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.

A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).

Original entry on oeis.org

4, 6, 8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Comments

Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=2 and cannot be eliminated by any sieve s >= 3. - R. J. Mathar, Jun 24 2009
After a(3)=8 all terms are 2*prime; for n > 3, a(n) = 2*prime(n-1) = 2*A000040(n-1). - Zak Seidov, Jul 18 2009
From Omar E. Pol, Jul 18 2009: (Start)
A classification of the natural numbers A000027.
=============================================================
Numbers k whose largest divisor <= sqrt(k) equals j
=============================================================
j Sequence Comment
=============================================================
1 ..... A008578 1 together with the prime numbers
2 ..... A161344 This sequence
3 ..... A161345
4 ..... A161424
5 ..... A161835
6 ..... A162526
7 ..... A162527
8 ..... A162528
9 ..... A162529
10 .... A162530
11 .... A162531
12 .... A162532
... And so on. (End)
The numbers k whose largest divisor <= sqrt(k) is j are exactly those numbers j*m where m is either a prime >= k or one of the numbers in row j of A163925. - Franklin T. Adams-Watters, Aug 06 2009
See also A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Also A100484 UNION 8. - Omar E. Pol, Nov 29 2012 (after Seidov and Hasler)
Is this the union of {4} and A073582? - R. J. Mathar, May 30 2025

Links

Crossrefs

Cf. A000005, A018253, A160811, A160812, A161205, A161346, A033676, A008578, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532, A163925.
Second column of array in A163280. Also, second row of array in A163990.

Programs

  • Maple
    isA := proc(n,s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161344 := proc(n) for s from 3 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,2) ; end: for n from 1 to 3000 do if isA161344(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009
  • Mathematica
    a[n_] := If[n <= 3, 2n+2, 2*Prime[n-1]]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Nov 26 2012, after Zak Seidov *)
  • PARI
    a(n)=if(n>3,prime(n-1),n+1)*2 \\ M. F. Hasler, Nov 27 2012

Formula

Equals 2*A000040 union {8}. - M. F. Hasler, Nov 27 2012
a(n) = 2*A046022(n+1) = 2*A175787(n). - Omar E. Pol, Nov 27 2012

Extensions

More terms from R. J. Mathar, Jun 24 2009
Definition added by R. J. Mathar, Jun 28 2009

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

Links

Crossrefs

Cf. A060679, A009530, A098764.
Equals {2} UNION {A100484}.

Programs

  • GAP
    Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019
  • Magma
    [2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019
  • Mathematica
    Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)
  • PARI
    print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012
  • Sage
    [2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

Formula

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A096014 a(n) = (smallest prime factor of n) * (least prime that is not a factor of n), with a(1)=2.

Original entry on oeis.org

2, 6, 6, 6, 10, 10, 14, 6, 6, 6, 22, 10, 26, 6, 6, 6, 34, 10, 38, 6, 6, 6, 46, 10, 10, 6, 6, 6, 58, 14, 62, 6, 6, 6, 10, 10, 74, 6, 6, 6, 82, 10, 86, 6, 6, 6, 94, 10, 14, 6, 6, 6, 106, 10, 10, 6, 6, 6, 118, 14, 122, 6, 6, 6, 10, 10, 134, 6, 6, 6, 142, 10, 146, 6, 6, 6, 14, 10, 158, 6, 6, 6
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 15 2004

Keywords

Links

Crossrefs

Cf. A001747, A020639, A053669, A073582, A096015.

Programs

  • Maple
    f:= proc(n) local p;
p:= 3;
if n::even then
  while type(n/p,integer) do p:= nextprime(p) od;
else
  while not type(n/p,integer) do p:= nextprime(p) od:
fi;
2*p;
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Jun 22 2018
  • Mathematica
    PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[n]]; f[1] = 2; f[n_] := Block[ {k = 1}, While[ Mod[ n, Prime[k]] == 0, k++ ]; Prime[k]PrimeFactors[n][[1]]]; Table[ f[n], {n, 83}] (* Robert G. Wilson v, Jun 15 2004 *)
spfn[n_]:=Module[{fi=FactorInteger[n][[;;,1]],k=2},While[MemberQ[fi,k],k=NextPrime[k]];fi[[1]]*k]; Array[spfn,90] (* Harvey P. Dale, Sep 22 2024 *)
  • PARI
    dnd(n) = forprime(p=2, , if (n % p, return(p)));
lpf(n) = if (n==1, 1, forprime(p=2, , if (!(n % p), return(p))));
a(n) = dnd(n)*lpf(n); \\ Michel Marcus, Jun 22 2018

Formula

a(n) = A020639(n)*A053669(n);
A096015(n) = a(n)/2.
If n (mod 6) = 2, 3 or 4, then a(n) = 6. If n (mod 6) = 0, 1 or 5, then a(n) belongs to A001747 less the first three terms or belongs to A073582 less the first two terms. - Robert G. Wilson v, Jun 15 2004
From Bill McEachen, Jul 26 2024: (Start)
a(n) <= 2*n, except when n = 2.
a(n) = 2*n for n an odd prime. (End)

A196226 m such that A054024(m) (sum of divisors of m reduced modulo m) is 3 + m/2.

