A019565 -id:A019565 - cp's OEIS frontend

A339815 Let x = A019565(2*n); a(n) is the difference between 2-adic valuations of phi(x) and (x-1).

0, 0, 2, 0, 0, 2, 1, 0, -3, 2, 2, 0, 2, -3, 4, 0, 2, -2, 4, 2, 0, 4, 4, 2, 2, 4, 1, 1, 4, 4, 6, 0, 4, 4, 6, 4, 4, 6, 5, 4, 2, 6, 6, 4, 6, 4, 8, 4, 6, 4, 8, 6, 3, 8, 8, 6, 6, 8, 6, 5, 8, 8, 10, 0, -1, 2, 2, 0, 2, 1, 4, -2, 2, 2, 4, 2, 2, 4, 3, 2, 2, 4, 3, -2, 4, 4, 6, 2, 4, 2, 6, 4, 1, 6, 6, 4, 3, 6, 6, 4, 6, 5, 8, 1, 6

Offset: 1

Antti Karttunen, Dec 18 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A007814, A019565, A339809, A339814, A339821, A339822.

Cf. A339816 (indices of terms < 1).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339815(n) = { my(x=A019565(2*n)); valuation(eulerphi(x),2)-valuation(x-1,2); };

a(n) = A339822(n) - A339814(n).

a(n) = A007814(A000010(A019565(2n))) - A007814(A019565(2n)-1).

A339899 a(n) = gcd(A019565(2n)-1, A000265(phi(A019565(2n)))).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 15, 1, 1, 1, 3, 5, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 3, 1, 1, 1, 27, 1, 1, 1, 1, 5, 3, 1, 1, 1, 81, 1, 1, 1, 3, 1, 1, 1, 9, 1, 3, 1

Offset: 0

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000010, A000265, A049559, A053575, A019565, A339809, A339821, A339971, A339898, A339901.

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); };
A339899(n) = { my(x=A019565(2*n)); gcd((x-1),A000265(eulerphi(x))); };

a(n) = gcd(A339809(2*n), A339971(n)), where A339971(n) = A053575(A019565(2n)).
a(n) = gcd(A339971(n), A339898(n)).
a(n) = A339971(n) / A339901(n).
a(n) = A000265(A049559(A019565(2*n))).

A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 16, 20, 32, 33, 34, 35, 38, 41, 50, 56, 64, 128, 176, 256, 259, 290, 512, 1024, 2048, 2056, 2081, 2089, 2096, 2180, 4096, 4130, 8192, 9218, 16384, 18436, 32768, 65536, 131072, 131140, 262144, 279552, 524288, 524308, 524546, 1048576, 1048736, 2097152, 4194304, 4194352, 4194420, 4196656, 4202499, 8388608

Offset: 1

Antti Karttunen, Dec 26 2020

nonn

Numbers k such that A339971(k) divides A339809(2k).

Union of {0}, A000079 and the terms of (1/2)*A048675(A339880(i)), for i >= 1, sorted into ascending order.

Antti Karttunen, Table of n, a(n) for n = 1..94

Positions of zeros in A339898, and of ones in A339901.
Cf. A000010, A000265, A048675, A339821, A339880, A339809, A339971, A339974.
Cf. A000079 (subsequence).
Cf. also A339816, A339906.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339971(n) = { my(x=A019565(2*n)); !((x-1)%A000265(eulerphi(x))); };

A351031 a(n) = Product_{d|n, dA019565(A304759(d)).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 6, 6, 12, 2, 18, 2, 2, 36, 90, 2, 180, 2, 180, 6, 4, 2, 810, 12, 10, 180, 30, 2, 180, 2, 9450, 12, 20, 12, 56700, 2, 30, 30, 56700, 2, 420, 2, 12, 1080, 10, 2, 1275750, 2, 120, 60, 30, 2, 31500, 24, 9450, 90, 20, 2, 238140, 2, 4, 2520, 10914750, 60, 84, 2, 420, 30, 31500, 2, 2946982500, 2, 6

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A019565, A048673, A289813, A304759, A351030, A351032, A351033 (rgs-transform).

Cf. also A293221.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/2; };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
A351031(n) = { my(m=1); fordiv(n,d,if(dA019565(A304759(d)))); (m); };

A351032 a(n) = Product_{d|n, dA019565(A291759(d)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 24, 1, 6, 1, 8, 1, 12, 1, 8, 3, 6, 1, 480, 1, 2, 1, 72, 1, 120, 1, 16, 3, 2, 3, 480, 1, 2, 1, 32, 1, 216, 1, 120, 5, 6, 1, 13440, 3, 60, 1, 120, 1, 168, 3, 1440, 1, 6, 1, 144000, 1, 10, 3, 32, 1, 1080, 1, 8, 3, 72, 1, 26880, 1, 10, 75, 24, 9, 1080, 1, 128, 7, 10, 1, 86400, 1, 30, 3

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A019565, A048673, A289814, A291759, A351030, A351031, A351034 (rgs-transform).

Cf. also A293222.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/2; };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A351032(n) = { my(m=1); fordiv(n,d,if(dA019565(A291759(d)))); (m); };

A351556 a(n) = gcd(n, A019565(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 7, 15, 1, 1, 3, 1, 5, 1, 11, 1, 1, 1, 1, 3, 7, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 1, 21, 1, 1, 5, 1, 1, 1, 1, 1, 3, 13, 1, 3, 55, 7, 1, 1, 1, 5, 1, 1, 21, 1, 1, 3, 1, 17, 1, 5, 1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 1, 1, 1, 1, 85, 1, 3, 11, 1, 3, 7, 1, 1, 1, 5, 1, 1, 1, 3, 5, 1, 51, 1, 13, 7

Offset: 0

Antti Karttunen, Feb 19 2022

Cf. A007947, A019565.

