A032742 -id:A032742 - cp's OEIS frontend

A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

1, 1, 3, 2, 5, 3, 7, 4, 7, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 15, 11, 23, 12, 21, 13, 27, 14, 29, 15, 31, 16, 31, 17, 31, 18, 37, 19, 31, 20, 41, 21, 43, 22, 31, 23, 47, 24, 47, 25, 51, 26, 53, 27, 47, 28, 55, 29, 59, 30, 61, 31, 63, 32, 61, 33, 67, 34, 63, 35, 71, 36, 73, 37, 59, 38, 75, 39, 79, 40, 63, 41, 83, 42

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003986, A032742, A060681, A318506, A318514, A318517, A318518.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318516(n) = bitor(A032742(n),n-A032742(n));

a(n) = A003986(A032742(n), A060681(n)).

a(n) = n - A318518(n).

A318518 a(n) = A032742(n) AND n-A032742(n), where AND is bitwise-and (A004198) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 4, 2, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 6, 11, 0, 12, 4, 13, 0, 14, 0, 15, 0, 16, 2, 17, 4, 18, 0, 19, 8, 20, 0, 21, 0, 22, 14, 23, 0, 24, 2, 25, 0, 26, 0, 27, 8, 28, 2, 29, 0, 30, 0, 31, 0, 32, 4, 33, 0, 34, 6, 35, 0, 36, 0, 37, 16, 38, 2, 39, 0, 40, 18, 41, 0, 42, 0, 43, 24, 44, 0, 45, 12, 46, 30, 47, 0, 48, 0, 49, 0, 50, 0, 51, 0

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004198, A032742, A060681, A318515, A318516, A318517.

Cf. also A318508.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318518(n) = bitand(A032742(n),n-A032742(n));

a(n) = A004198(A032742(n), A060681(n)).

a(n) = n - A318516(n) = (n - A318517(n))/2.

A319714 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 3, 7, 8, 9, 3, 10, 5, 11, 12, 13, 5, 14, 3, 15, 16, 17, 3, 18, 19, 20, 21, 22, 5, 23, 3, 24, 25, 26, 27, 28, 5, 29, 30, 31, 5, 32, 3, 33, 34, 35, 3, 36, 16, 37, 38, 39, 5, 40, 41, 42, 43, 44, 3, 45, 5, 46, 47, 48, 49, 50, 3, 51, 52, 53, 3, 54, 5, 55, 56, 57, 25, 58, 3, 59, 60, 61, 3, 62, 63, 64, 65, 66, 5, 67, 30, 68, 69, 70, 71, 72, 5, 73

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of A286474, or equally, of A286473.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A010873, A032742, A286473, A286474, A319704.

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; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286474(n) = if(1==n,n,(4*A032742(n) + (n % 4)));
v319714 = rgs_transform(vector(up_to,n,A286474(n)));
A319714(n) = v319714[n];

A326068 a(n) = n - sigma(A032742(n)), where sigma is the sum of divisors of n and A032742 gives the largest proper divisor of n.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 1, 5, 4, 10, 0, 12, 6, 9, 1, 16, 5, 18, 2, 13, 10, 22, -4, 19, 12, 14, 4, 28, 6, 30, 1, 21, 16, 27, -3, 36, 18, 25, -2, 40, 10, 42, 8, 21, 22, 46, -12, 41, 19, 33, 10, 52, 14, 43, 0, 37, 28, 58, -12, 60, 30, 31, 1, 51, 18, 66, 14, 45, 22, 70, -19, 72, 36, 44, 16, 65, 22, 78, -10, 41, 40, 82, -12, 67, 42, 57, 4, 88, 12

Offset: 1

Antti Karttunen, Jun 06 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000203, A013661, A032742, A033879, A326065, A326066, A326067, A326069.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326065(n) = sigma(A032742(n));
A326068(n) = (n - A326065(n));

a(n) = n - A326065(n) = n - A000203(A032742(n)).
a(n) = A326067(n) + A033879(n).
Sum_{k=1..n} a(k) ~ ((1 - zeta(2) * c)/2) * n^2, where c = Sum_{p prime} ((p/((p-1)^2*(p+1))) * Product_{prime q <= p} ((q-1)^2*(q+1)/q^3)) = 0.307613599749... . - Amiram Eldar, Dec 21 2024

A326139 a(n) = gcd(A032742(n), A302042(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 1, 11, 1, 12, 5, 13, 1, 14, 1, 15, 1, 16, 1, 17, 7, 18, 1, 19, 1, 20, 1, 21, 1, 22, 3, 23, 1, 24, 7, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 11, 39, 1, 40, 3, 41, 1, 42, 1, 43, 1, 44, 1, 45, 1, 46, 1, 47, 1, 48, 1, 49, 1, 50, 1, 51, 1, 52, 1

Offset: 1

Antti Karttunen, Aug 23 2019

nonn

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

Cf. A032742, A302042, A323888, A326075, A326189, A326190, A326191.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A032742(n) = (n/A020639(n));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A326139(n) = gcd(A032742(n),A302042(n));

a(n) = gcd(A032742(n), A302042(n)).

