A019565 -id:A019565 - cp's OEIS frontend

A103786 a(n) is the minimum k that makes primorial P(n)/A019565(k)+A019565(k) prime, k>=0, n>0.

0, 0, 0, 0, 0, 1, 3, 3, 5, 1, 0, 1, 6, 6, 1, 11, 1, 3, 3, 4, 2, 14, 5, 2, 9, 22, 5, 8, 1, 45, 23, 13, 10, 2, 13, 24, 42, 7, 20, 9, 8, 10, 114, 5, 31, 5, 33, 1, 6, 19, 22, 6, 7, 4, 20, 59, 65, 4, 29, 15, 3, 6, 1, 12, 32, 17, 26, 34, 8, 59, 115, 32, 33, 26, 0, 25, 1, 35, 71, 27, 65, 75, 71, 5

Offset: 1

Lei Zhou, Feb 15 2005

nonn

This is the k value of A103785. Conjecture: sequence is defined for all n>=1.

			for n=1, P(1)/A019565(0)+A019565(0)=2/1+1=3 is prime, so a(1)=0;
for n=7, P(7)/A019565(3)+A019565(3)=510510/6+6=85091 is prime, so a(7)=3;

Cf. A019565, A002110, A103785.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tn = 0; tt = 1; cp = npd/tt + tt; While[(IntegerQ[cp]) && (! (PrimeQ[cp])), tn = tn + 1; tt = 1; k1 = tn; o = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; cp = npd/tt + tt]; Print[tn]; n = n + 1; npd = npd*Prime[n]]

A103796 Indices of n such that A019565(n)+1 is prime.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 31, 35, 37, 43, 45, 47, 51, 57, 59, 67, 73, 79, 83, 85, 97, 99, 107, 111, 119, 123, 133, 135, 145, 151, 153, 155, 159, 163, 167, 173, 175, 185, 193, 201, 203, 211, 213, 215, 233, 245, 251, 253, 257, 259, 263, 267, 271, 277

Offset: 0

Lei Zhou, Feb 22 2005

nonn

			A019565(0)=1, 1+1=2 is prime, so a(1)=0;
A019565(1)=2, 2+1=3 is prime, so a(2)=1;
A019565(2)=3, 3+1=4 is not prime
A019565(3)=6, 6+1=7 is prime, so a(3)=3;

Cf. A019565.

A019565 = Function[tn, k1 = tn; o = 1; tt = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; tt]; Do[cp = A019565[n] + 1; If[PrimeQ[cp], Print[n]], {n, 0, 1000} ]

A285323 a(n) = A065642(A065642(A019565(n))) / A019565(n).

Original entry on oeis.org

1, 4, 9, 3, 25, 4, 5, 3, 49, 4, 7, 3, 7, 4, 5, 3, 121, 4, 9, 3, 11, 4, 5, 3, 11, 4, 7, 3, 7, 4, 5, 3, 169, 4, 9, 3, 13, 4, 5, 3, 13, 4, 7, 3, 7, 4, 5, 3, 13, 4, 9, 3, 11, 4, 5, 3, 11, 4, 7, 3, 7, 4, 5, 3, 289, 4, 9, 3, 17, 4, 5, 3, 17, 4, 7, 3, 7, 4, 5, 3, 17, 4, 9, 3, 11, 4, 5, 3, 11, 4, 7, 3, 7, 4, 5, 3, 17, 4, 9, 3, 13, 4, 5, 3, 13, 4, 7, 3, 7, 4, 5, 3

Offset: 0

Antti Karttunen, Apr 19 2017

nonn

After the initial a(0)=1, the third row of array A285321 divided by its first row. After 1, all terms are either primes or squares of primes. See A285110.

The sequence is completely determined by the positions of two least significant 1-bits of n: After initial zero, if n is a power of two (only one 1-bit present) or if prime(1+A285099(n)) > prime(1+A007814(n))^2, a(n) = prime(1+A007814(n))^2 = A020639(A019565(n))^2, otherwise a(n) = prime(1+A285099(n)) = A014673(A019565(n)).

