A336853 -id:A336853 - cp's OEIS frontend

A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

1, -1, -2, -4, -2, -5, -4, -10, -12, -7, -2, -1, -4, -11, -12, -16, -2, -1, -4, -7, -18, -13, -6, 42, -20, -17, -42, -5, -2, 21, -6, -4, -24, -19, -26, 106, -4, -23, -30, 38, -2, 45, -4, -25, -10, -29, -6, 196, -56, -17, -36, -23, -6, 123, -28, 82, -42, -31, -2, 225, -6, -37, 4, 80, -38, 15, -4, -43, -52, 39, -2, 413

Offset: 1

Antti Karttunen, Aug 19 2021

sign

Dirichlet inverse of the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...).

Cf. A000040, A001223, A003961, A336853, A347099, A347100.

Cf. also A346248, A347096.

f[p_, e_] := NextPrime[p]^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d).

For all n >= 1, a(A000040(n)) = -A001223(n).

A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 9, 4, 4, 0, 32, 0, 8, 8, 49, 0, 56, 0, 36, 16, 4, 0, 153, 4, 8, 56, 66, 0, 96, 0, 207, 8, 4, 16, 295, 0, 8, 16, 187, 0, 168, 0, 48, 120, 12, 0, 553, 16, 80, 8, 78, 0, 444, 8, 323, 16, 4, 0, 480, 0, 12, 216, 745, 16, 144, 0, 60, 24, 200, 0, 1016, 0, 8, 152, 90, 16, 216, 0, 723, 472, 4, 0, 786, 8, 8, 8, 289

Offset: 1

Antti Karttunen, Aug 19 2021

sign

Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.

The first negative term is a(720) = -6306.

Cf. A001223, A001248, A003961, A063524, A336853, A347098.

Cf. also A346250, A347097.

up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];
A347099(n) = if(1==n,2,A336853(n)+A347098(n));

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A336853(d) * A347098(n/d).

For all n >= 1, a(A001248(n)) = A001223(n)^2.

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 14 1999

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;

Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014

Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

A336852 a(n) = sigma(A003961(n)) - sigma(n).

Original entry on oeis.org

0, 1, 2, 6, 2, 12, 4, 25, 18, 14, 2, 50, 4, 24, 24, 90, 2, 85, 4, 62, 40, 20, 6, 180, 26, 30, 116, 100, 2, 120, 6, 301, 36, 26, 48, 312, 4, 36, 52, 230, 2, 192, 4, 98, 170, 48, 6, 602, 76, 135, 48, 136, 6, 504, 40, 360, 64, 38, 2, 456, 6, 56, 268, 966, 60, 192, 4, 134, 84, 240, 2, 1045, 6, 54, 218, 172, 72, 264, 4, 782

Offset: 1

Antti Karttunen, Aug 05 2020

nonn

Inverse Möbius transform of A336853(n) = (A003961(n) - n).

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A003973, A286385, A336851, A336853.

Cf. A001105 (positions of odd terms), A001359 (positions of 2's).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336852(n) = (sigma(A003961(n)) - sigma(n));

PARI
```
A336852(n) = sumdiv(n,d,A003961(d)-d);
```

a(n) = Sum_{d|n} (A003961(d)-d).
a(n) = A003973(n) - A000203(n) = A000203(A003961(n)) - A000203(n).
a(n) = A336851(n) + A286385(n).

A337549 a(n) = A003972(n) - n.

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 3, 10, 11, 2, 1, 12, 3, 6, 9, 38, 1, 22, 3, 16, 19, 2, 5, 48, 17, 6, 73, 32, 1, 18, 5, 130, 15, 2, 25, 84, 3, 6, 25, 68, 1, 38, 3, 28, 75, 10, 5, 168, 61, 34, 21, 44, 5, 146, 17, 124, 31, 2, 1, 84, 5, 10, 137, 422, 31, 30, 3, 40, 43, 50, 1, 288, 5, 6, 93, 56, 43, 50, 3, 244, 419, 2, 5, 156, 23, 6, 33

Offset: 1

Antti Karttunen, Sep 21 2020

nonn

Möbius transform of A286385.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A003972, A286385, A336853.

A003972(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); eulerphi(factorback(f)); };
A337549(n) = (A003972(n) - n);

a(n) = A003972(n) - n.