A326190 Length of the shortest path to 1 when starting from x=n and on each iteration step one may always choose either transition x -> A032742(x) or x -> A302042(x).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 3, 2, 4, 2, 2, 1, 4, 1, 2, 2, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 2, 3, 2, 3, 1, 5, 3, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 2, 4, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Aug 23 2019

nonn

			The directed acyclic graph whose unique root is 153 (illustrated below), spans the following seven numbers [1, 5, 17, 25, 51, 75, 153], as A032742(153) = 51, A302042(153) = 75, A032742(51) = 17, A302042(51) = 25, A032742(75) = 25, A302042(75) = 15, A032742(25) = A302042(25) = 5, and A032742(17) = A302042(17) = A032742(5) = A302042(5) = 1. The length of shortest path(s) from 153 to 1 is 3 (there are actually two shortest paths: 153 -> 51 -> 17 -> 1 and 153 -> 75 -> 17 -> 1), thus a(153) = 3.
.
        153
       /  \
      /    \
     51    75
     / \  /  \
    /   17    \
    \    |    /
     \   1   /
      \     /
       \   /
        25
         |
         5
         |
         1

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

Cf. A032742, A253557, A302042, A323888, A326075, A326139, A326089, A326191.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A032742(n) = (n/A020639(n));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A326190(n) = if(1==n,0,1+min(A326190(A032742(n)), A326190(A302042(n))));
\\ Somewhat faster version:
memo302042 = Map();
A302042(n) = if((1==n)||isprime(n),1,my(v); if(mapisdefined(memo302042, n, &v), v, my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); v=(k*p); mapput(memo302042,n,v); (v)));
A326190list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2,up_to, v[n] = 1+min(v[A032742(n)], v[A302042(n)])); (v); };
v326190 = A326190list(up_to);
A326190(n) = v326190[n];

a(1) = 0; for n > 1, a(n) = 1 + min(a(A032742(n)), a(A302042(n))).

a(n) <= min(A001222(n),A253557(n)) <= max(A001222(n),A253557(n)) <= A326191(n) <= A326189(n).

A326191 Length of the longest path to 1 when starting from x=n and on each iteration step one may always choose either transition x -> A032742(x) or x -> A302042(x).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 3, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 4, 2, 1, 5, 2, 3, 3, 3, 1, 4, 3, 4, 4, 2, 1, 4, 1, 2, 4, 6, 2, 4, 1, 3, 4, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 5, 3, 2, 3, 4, 1, 5, 3, 3, 5, 2, 2, 6, 1, 3, 4, 4, 1, 4, 1, 4, 4

Offset: 1

Antti Karttunen, Aug 23 2019

nonn

			The directed acyclic graph whose unique root is 153 (illustrated below), spans the following seven numbers [1, 5, 17, 25, 51, 75, 153], as A032742(153) = 51, A302042(153) = 75, A032742(51) = 17, A302042(51) = 25, A032742(75) = 25, A302042(75) = 15, A032742(25) = A302042(25) = 5, and A032742(17) = A302042(17) = A032742(5) = A302042(5) = 1. The length of longest path(s) from 153 to 1 is 4 (there are actually two longest paths: 153 -> 51 -> 25 -> 5 -> 1 and 153 -> 75 -> 25 -> 5 -> 1), thus a(153) = 4.
.
        153
       /  \
      /    \
     51    75
     / \  /  \
    /   17    \
    \    |    /
     \   1   /
      \     /
       \   /
        25
         |
         5
         |
         1

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

Cf. A001222, A032742, A253557, A302042, A323888, A326075, A326139, A326189, A326190.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A032742(n) = (n/A020639(n));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A326191(n) = if(1==n,0,1+max(A326191(A032742(n)), A326191(A302042(n))));
\\ Slightly faster:
memo302042 = Map();
A302042(n) = if((1==n)||isprime(n),1,my(v); if(mapisdefined(memo302042, n, &v), v, my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); v=(k*p); mapput(memo302042,n,v); (v)));
A326191list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2,up_to, v[n] = 1+max(v[A032742(n)], v[A302042(n)])); (v); };
v326191 = A326191list(up_to);
A326191(n) = v326191[n];

a(1) = 0; for n > 1, a(n) = 1 + max(a(A032742(n)), a(A302042(n))).

A326189(n) >= a(n) >= max(A001222(n), A253557(n)) >= min(A001222(n), A253557(n)) >= A326190(n).

A245729 Composite numbers n = A020639(n) * A032742(n) where the greatest proper divisor A032742(n) is greater than the square of the smallest prime factor A020639(n), and that greatest proper divisor A032742(n) is either a prime or satisfies the same condition (i.e., is itself the term of this sequence).