Cf. also A324198, A351549, A351557, A351558.

Table[GCD[n, Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 0, 105}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A351556(n) = gcd(n, A019565(n));

a(n) = gcd(n, A019565(n)) = gcd(A007947(n), A019565(n)).
a(n) = A007947(a(n)).
a(n) = A019565(A351558(n)).

A351557 a(n) = gcd(sigma(n), A019565(n)).

Original entry on oeis.org

1, 3, 2, 1, 2, 3, 2, 1, 1, 3, 6, 7, 14, 3, 6, 1, 2, 3, 2, 1, 2, 3, 6, 1, 1, 21, 2, 7, 10, 3, 2, 1, 2, 3, 6, 13, 2, 15, 2, 1, 14, 3, 2, 7, 26, 3, 6, 1, 1, 3, 6, 1, 2, 15, 6, 1, 2, 3, 6, 7, 2, 3, 26, 1, 2, 3, 34, 1, 2, 3, 6, 1, 2, 3, 2, 35, 2, 21, 10, 1, 11, 3, 6, 1, 2, 33, 30, 1, 2, 3, 14, 7, 2, 3, 30, 1, 2, 3, 78

Offset: 1

Antti Karttunen, Feb 19 2022

nonn

Cf. A000203, A007947, A019565, A080398.

Cf. also A324644, A351549, A351556, A351559.

Table[GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 99}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A351557(n) = gcd(sigma(n), A019565(n));

a(n) = gcd(A000203(n), A019565(n)) = gcd(A080398(n), A019565(n)).
a(n) = A007947(a(n)).
a(n) = A019565(A351559(n)).

A351558 a(n) = A048675(gcd(n, A019565(n))).

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 8, 6, 0, 0, 2, 0, 4, 0, 16, 0, 0, 0, 0, 2, 8, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 34, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 2, 32, 0, 2, 20, 8, 0, 0, 0, 4, 0, 0, 10, 0, 0, 2, 0, 64, 0, 4, 0, 0, 0, 0, 2, 0, 8, 2, 0, 0, 0, 0, 0, 0, 68, 0, 2, 16, 0, 2, 8, 0, 0, 0, 4, 0, 0, 0, 2, 4, 0, 66

Offset: 0

Antti Karttunen, Feb 19 2022

Cf. A004198, A007947, A019565, A087207, A351556.

Cf. also A324198, A351559.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[n, Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 102}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351558(n) = A048675(gcd(n, A019565(n)));

a(n) = A048675(A351556(n)) = A048675(gcd(n, A019565(n))).

a(n) = n AND A087207(n), where AND is bitwise-and, A004198.

A376407 a(0) = 0, and for n > 0, a(n) = a(n-1) + A019565(a(n-1)), where A019565 is the base-2 exp-function.

Original entry on oeis.org

0, 1, 3, 9, 23, 353, 10519, 12086209, 1174153011340170531, 73582975079922326904310062621361286634299329277087298285

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(10) has 272 digits and a(11) has 1523 digits.

By induction, it is easy to see that formula a(n) = A048675(A376406(n)) implies that from the second term onward, this sequence gives the partial sums of A376406. See comments and examples in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..10
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A376406, A376409.

Cf. also A376403 (an analogous sequence for A276076).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376407(n) = if(!n,0,my(x=A376407(n-1)); x+A019565(x));

a(n) = A048675(A376406(n)).

a(0) = 0; and for n > 0, a(n) = a(n-1) + A376406(n-1) = Sum_{i=0..n-1} A376406(i).

A379496 a(n) = A019565(1+n) - A019565(A001065(n)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

Original entry on oeis.org

2, 4, 3, 4, 13, 15, 5, -16, 16, 35, 33, 59, 103, 189, -3, -188, 31, -44, 53, -55, 123, 225, 75, 89, 216, 451, 315, 385, 1153, 2037, 11, -2284, -171, 23, -5, -4160, 193, 225, 69, -247, 271, -1599, 453, 819, 1339, 2499, 141, -309, 422, 312, 605, 65, 2143, 4239, 979, 1985, 2673, 5993, 5003, 2275, 15013, 29991, -165

Offset: 1

Antti Karttunen, Jan 05 2025

sign

Cf. A001065, A019565, A379495, A379501 [= a(A016754(n))].

Cf. also A379498.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A379496(n) = (A019565(1+n) - A019565(sigma(n)-n));

a(n) = A019565(1+n) - A379495(n).
For even n, a(n) = 2*A019565(n) - A379495(n).
For n of the form 4k+1, a(n) = (3/2)*A019565(n) - A379495(n).

cp's OEIS Frontend

A339815 Let x = A019565(2*n); a(n) is the difference between 2-adic valuations of phi(x) and (x-1).

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A339899 a(n) = gcd(A019565(2n)-1, A000265(phi(A019565(2n)))).

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A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

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A351031 a(n) = Product_{d|n, dA019565(A304759(d)).

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A351032 a(n) = Product_{d|n, dA019565(A291759(d)).

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A351556 a(n) = gcd(n, A019565(n)).

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A351557 a(n) = gcd(sigma(n), A019565(n)).

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A351558 a(n) = A048675(gcd(n, A019565(n))).

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A376407 a(0) = 0, and for n > 0, a(n) = a(n-1) + A019565(a(n-1)), where A019565 is the base-2 exp-function.

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