A336855 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(p) = p-nextprime(p) for primes p, and f(n) = n for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 09 2020

nonn

Restricted growth sequence transform of function f defined as: f(n) = -{distance to the next larger prime} when n is a prime, otherwise f(n) = -n.
For all i, j:
  a(i) = a(j) => A305801(i) = A305801(j),
  a(i) = a(j) => A336852(i) = A336852(j),
  a(i) = a(j) => A336853(i) = A336853(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for primes, gaps between

Cf. also A001359 (positions of 3's), A305801, A319704, A331304, A336852, A336853.

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; };
A336855aux(n) = if(isprime(n),n-nextprime(1+n),n);
v336855 = rgs_transform(vector(up_to,n,A336855aux(n)));
A336855(n) = v336855[n];

A347100 a(n) = phi(A003961(n)) - phi(n), where A003961 is the prime shift towards larger primes, and phi is Euler totient function.

Original entry on oeis.org

0, 1, 2, 4, 2, 6, 4, 14, 14, 8, 2, 20, 4, 14, 16, 46, 2, 34, 4, 28, 28, 14, 6, 64, 22, 20, 82, 48, 2, 40, 6, 146, 28, 20, 36, 108, 4, 26, 40, 92, 2, 68, 4, 52, 96, 34, 6, 200, 68, 64, 40, 72, 6, 182, 32, 156, 52, 32, 2, 128, 6, 42, 164, 454, 48, 76, 4, 76, 68, 96, 2, 336, 6, 44, 128, 96, 60, 104, 4, 292, 446, 44, 6

Offset: 1

Antti Karttunen, Aug 19 2021

nonn

Möbius transform of A336853.

Möbius transform of A336853.
Cf. A000010, A000040, A001223, A003961, A003972, A008683, A051953, A337549.
Cf. also A346249, A347098.

f[p_, e_] := NextPrime[p]^e; a[n_] := EulerPhi[Times @@ f @@@ FactorInteger[n]] - EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A347100(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (eulerphi(factorback(f))-eulerphi(n)); };

A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
A347100(n) = sumdiv(n,d,moebius(n/d)*A336853(d));

a(n) = A003972(n) - A000010(n).
a(n) = A337549(n) + A051953(n).
a(n) = Sum_{d|n} A008683(n/d) * A336853(d).
For all n >= 1, a(A000040(n)) = A001223(n).

A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 10, 15, 19, 14, 12, 29, 16, 24, 27, 49, 18, 54, 22, 39, 45, 26, 28, 91, 39, 36, 98, 67, 30, 74, 36, 163, 51, 38, 65, 165, 40, 48, 69, 121, 42, 124, 46, 69, 136, 62, 52, 293, 107, 102, 75, 97, 58, 294, 75, 205, 93, 62, 60, 223, 66, 78, 224, 537, 101, 134, 70, 99, 119, 172, 72, 519, 78, 84, 190, 127

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A003961, A168036, A286385, A336853, A343224, A347964, A347965.

f1[p_, e_] := e/p; f2[p_, e_] := NextPrime[p]^e; a[n_] := Times @@ f2 @@@ (f = FactorInteger[n]) - n * Plus @@ f1 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348937(n) = (A003961(n) - A003415(n));

a(n) = A003961(n) - A003415(n).
a(n) = A336853(n) - A168036(n).
a(n) = A286385(n) + A343224(n).

A347378 a(n) = A250469(n) - n.

Original entry on oeis.org

0, 1, 2, 5, 2, 9, 4, 13, 16, 17, 2, 21, 4, 25, 20, 29, 2, 33, 4, 37, 34, 41, 6, 45, 24, 49, 38, 53, 2, 57, 6, 61, 52, 65, 42, 69, 4, 73, 56, 77, 2, 81, 4, 85, 70, 89, 6, 93, 72, 97, 74, 101, 6, 105, 36, 109, 88, 113, 2, 117, 6, 121, 92, 125, 54, 129, 4, 133, 106, 137, 2, 141, 6, 145, 110, 149, 66, 153, 4, 157, 124

Offset: 1

Antti Karttunen, Sep 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A001223, A250469, A280692, A336853, A346476.

PARI
```
A347378(n) = (A250469(n)-n);
```

a(n) = A250469(n) - n.
a(n) = n - A346476(n).
a(n) = A336853(n) - A280692(n).
For all n >= 1, a(A000040(n)) = A001223(n).

cp's OEIS Frontend

A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

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A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

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A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

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A336852 a(n) = sigma(A003961(n)) - sigma(n).

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A337549 a(n) = A003972(n) - n.

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A336855 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(p) = p-nextprime(p) for primes p, and f(n) = n for all other numbers.

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A347100 a(n) = phi(A003961(n)) - phi(n), where A003961 is the prime shift towards larger primes, and phi is Euler totient function.

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A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.

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