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000040, A007814, A014673, A020639, A019565, A065642, A285099, A285110, A285321, A285327.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285323(n) = A065642(A065642(A019565(n))) / A019565(n);

from operator import mul
from sympy import prime, primefactors
from functools import reduce
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n += r
    while a007947(n)!=r:
        n+=r
    return n
def a(n): return a065642(a065642(a019565(n)))//a019565(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 20 2017

(define (A285323 n) (/ (A065642 (A065642 (A019565 n))) (A019565 n)))
(define (A285323 n) (cond ((zero? n) 1) ((or (= 1 (A000120 n)) (> (A000040 (+ 1 (A285099 n))) (A000290 (A000040 (+ 1 (A007814 n)))))) (A000290 (A000040 (+ 1 (A007814 n))))) (else  (A000040 (+ 1 (A285099 n))))))

a(n) = A065642(A065642(A019565(n))) / A019565(n).

A294931 Multiplicative with a(p^e) = A019565(A289813(e)).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 6, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 3, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 12, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 12, 6, 4, 2, 4, 4, 4, 4, 6, 2, 4, 4, 2, 4, 4, 4, 6, 2, 2, 2, 1, 2, 8, 2, 6, 8, 4, 2, 3, 2, 8, 4, 12, 2, 8, 4, 2, 2, 4, 4

Offset: 1

Antti Karttunen, Nov 11 2017

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

Cf. A019565, A289813, A293442, A294932, A294933.

(define (A294931 n) (if (= 1 n) n (* (A019565 (A289813 (A067029 n))) (A294931 (A028234 n)))))

A303778 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A303775(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 10, 30, 210, 105, 35, 7, 70, 14, 42, 21, 1155, 11, 22, 66, 33, 330, 55, 110, 2310, 165, 2145, 429, 143, 13, 858, 286, 26, 4290, 1430, 715, 65, 5005, 385, 77, 770, 154, 462, 231, 30030, 10010, 2002, 1001, 91, 15015, 3003, 39, 6006, 78, 390, 195, 1365, 455, 910, 130, 2730, 182, 546, 273, 4641, 17, 1870, 935, 187, 2805, 561, 51, 5610, 374, 34

Offset: 0

Antti Karttunen, May 06 2018

nonn

Each a(n+1) is either a divisor or a multiple of a(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A005117, A019565, A303775.

Cf. also A304087.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A303778(n) = A019565(A303775(n)); \\ Needs also code from A303775.

a(n) = A019565(A303775(n)).

A320017 a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[moebius(d) = +1].

Original entry on oeis.org

1, 2, 2, 2, 2, 30, 2, 2, 2, 42, 2, 30, 2, 210, 420, 2, 2, 30, 2, 42, 220, 330, 2, 30, 2, 462, 2, 210, 2, 132300, 2, 2, 52, 78, 156, 30, 2, 390, 780, 42, 2, 346500, 2, 330, 420, 2730, 2, 30, 2, 42, 1716, 462, 2, 30, 8580, 210, 4004, 6006, 2, 132300, 2, 30030, 220, 2, 68, 128700, 2, 78, 340, 343980, 2, 30, 2, 714, 420, 390, 2380, 2702700, 2, 42, 2

Offset: 1

Antti Karttunen, Nov 08 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A019565, A320018 (rgs-transform).

Cf. also A300831, A300832.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A320017(n) = if(1==n,n,my(m=1); fordiv(n,d,if(1==moebius(d), m *= A019565(d))); (m));

a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[A008683(d) > 0].

For n >= 2, A048675(a(n)) = A318674(n).

A339820 a(n) = phi(A019565(n)), where phi is Euler totient function.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 6, 6, 12, 12, 24, 24, 48, 48, 10, 10, 20, 20, 40, 40, 80, 80, 60, 60, 120, 120, 240, 240, 480, 480, 12, 12, 24, 24, 48, 48, 96, 96, 72, 72, 144, 144, 288, 288, 576, 576, 120, 120, 240, 240, 480, 480, 960, 960, 720, 720, 1440, 1440, 2880, 2880, 5760, 5760, 16, 16, 32, 32, 64, 64, 128, 128

Offset: 0

Antti Karttunen, Dec 18 2020

nonn

Cf. A000010, A019565, A339821 (bisection).

