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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A318517 a(n) = A032742(n) XOR n-A032742(n), where XOR is bitwise-xor (A003987) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 0, 3, 0, 5, 0, 7, 0, 5, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 9, 0, 23, 0, 17, 0, 27, 0, 29, 0, 31, 0, 29, 0, 27, 0, 37, 0, 23, 0, 41, 0, 43, 0, 17, 0, 47, 0, 45, 0, 51, 0, 53, 0, 39, 0, 53, 0, 59, 0, 61, 0, 63, 0, 57, 0, 67, 0, 57, 0, 71, 0, 73, 0, 43, 0, 73, 0, 79, 0, 45, 0, 83, 0, 85, 0, 39, 0, 89, 0, 67, 0, 33, 0, 95
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2018

Keywords

Links

Crossrefs

Cf. A003987, A032742, A060681, A106409, A318507, A318516, A318518.

Programs

Formula

a(n) = A003987(A032742(n), A060681(n)).
a(n) = n - 2*A318518(n).

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Antti Karttunen, Oct 04 2018

Keywords

Comments

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].
In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.
Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.
Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.
Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.
There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.
For all i, j:
a(i) = a(j) => A002324(i) = A002324(j),
a(i) = a(j) => A067029(i) = A067029(j),
a(i) = a(j) => A071178(i) = A071178(j),
a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

Examples

			For n = 65 = 5*13 and 143 = 11*13, the smallest prime factor is of the form 6k+5,  doesn't occur more than once in the factorization, and the largest proper divisor is the same number (13) in both cases, thus a(65) = a(143) (= 51, a running count value allotted by rgs-transform for this equivalence class).
For n = 1805 (5*19^2), 3971 (11*19^2), 6137 (17*19^2), it's like above, but the largest proper divisor is in all three cases 361 = 19^2, thus a(1805) = a(3971) = a(6137) (= 1405).
Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3, and also, they may not contain numbers from A283050. See the comments section.

Links

Crossrefs

Cf. A101296, A319707, A319716, A319996.
Cf. also A320004 (analogous sequence for modulo 4 residues).
Differs from A319707 for the first time at n=143, where a(143) = 51, differs from A319716 for the first time at n=121, where a(121) = 95.

Programs

  • PARI
    up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v319717 = rgs_transform(vector(up_to,n,[A286476(n),A319710(n)]));
A319717(n) = v319717[n];

A323888 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A032742(n),A302042(n)] for all n > 1, with f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 4, 6, 2, 7, 2, 8, 6, 9, 2, 10, 2, 11, 12, 13, 2, 14, 6, 15, 16, 17, 2, 18, 2, 19, 20, 21, 8, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 8, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 2, 40, 41, 42, 24, 43, 2, 44, 45, 46, 2, 47, 2, 48, 49, 50, 13, 51, 2, 52, 53, 54, 2, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 2, 67, 68, 69, 2, 70, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 08 2019

Keywords

Comments

For all i, j:
A305800(i) = A305800(j) => a(i) = a(j),
a(i) = a(j) => A001222(i) = A001222(j),
a(i) = a(j) => A253557(i) = A253557(j).

Links

Crossrefs

Cf. A032742, A302042.
Cf. also A001222, A253557, A300247, A305800.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A032742(n) = (n/A020639(n));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A323888aux(n) = if(1==n, 0, [A032742(n),A302042(n)]);
v323888 = rgs_transform(vector(up_to, n, A323888aux(n)));
A323888(n) = v323888[n];

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2019

Keywords

Comments

Nineteen initial terms factored:
n a(n) factorization A060681(a(n))/A318505(a(n))
1: 117 = 3^2 * 13, (3)
2: 775 = 5^2 * 31, (10)
3: 10309 = 13^2 * 61, (39)
4: 56347 = 29^2 * 67, (58)
5: 88723 = 17^2 * 307, (136)
6: 2896363 = 41^2 * 1723, (820)
7: 9597529 = 73^2 * 1801, (1314)
8: 12326221 = 59^2 * 3541, (1711)
9: 12654079 = 113^2 * 991, (904)
10: 25774633 = 71^2 * 5113, (2485)
11: 29817121 = 97^2 * 3169, (2328)
12: 63455131 = 89^2 * 8011, (3916)
13: 105100903 = 101^2 * 10303, (5050)
14: 203822581 = 157^2 * 8269, (6123)
15: 261019543 = 349^2 * 2143, (2094)
16: 296765173 = 131^2 * 17293, (8515)
17: 422857021 = 233^2 * 7789, (6757)
18: 573332713 = 331^2 * 5233, (4965)
19: 782481673 = 167^2 * 28057, (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Links

Crossrefs

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Programs

  • Mathematica
    Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)
  • PARI
    A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