Original entry on oeis.org

8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514
Offset: 1

Views

Author

John W. Layman, Sep 29 2011

Keywords

Comments

This sequence appears to be identical to A073582 with its first term omitted and to A161344 with its first two terms omitted.
Conjectures. (1) If m>=14 is a term of this sequence, then sigma(2,m) is congruent to 5 + m/2 modulo m; (2) If m>=22 is a term of this sequence, then sigma(3,m) is congruent to 9 + m/2 modulo m; If m>=38 is a term of this sequence, then sigma(4,m) is congruent to 17 + m/2 modulo m. (sigma(k,m) denotes the sum of the k-th powers of the divisors of m.)
Similar conjectures can be made about sigma(k,m) congruent to 2^k+1 + m/2 modulo m, for m a sufficiently large term of this sequence..
The even semiprimes (A100484) m= 2*p with p>3, with sigma(2*p)= 3+p (mod 2p), are a subsequence. - R. J. Mathar, Oct 02 2011
The terms in this sequence which are not even semiprimes are 8, 690, 12978, 176946, ... - R. J. Mathar, Aug 24 2023

Crossrefs

Cf. A054024, A073582, A161344.

Programs

  • Maple
    isA196226 := proc(n)
    sigmar := modp(numtheory[sigma](n),n) ;
    if sigmar = 3+n/2 then
        true;
    else
        false;
    end if;
end proc:
A196226 := proc(n)
     option remember;
     if n =1 then
        8;
    else
        for a from procname(n-1)+1 do
            if isA196226(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A196226(n),n=1..100) ; # R. J. Mathar, Aug 24 2023
  • PARI
    lista(nn) = {for(n=1, nn, if ((sigma(n) % n) == (3 + n/2), print1(n, ", ")););} \\ Michel Marcus, Jul 12 2014

A309491 Let gcd_2(b,c) be the second-largest common divisor of non-coprime integers b and c; then a(n) = Sum_{k=1..n} gcd_2(k,n). If b and c are coprime, then gcd_2(b,c) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 8, 5, 10, 1, 17, 1, 14, 11, 20, 1, 26, 1, 29, 15, 22, 1, 44, 9, 26, 21, 41, 1, 56, 1, 48, 23, 34, 17, 73, 1, 38, 27, 76, 1, 80, 1, 65, 51, 46, 1, 108, 13, 74, 35, 77, 1, 102, 25, 108, 39, 58, 1, 157, 1
Offset: 1

Views

Author

Lechoslaw Ratajczak, Aug 04 2019

Keywords

Comments

The first even solution of the equation a(n) = n which divided by 2 is not a prime number is 8. Is there a larger such number? If such a number exists, is greater than 5*10^4. If no such number exists, consecutive even solutions of the above equation are consecutive terms of A073582.

Examples

			a(4) = gcd_2(1,4) + gcd_2(2,4) + gcd_2(3,4) + gcd_2(4,4) = 0 + 1 + 0 + 2 = 3.

Links

Crossrefs

Cf. A018804, A073582.

Programs

  • Mathematica
    r[n_] := If[n==1, 0, n/FactorInteger[n][[1, 1]]]; a[n_] := Sum[r[GCD[n, k]], {k, 1, n}]; Array[a, 70] (* Amiram Eldar, Aug 06 2019 *)
  • PARI
    A032742with_a1_0(n) = if(1==n,0,n/vecmin(factor(n)[,1]));
gcd_2(a,b) = A032742with_a1_0(gcd(a,b));
A309491(n) = sum(k=1,n,gcd_2(k,n)); \\ Antti Karttunen, Dec 28 2019

Extensions

Definition clarified by Antti Karttunen, Dec 28 2019

A385350 Numbers j such that the product of odd proper divisors of j is j.

Original entry on oeis.org

1, 15, 21, 27, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 26 2025

Keywords

Comments

Fixed points of A385349.
Odd terms in A007422.
Also 1 with odd numbers with exactly 4 divisors. - David A. Corneth, Jun 26 2025

Links

Crossrefs

Cf. A007422, A030513, A073582, A385349.

Programs

  • Maple
    q:= n-> n=1 or n::odd and numtheory[tau](n)=4:
select(q, [$1..500])[];  # Alois P. Heinz, Jun 26 2025
  • Mathematica
    A385349[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Select[Range[300], A385349[#] == # &]
  • PARI
    isok(k) = vecprod(select((x->((x%2)==1) && (xMichel Marcus, Jun 26 2025
  • PARI
    is(n) = (n == 1) || (bitand(n, 1) && numdiv(n) == 4) \\ David A. Corneth, Jun 26 2025
  • Python
    from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A385350(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jun 27 2025
Showing 1-6 of 6 results.

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

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