Cf. also A324650, A339809.

A339820(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= (p-1)); n >>= 1); (m); };

If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A006093(e1) * A006093(e2) * ... * A006093(ek).

a(n) = A000010(A019565(n)).

A339970 a(n) = A329697(A019565(2n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 5, 6, 2, 3, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 7, 8, 1, 2, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 6, 7, 3, 4, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 8, 9, 5, 6, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 10, 11, 4, 5, 5, 6, 6, 7, 7, 8, 6, 7

Offset: 0

Antti Karttunen, Dec 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000040, A000120, A003961, A019565, A329697, A339971.

Differs from A106486 for the first time at n=32, where a(32) = 1, while A106486(32) = 2.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A339970(n) = A329697(A019565(2*n));

A339970(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += A329697(p)); n >>= 1); (s); };

a(n) = A329697(A019565(2*n)) = A329697(A003961(A019565(n))).
a(n) = A329697(A339971(n)) + A000120(n).
If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A329697(A000040(e1)) + A329697(A000040(e2)) + ... + A329697(A000040(ek)).

A342921 a(n) = A003415(A019565(n)).

Original entry on oeis.org

0, 1, 1, 5, 1, 7, 8, 31, 1, 9, 10, 41, 12, 59, 71, 247, 1, 13, 14, 61, 16, 87, 103, 371, 18, 113, 131, 493, 167, 719, 886, 2927, 1, 15, 16, 71, 18, 101, 119, 433, 20, 131, 151, 575, 191, 837, 1028, 3421, 24, 191, 215, 859, 263, 1241, 1504, 5153, 311, 1623, 1934, 6871, 2556, 10117, 12673, 40361, 1, 19, 20, 91, 22, 129

Offset: 0

Antti Karttunen, Apr 06 2021

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A003415, A019565, A069359, A276156, A327860, A329029, A342002.

Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[#, 2], 1]] &, 70, 0] (* Michael De Vlieger, Apr 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A342921(n) = A003415(A019565(n));

a(n) = A003415(A019565(n)) = A069359(A019565(n)).

a(n) = A327860(A276156(n)) = A329029(A276156(n)) = A342002(A276156(n)).

A351081 a(n) = Product_{d|n} A019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of the divisors of n.

Original entry on oeis.org

2, 2, 6, 12, 6, 6, 4, 12, 30, 60, 10, 540, 60, 60, 90, 120, 10, 30, 4, 360, 36, 60, 6, 540, 12, 60, 210, 5040, 14, 18900, 84, 2520, 210, 140, 84, 94500, 140, 140, 18900, 75600, 210, 18900, 140, 12600, 3150, 84, 14, 113400, 168, 2520, 210, 5040, 14, 210, 60, 5040, 36, 84, 6, 1701000, 4, 84, 900, 25200, 900, 18900

Offset: 1

Antti Karttunen, Jan 31 2022

Cf. A019565, A289813, A293221, A351082, A351091.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A351081(n) = { my(m=1); fordiv(n,d,m *= A019565(A289813(d))); (m); };

a(n) = A019565(A289813(n)) * A293221(n).

cp's OEIS Frontend

A103786 a(n) is the minimum k that makes primorial P(n)/A019565(k)+A019565(k) prime, k>=0, n>0.

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A103796 Indices of n such that A019565(n)+1 is prime.

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A285323 a(n) = A065642(A065642(A019565(n))) / A019565(n).

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A294931 Multiplicative with a(p^e) = A019565(A289813(e)).

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A303778 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A303775(n)).

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A320017 a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[moebius(d) = +1].

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A339820 a(n) = phi(A019565(n)), where phi is Euler totient function.

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A339970 a(n) = A329697(A019565(2n)).

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A342921 a(n) = A003415(A019565(n)).

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