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 33, 34, 38, 39, 40, 44, 46, 51, 52, 56, 57, 58, 62, 66, 68, 69, 74, 76, 78, 80, 82, 86, 87, 88, 92, 93, 94, 99, 102, 104, 106, 111, 112, 114, 116, 117, 118, 122, 123, 124, 129, 132, 134, 136, 138, 141, 142, 145, 146, 148, 152, 153, 155, 156, 158, 159, 160, 164, 166, 171, 172, 174, 176, 177

Offset: 1

Antti Karttunen and Reinhard Zumkeller, Jan 09 2015

nonn

If n is present, then so is also 2*n.

If n = p_1^e_1*p_2^e_2*... with p_1 > p_2 > ..., then n is in this sequence iff p_1^2 > p_2 and e_1 = 1. - Charlie Neder, Jun 13 2019

			10 = 2*5 is present, because 2^2 < 5 and 5 is a prime.
20 = 2*10 is present, because 2^2 < 10, and 10 itself is present in the sequence.

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

Subsequence of A088381 and A251727.
Subsequences: A138511, A253569.
Cf. A001222, A000290, A010051, A020639, A032742.

a245729 n = a245729_list !! (n-1)
a245729_list = filter f [1..] where
                      f x = p ^ 2 < q && (a010051' q == 1 || f q)
                            where q = div x p; p = a020639 x
-- Antti Karttunen after Reinhard Zumkeller's code for A138511, Jan 09 2015

;; With Antti Karttunen's IntSeq-library.
(define (charfun_for_A245729 n) (if (and (> (A001222 n) 1) (> (A032742 n) (A000290 (A020639 n)))) (+ (A010051 (A032742 n)) (charfun_for_A245729 (A032742 n))) 0))
(define A245729 (NONZERO-POS 1 1 charfun_for_A245729))
;; Antti Karttunen, Jan 16 2015

A280495 a(n) = A032742(A250245(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 9, 11, 1, 12, 5, 13, 7, 14, 1, 15, 1, 16, 13, 17, 7, 18, 1, 19, 21, 20, 1, 27, 1, 22, 11, 23, 1, 24, 7, 25, 25, 26, 1, 21, 13, 28, 33, 29, 1, 30, 1, 31, 19, 32, 19, 39, 1, 34, 37, 35, 1, 36, 1, 37, 17, 38, 11, 63, 1, 40, 15, 41, 1, 54, 31, 43, 45, 44, 1, 33, 17, 46, 57, 47, 37, 48, 1, 49, 27, 50, 1

Offset: 1

Antti Karttunen, Jan 09 2017

nonn

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

Cf. A020639, A032742, A250245.
Differs from related A280496 and A280498 for the first time at n=33, where a(33) = 13, while A280496(33) = A280498(33) = 15.
Differs from related A280497 for the first time at n=42, where a(42) = 27, while A280497(42) = 21.

(define (A280495 n) (A032742 (A250245 n)))

a(n) = A032742(A250245(n)).

a(n) = A250245(n) / A020639(n). [Because A250245 preserves the smallest prime factor of n.]

A283463 a(n) = A032742(A266645(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 3, 4, 1, 7, 1, 11, 5, 6, 1, 13, 1, 17, 7, 10, 1, 19, 5, 9, 11, 8, 1, 23, 1, 29, 13, 14, 7, 15, 1, 31, 17, 22, 1, 37, 1, 41, 19, 12, 1, 43, 7, 25, 9, 26, 1, 47, 11, 21, 23, 34, 1, 53, 1, 59, 29, 20, 13, 33, 1, 61, 15, 38, 1, 67, 1, 71, 31, 18, 11, 35, 1, 73, 37, 16, 1, 79, 17, 39, 41, 46, 1, 83, 13, 55, 43, 58, 19, 51

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A020639, A032742, A266645, A280497, A280498, A283464, A283465.

f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Function[{m, n}, If[# == 1, 1, Divisors[#][[-2]]] &@ f[Lookup[s, g[n] + 1][[m]] - Boole[n == 1]]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)

(define (A283463 n) (A032742 (A266645 n)))

a(n) = A032742(A266645(n)).

a(n) = A266645(n) / A020639(n). [Because A266645 preserves the smallest prime factor of n.]

cp's OEIS Frontend

A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

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A318518 a(n) = A032742(n) AND n-A032742(n), where AND is bitwise-and (A004198) and A032742 = the largest proper divisor of n.

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A319714 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873).

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A326068 a(n) = n - sigma(A032742(n)), where sigma is the sum of divisors of n and A032742 gives the largest proper divisor of n.

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A326139 a(n) = gcd(A032742(n), A302042(n)).

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A326190 Length of the shortest path to 1 when starting from x=n and on each iteration step one may always choose either transition x -> A032742(x) or x -> A302042(x).

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A326191 Length of the longest path to 1 when starting from x=n and on each iteration step one may always choose either transition x -> A032742(x) or x -> A302042(x).

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Haskell

Scheme

A280495 a(n) = A032742(A250245(n)).

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