

Onno M. Cain, Sep 06 2019

Keywords

Comments

Each term is only conjectured and has been verified up to 10^6.
Note a(2) is undefined if there are infinitely many Mersenne primes.

Examples

			For example, a(3) = 16 because 16 * 17 * 18 = 2^5 * 3^2 * 17 admits only three prime divisors (2, 3, and 17) and appears to be the largest product of three consecutive integers with the property.

Crossrefs

Cf. A002182, A045619, A163264, A164799, A239035, A244656, A006549.

Programs

  • SageMath
    for r in range(3, 100):
  history = []
  M = 0
  for n in range(1, 100000):
    primes = {p for p, _ in factor(n)}
    history.append(primes)
    history = history[-r:]
    total = set()
    for s in history: total |= s
    # Skip if too many primes.
    if len(total) > r: continue
    if n > M: M = n
  print(r, M-r+1)
Previous Showing 51-57 of 57 results.

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

Content is available under The OEIS End-User License Agreement.