Extensions

More terms from Amiram Eldar, Dec 24 2020

A326065 Sum of divisors of the largest proper divisor of n: a(n) = sigma(A032742(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 12, 1, 8, 6, 15, 1, 13, 1, 18, 8, 12, 1, 28, 6, 14, 13, 24, 1, 24, 1, 31, 12, 18, 8, 39, 1, 20, 14, 42, 1, 32, 1, 36, 24, 24, 1, 60, 8, 31, 18, 42, 1, 40, 12, 56, 20, 30, 1, 72, 1, 32, 32, 63, 14, 48, 1, 54, 24, 48, 1, 91, 1, 38, 31, 60, 12, 56, 1, 90, 40, 42, 1, 96, 18, 44, 30, 84, 1, 78, 14
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2019

Keywords

Links

Crossrefs

Cf. A000203, A013661, A020639, A032742, A067029, A143112, A326066, A326067, A326068, A326069, A326135.

Programs

Formula

a(n) = A000203(A032742(n)) = A000203(n) - A326066(n).
a(n) = A326135(n) * A000203(A020639(n)^(A067029(n)-1)).
Sum_{k=1..n} a(k) ~ (zeta(2)/2) * c * n^2, where c = Sum_{p prime} ((p/((p-1)^2*(p+1))) * Product_{primes q <= p} ((q-1)^2*(q+1)/q^3)) = 0.3076135997... . - Amiram Eldar, Dec 21 2024

A326066 a(n) = sigma(n) - sigma(A032742(n)), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

0, 2, 3, 4, 5, 8, 7, 8, 9, 12, 11, 16, 13, 16, 18, 16, 17, 26, 19, 24, 24, 24, 23, 32, 25, 28, 27, 32, 29, 48, 31, 32, 36, 36, 40, 52, 37, 40, 42, 48, 41, 64, 43, 48, 54, 48, 47, 64, 49, 62, 54, 56, 53, 80, 60, 64, 60, 60, 59, 96, 61, 64, 72, 64, 70, 96, 67, 72, 72, 96, 71, 104, 73, 76, 93, 80, 84, 112, 79, 96, 81, 84, 83
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2019

Keywords

Links

Crossrefs

Cf. A000203, A013661, A020639, A032742, A246655 (positions of fixed points), A247180, A326065, A326067, A326135, A326136.

Programs

  • Mathematica
    Join[{0},Table[DivisorSigma[1,n]-DivisorSigma[1,Divisors[n][[-2]]],{n,2,100}]] (* Harvey P. Dale, Jan 12 2022 *)
  • PARI
    A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326065(n) = sigma(A032742(n));
A326066(n) = (sigma(n) - sigma(A032742(n)));

Formula

a(n) = A000203(n) - A326065(n) = A000203(n) - A000203(A032742(n)).
a(1) = 0; for n > 1, if n is of the form p^k (p prime and exponent k >= 1), then a(n) = n, otherwise a(n) > n.
For terms in A247180, i.e., when n = A020639(n) * A032742(n), with the smallest prime factor A020639(n) unitary, a(n) = A020639(n) * A326065(n).
Sum_{k=1..n} a(k) ~ (zeta(2)/2) * (1 - c) * n^2, where c is defined in the corresponding formula in A326065. . - Amiram Eldar, Dec 21 2024

A326067 a(n) = sigma(n) - sigma(A032742(n)) - n, where A032742 gives the largest proper divisor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

-1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 3, 0, 0, 8, 0, 4, 3, 2, 0, 8, 0, 2, 0, 4, 0, 18, 0, 0, 3, 2, 5, 16, 0, 2, 3, 8, 0, 22, 0, 4, 9, 2, 0, 16, 0, 12, 3, 4, 0, 26, 5, 8, 3, 2, 0, 36, 0, 2, 9, 0, 5, 30, 0, 4, 3, 26, 0, 32, 0, 2, 18, 4, 7, 34, 0, 16, 0, 2, 0, 44, 5, 2, 3, 8, 0, 66, 7, 4, 3, 2, 5, 32, 0, 16, 9, 24, 0, 42, 0, 8, 39
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2019

Keywords

Links

Crossrefs

Cf. A000203, A013661, A032742, A033879, A246655 (positions of zeros), A326065, A326066, A326068, A326069.

Programs

Formula

a(n) = A326066(n) - n = A000203(n) - A000203(A032742(n)) - n.
a(n) = A326068(n) - A033879(n).
a(p^k) = 0 for all primes and all exponents k >= 1.
Sum_{k=1..n} a(k) ~ ((zeta(2) * (1 - c) - 1)/2) * n^2, where c is defined in the corresponding formula in A326065. . - Amiram Eldar, Dec 21 2024

A117818 a(n) = n if n is 1 or a prime, otherwise a(n) = n divided by the least prime factor of n (A032742(n)).

