A340075 -id:A340075 - cp's OEIS frontend

A340076 Positions of ones in A340075.

1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 26, 29, 31, 37, 39, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 78, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 138, 139, 149, 151, 157, 163, 167, 173, 179, 181, 182, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 259, 263, 269, 271, 277, 281, 283

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

Subsequence of A340150 from which this differs for the first time at n=98, as this lacks the term 445 which is present in A340076.

If one applies prime shift (A003961) to each term, and then sorts the results into the ascending order, one gets A340077.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A340075, A340077.

Subsequence of A340150.

isA340076(n) = (1==A340075(n)); \\ See A340075 for the other needed code.

A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 15, 3, 5, 15, 1, 3, 3, 3, 1, 9, 9, 9, 15, 15, 5, 15, 9, 45, 5, 45, 1, 1, 1, 1, 3, 3, 1, 3, 5, 1, 5, 5, 5, 15, 15, 15, 3, 3, 1, 3, 9, 9, 3, 9, 1, 15, 15, 15, 15, 9, 45, 45, 1, 9, 9, 9, 9, 27, 27, 27, 45, 45, 5, 45, 135, 135, 45, 135, 9, 27, 27, 27, 3, 81, 81, 81, 135, 27, 45, 135, 405

Offset: 0

Antti Karttunen, Dec 28 2020

Compare also to the scatter plot of A339898.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000265, A053575, A019565, A160595, A339809, A339821, A339971, A339898, A339899, A340075, A340085.

Cf. A339973 (positions of ones).

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339901(n) = { my(x=A019565(2*n), y=A000265(eulerphi(x))); y/gcd((x-1),y); };

a(n) = A339971(n) / A339899(n).
a(n) = A000265(A160595(A019565(2*n))).
a(n) = A340075(A019565(n)) = A340085(A019565(2*n)).

A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 9, 5, 3, 1, 6, 1, 5, 12, 27, 1, 20, 1, 18, 20, 12, 1, 36, 7, 16, 25, 30, 1, 6, 1, 81, 3, 9, 15, 15, 1, 11, 16, 27, 1, 20, 1, 18, 20, 28, 1, 54, 11, 42, 36, 12, 1, 100, 4, 45, 44, 15, 1, 72, 1, 36, 100, 243, 48, 48, 1, 54, 7, 12, 1, 180, 1, 40, 42, 66, 60, 64, 1, 162, 125, 21, 1, 120, 9, 23, 60, 108

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A160595.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A160595, A253885, A340071, A340073, A340075 (gives the odd part).

Cf. also A340082.

f:= proc(n) local F,x,p,t;
  F:= ifactors(n)[2];
  x:= mul(nextprime(t[1])^t[2],t=F);
  p:= numtheory:-phi(x);
  p/igcd(x-1,p)
end proc:
map(f,[$1..100]); # Robert Israel, Dec 28 2020

a[n_] := Module[{x, p, e, phi}, x = Product[{p, e} = pe; NextPrime[p]^e, {pe, FactorInteger[n]}]; phi = EulerPhi[x]; phi/GCD[x-1, phi]];
Array[a, 100] (* Jean-François Alcover, Jan 04 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };

a(n) = A160595(A003961(n)).

a(n) = A003972(n) / A340071(n).

A340149 Odd part of A340147: a(n) = A000265(A247074(A003961(n))).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 5, 3, 1, 3, 1, 5, 3, 27, 1, 5, 1, 9, 5, 3, 1, 9, 7, 1, 25, 15, 1, 3, 1, 81, 3, 9, 15, 15, 1, 11, 1, 27, 1, 5, 1, 9, 5, 7, 1, 27, 11, 21, 9, 3, 1, 25, 1, 45, 11, 15, 1, 9, 1, 9, 25, 243, 3, 3, 1, 27, 7, 3, 1, 45, 1, 5, 21, 33, 15, 1, 1, 81, 125, 21, 1, 15, 3, 23, 15, 27, 1, 15, 5, 21, 9, 13, 33

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Each term a(n) is a divisor of A340075(n), at n=85 occurs the first proper divisor.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A247074, A340147, A340150 (positions of ones).

Differs from related A340075 for the first time at n=85, where a(85) = 3, while A340075(85) = 9.

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340149(n) = A000265(A247074(A003961(n)));

a(n) = A000265(A340147(n)) = A000265(A247074(A003961(n))).

A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 5, 9, 5, 3, 3, 3, 1, 5, 3, 27, 9, 5, 11, 9, 5, 3, 7, 9, 21, 1, 25, 15, 15, 3, 9, 81, 3, 9, 15, 15, 5, 11, 1, 27, 21, 5, 23, 9, 15, 7, 13, 27, 55, 21, 9, 3, 29, 25, 9, 45, 11, 15, 15, 9, 33, 9, 25, 243, 3, 3, 35, 27, 7, 15, 9, 45, 39, 5, 21, 33, 15, 1, 41, 81, 125, 21, 11, 15, 27, 23, 15, 27, 3, 15

Offset: 1

Antti Karttunen, Dec 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A000265, A003961, A003972, A005117, A053575, A151800, A339903, A340075.

