A339904 -id:A339904 - cp's OEIS frontend

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

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).

A340075 The odd part of A340072(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, 9, 23, 15, 27, 1, 15, 5, 21, 9, 13, 33, 81

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Each term a(n) is a multiple of A340149(n), therefore, as both sequences have only positive terms, it follows that if a(n) = 1 then A340149(n) = 1 also, but not necessarily vice versa.

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, A019565, A339901, A339904, A340072, A340074, A340076 (positions of ones), A340149 (differs from the first time at n=85).

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); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };
A340075(n) = A000265(A340072(n));

a(n) = A000265(A340072(n)).
a(n) = A339904(n) / A340074(n) = A339904(n) / gcd(A003961(n)-1, A339904(n)).
For all n >= 0, a(A019565(n)) = A339901(n).

A339903 Fully multiplicative with a(p) = A000265(q-1), where q = A151800(p), the next prime > p.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 3, 1, 1, 5, 3, 1, 9, 1, 11, 3, 5, 3, 7, 1, 9, 1, 1, 5, 15, 3, 9, 1, 3, 9, 15, 1, 5, 11, 1, 3, 21, 5, 23, 3, 3, 7, 13, 1, 25, 9, 9, 1, 29, 1, 9, 5, 11, 15, 15, 3, 33, 9, 5, 1, 3, 3, 35, 9, 7, 15, 9, 1, 39, 5, 9, 11, 15, 1, 41, 3, 1, 21, 11, 5, 27, 23, 15, 3, 3, 3, 5, 7, 9, 13, 33, 1, 25, 25

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. A000265, A003958, A003961, A005117, A151800, A339904.

A000265(n) = (n>>valuation(n,2));
A339903(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = nextprime(1+f[i,1])-1); A000265(factorback(f)));

a(n) = A336466(A003961(n)) = A000265(A003958(A003961(n))).

For all squarefree numbers k, a(k) = A339904(k).

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)));

cp's OEIS Frontend

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

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A340075 The odd part of A340072(n).

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A339903 Fully multiplicative with a(p) = A000265(q-1), where q = A151800(p), the next prime > p.

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

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PARI