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 8, 17, 9, 19, 10, 7, 11, 23, 12, 5, 13, 9, 14, 29, 15, 31, 16, 11, 17, 7, 18, 37, 19, 13, 20, 41, 21, 43, 22, 15, 23, 47, 24, 7, 25, 17, 26, 53, 27, 11, 28, 19, 29, 59, 30, 61, 31, 21, 32, 13, 33, 67, 34, 23, 35, 71, 36, 73, 37, 25, 38
Offset: 1

Views

Author

Roger L. Bagula, Apr 30 2006

Keywords

Comments

A026741 generalized to give either a prime or the largest proper divisor of a nonprime.
Sometimes called "Conway's subprime function", although it surely predates John Conway. - N. J. A. Sloane, Sep 29 2017

Links

Crossrefs

Cf. A026741, A032742, A292772.

Programs

  • Haskell
    a117818 n = if a010051 n == 1 then n else a032742 n
-- Reinhard Zumkeller, Jun 24 2013
  • Maple
    A117818 := proc(n)
    local a,d;
    if isprime(n) or n =1 then
        return n;
    end if;
    a := -1 ;
    for d in numtheory[divisors](n) do
        if d < n and d> a then
            a := d ;
        end if;
    end do:
    a ;
end proc:
seq(A117818(n),n=1..100) ; # R. J. Mathar, Apr 30 2024
  • Mathematica
    Table[If[PrimeQ[n], n, If[n == 1, 1, n/FactorInteger[n][[1, 1]]]], {n, 1, 76}]
Table[Which[n==1,1,PrimeQ[n],1,True,Divisors[n][[-2]]],{n,80}] (* Harvey P. Dale, Feb 02 2022 *)
  • Python
    import sympy
def A117818(n):
    if n == 1:
        return 1
    else:
        =sympy.ntheory.factor.primefactors(n)
        return _[-1]
print([A117818(n) for n in range(1,100)])
# R. J. Mathar, May 24 2024

Extensions

Edited by Stefan Steinerberger, Jul 22 2007
Extended by Charles R Greathouse IV, Jul 28 2010

A280496 a(n) = A032742(A250246(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 9, 11, 1, 12, 5, 13, 7, 14, 1, 15, 1, 16, 15, 17, 7, 18, 1, 19, 11, 20, 1, 27, 1, 22, 27, 23, 1, 24, 7, 25, 25, 26, 1, 21, 25, 28, 21, 29, 1, 30, 1, 31, 13, 32, 11, 45, 1, 34, 45, 35, 1, 36, 1, 37, 17, 38, 11, 33, 1, 40, 33, 41, 1, 54, 35, 43, 35, 44, 1, 81, 49, 46, 81, 47, 13, 48, 1, 49, 19, 50, 1
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2017

Keywords

Links

Crossrefs

Cf. A020639, A032742, A250246.
Differs from related A280495 and A280497 for the first time at n=33, where a(33) = 15, while A280495(33) = A280497(33) = 13.
Differs from related A280498 for the first time at n=42, where a(42) = 27, while A280498(42) = 21.

Programs

Formula

a(n) = A032742(A250246(n)).
a(n) = A250246(n) / A020639(n). [Because A250246 preserves the smallest prime factor of n.]

A280689 a(n) = A000045(A032742(n)) / A000045(A054576(n)), where A000045(n) gives the n-th Fibonacci number, A032742(n) = the largest proper divisor of n, and A054576(n) = A032742(A032742(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 4, 1, 13, 5, 7, 1, 17, 1, 11, 13, 89, 1, 18, 5, 233, 17, 29, 1, 122, 1, 47, 89, 1597, 13, 76, 1, 4181, 233, 123, 1, 842, 1, 199, 122, 28657, 1, 322, 13, 15005, 1597, 521, 1, 5777, 89, 843, 4181, 514229, 1, 1364, 1, 1346269, 842, 2207, 233, 39602, 1, 3571, 28657, 709805, 1, 5778, 1, 24157817, 15005, 9349, 89, 271442, 1, 15127, 5777
Offset: 1

Views

Author

Antti Karttunen, Jan 11 2017

Keywords

Comments

A000045 is a divisibility sequence, which guarantees that the result of the division is an integer.

Links

Crossrefs

Cf. A000045, A032742, A054576, A105800, A280688, A280690, A280691.

Programs

Formula

a(n) = A105800(n) / A280688(n) = A105800(n) / A105800(A032742(n)).
a(n) = A000045(A032742(n)) / A000045(A054576(n)).
a(n) = A280690(A032742(n)).
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