A000265(n) = (n>>valuation(n,2));
A339904(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,my(q=nextprime(1+f[i,1])); A000265(q-1)*(q^(f[i,2]-1))));

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339903(n) = A000265(eulerphi(A003961(n)));

Multiplicative with a(p^e) = A000265(q-1) * q^(e-1), where q = A151800(p), the next prime larger than p.
a(n) = A000265(A003972(n)) = A053575(A003961(n)) = A000265(A000010(A003961(n))).
For all squarefree numbers k, a(k) = A339903(k).

A340077 Odd numbers k for which k-1 is a multiple of A053575(k) [the odd part of phi(k)].

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 255, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

Sequence A003961(A340076(i)), i = 1.., sorted into ascending order. In other words, this sequence consists of such odd numbers k that A064989(k) is in A340076.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064989, A340075, A340076.
Subsequence of A339879.
Subsequences: A065091, A339880 (composite terms), A339869, A339870 (and their further subsequences).

A000265(n) = (n>>valuation(n, 2));
isA340077(n) = ((n%2)&&!((n-1)%A000265(eulerphi(n))));

A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
isA340077(n) = ((n%2)&&(1==A340075(A064989(n)))); \\ Needs also code from A340075.

A340091 Odd numbers k such that A064989(k) is in A340151.

Original entry on oeis.org

679, 703, 1387, 1729, 1891, 2047, 2509, 2701, 2821, 3277, 3367, 5551, 7471, 7735, 8119, 8827, 9997, 10963, 11305, 12403, 13021, 13747, 13981, 14491, 14701, 15841, 16471, 17563, 19951, 21349, 21907, 21931, 22015, 23959, 24727, 25669, 26281, 27511, 28939, 29341, 31417, 32407, 38503, 39091, 39831, 39865, 40501, 41041

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Sequence A003961(A340151(i)), for i >= 1, sorted into ascending order.

By definition, this has no common terms with A340077 nor any of its subsequences like A339869 or A339880.

Antti Karttunen, Table of n, a(n) for n = 1..3290

Cf. A002997, A003961, A339869, A339880, A340077, A340151.

Cf. A340092 (Carmichael numbers in this sequence).

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339904(n) = A000265(eulerphi(A003961(n)));
A340075(n) = { my(u=A339904(n)); u/gcd(A003961(n)-1, u); };
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340149(n) = A000265(A247074(A003961(n)));
isA340151(n) = ((1!=A340075(n))&&(1==A340149(n)));
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
isA340091(n) = ((n%2)&&isA340151(A064989(n)));

A340074 a(n) = gcd(A003961(n)-1, A339904(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 11, 1, 1, 1, 7, 1, 3, 1, 1, 1, 15, 1, 9, 1, 1, 1, 1, 1, 5, 1, 1, 1, 21, 1, 23, 1, 3, 1, 13, 1, 5, 1, 1, 1, 29, 1, 9, 1, 1, 1, 15, 1, 33, 1, 1, 1, 1, 1, 35, 1, 1, 5, 9, 1, 39, 1, 1, 1, 1, 1, 41, 1, 1, 1, 11, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 25, 1, 3, 1, 51, 1, 53, 1, 3

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

Odd part of A340071(n).

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A339904, A340071, A340075.

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339904(n) = A000265(eulerphi(A003961(n)));
A340074(n) = gcd(A003961(n)-1, A339904(n));

a(n) = gcd(A003961(n)-1, A339904(n)).
a(n) = A000265(A340071(n)).
a(n) = A339904(n) / A340075(n).

A340151 Setwise difference A340150 \ A340076.

Original entry on oeis.org

445, 527, 935, 1207, 1577, 1595, 1705, 1711, 2101, 2145, 2201, 2507, 3245, 3315, 4895, 6045, 6631, 6931, 7511, 8371, 9707, 9845, 10189, 10295, 10505, 11023, 11895, 12137, 12194, 13145, 13571, 13845, 14101, 14245, 15042, 15281, 16235, 16643, 17355, 17701, 18559, 19567, 20119, 20865, 22703, 23347, 25123, 26581, 27101, 27695

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Positions k where A340149(k) = 1, but A340075(k) > 1.

Setwise difference A340150 \ A340076.
Cf. A340091 (gives the same terms prime shifted once and sorted into ascending order).
Cf. A340075, A340149.

isA340151(n) = ((1!=A340075(n))&&(1==A340149(n))); \\ Uses code from both A340075 and A340149

cp's OEIS Frontend

A340076 Positions of ones in A340075.

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A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

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A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

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A340149 Odd part of A340147: a(n) = A000265(A247074(A003961(n))).

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A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

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A340077 Odd numbers k for which k-1 is a multiple of A053575(k) [the odd part of phi(k)].

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A340091 Odd numbers k such that A064989(k) is in A340151.

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A340074 a(n) = gcd(A003961(n)-1, A339904(n)).

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