A253553 -id:A253553 - cp's OEIS frontend

A336121 a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + a(A253553(n)).

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 2, 0, 2, 0, 2, 0, 2, 1

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

Positions for the first occurrence of each n, for n >= 0, are: 1, 4, 16, 32, 144, 512, 2048, 6912, 20736, 62208, ...

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

Cf. A000120, A001222, A122111, A253553, A292377, A292383, A336120, A336123, A336124.

Cf. A336119 (positions of zeros).

A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336121(n) = if(1==n,0,(3==A336124(n))+A336121(A253553(n)));

a(1) = 0, and for n > 1, a(n) = [A336124(n) == 3] + a(A253553(n)).
a(n) = A000120(A336120(n)).
a(n) = A292377(A122111(n)).
a(n) = A001222(n) - A336123(n).

A336123 a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + a(A253553(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 4, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

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

Cf. A000120, A000244, A001222, A122111, A292375, A336121, A336124, A336125.

\\ Uses also code given in A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336123(n) = if(n<=2,n-1,(1==A336124(n))+A336123(A253553(n)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A336124(n) == 1] + a(A253553(n)).
a(n) = A000120(A336125(n)).
For n > 1, a(n) = A292375(A122111(n)).
a(n) = A001222(n) - A336121(n).
For all n >= 0, a(3^n) = n.

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 4, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 5, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 5, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

a(n) tells how many iterations of A252463 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A005940 and A163511.

Similarly for A253553 in trees A253563 and A253565. - Antti Karttunen, Apr 14 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
and largest hook
  o o o o o o
  o
  o
both of which have size 8, so a(715) = 8.
(End)

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

Cf. A000040, A000079, A001221, A001222, A005940, A029837, A061395, A156552 (A005941), A163511, A243071, A252461, A252463, A252735, A252736, A252759, A253553, A253563, A253565, A297113, A297155, A297167, A324870, A324872.
Right edge of irregular triangle A265146.
Cf. also A246348.
Cf. A052126, A056239, A093641, A112798, A257990, A325133, A325134, A325135.

Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]]-1,{n,100}] (* Gus Wiseman, Apr 02 2019 *)

A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = (bigomega(n) + A061395(n) - 1); \\ Antti Karttunen, Apr 14 2019

from sympy import primepi, primeomega, primefactors
def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023

a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
a(n) = A029837(1+A243071(n)). [a(n) = binary width of terms of A243071.]
a(n) = A029837(A005941(n)) = A029837(1+A156552(n)). [Also binary width of terms of A156552.]
Other identities. For all n >= 1:
a(A000040(n)) = n.
a(A001248(n)) = n+1.
a(A030078(n)) = n+2.
And in general, a(prime(n)^k) = n+k-1.
a(A000079(n)) = n. [I.e., a(2^n) = n.]
For all n >= 2:
a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).
a(n) = A325134(n) - 1. - Gus Wiseman, Apr 02 2019
From Antti Karttunen, Apr 14 2019: (Start)
a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
a(n) = A001221(n) + A297167(n) = A297113(n) + A297155(n).
(End).

A253550 Shift one instance of the largest prime one step towards larger primes: a(1) = 1, for n>1: a(n) = (n / prime(g)) * prime(g+1), where g = A061395(n), index of the greatest prime dividing n.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 12, 15, 14, 13, 20, 17, 22, 21, 24, 19, 30, 23, 28, 33, 26, 29, 40, 35, 34, 45, 44, 31, 42, 37, 48, 39, 38, 55, 60, 41, 46, 51, 56, 43, 66, 47, 52, 63, 58, 53, 80, 77, 70, 57, 68, 59, 90, 65, 88, 69, 62, 61, 84, 67, 74, 99, 96, 85, 78, 71, 76, 87, 110, 73, 120, 79, 82, 105, 92, 91, 102, 83, 112, 135, 86, 89

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

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

Inverse: A252462.
Cf. A102750 (same terms, but with 2 instead of 1, sorted into ascending order).
Cf. A003961, A052126, A065091, A061395, A122111, A253553, A253554, A253560, A253563, A253565.

(define (A253550 n) (if (= 1 n) n (* (A065091 (A061395 n)) (A052126 n))))

a(1) = 1; for n>1: a(n) = A065091(A061395(n)) * A052126(n).
Other identities. For all n >= 1:
A252462(a(n)) = n. [A252462 works as an inverse function for this injection.]
a(n) <= A253560(n).

A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

From Gus Wiseman, Aug 05 2021 and Antti Karttunen, Oct 13 2021: (Start)
Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
Proof: A108951(n) gives a number with the same largest prime factor (A006530) and its exponent (A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example.
(End)

			From _Gus Wiseman_, Aug 15 2021: (Start)
The list of all numbers with image 12 and their corresponding prime factors begins:
  144: (3,3,2,2,2,2)
  216: (3,3,3,2,2,2)
  240: (5,3,2,2,2,2)
  288: (3,3,2,2,2,2,2)
  336: (7,3,2,2,2,2)
  360: (5,3,3,2,2,2)
(End)
The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - _Antti Karttunen_, Oct 13 2021
Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - _Antti Karttunen_, Oct 15 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to Heinz numbers
Index entries for sequences related to primorial numbers

Cf. A000188, A108951, A319626, A329602, A329900.
A left inverse of A000290.
Positions of 1's are A008578.
Positions of primes are A168645.
The sum of prime indices of a(n) is A346700(n).
The odd version is A346701.
The odd non-reverse version is A346703.
The non-reverse version is A346704.
The version for standard compositions is A346705, odd A346702.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A001414 adds up prime factors, row sums of A027746.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A346633 adds up the even bisection of standard compositions.
A346698 adds up the even bisection of prime indices.
Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.

Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]],2],{n,100}] (* Gus Wiseman, Oct 13 2021 *)

PARI
```
A329888(n) = A329900(A329602(n));
```

A329888(n) = if(1==n,n,my(f=factor(n),m=1,p=0); forstep(k=#f~,1,-1,while(f[k,2], m *= f[k,1]^(p%2); f[k,2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - Antti Karttunen, Sep 21 2021

a(n) = A329900(A329602(n)) = A329900(A000188(A108951(n))).
A108951(a(n)) = A329602(n).
a(n^2) = n for all n >= 1.
a(n) * A346701(n) = n. - Gus Wiseman, Aug 07 2021
A056239(a(n)) = A346700(n). - Gus Wiseman, Aug 07 2021
Antti Karttunen, Sep 21 2021
From Antti Karttunen, Oct 13 2021: (Start)
a(n) = A319626(A329602(n)) = A319626(A000188(A108951(n))).
For all x in A102750, a(x) = a(A253553(x)). (End)

Name amended with Gus Wiseman's new interpretation - Antti Karttunen, Oct 13 2021

A336120 a(n) = A292383(A122111(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 4, 0, 8, 0, 5, 0, 5, 0, 8, 0, 16, 0, 10, 1, 32, 0, 16, 0, 10, 0, 11, 0, 64, 2, 8, 0, 128, 0, 20, 0, 20, 0, 32, 0, 256, 0, 22, 0, 8, 0, 64, 0, 11, 4, 40, 0, 512, 0, 16, 0, 1024, 0, 22, 8, 40, 0, 128, 0, 16, 0, 20, 0, 2048, 1, 256, 0, 80, 0, 44, 0, 4096, 0, 32, 16, 8192, 0, 80, 0, 22, 0, 512, 0, 16384, 32, 44, 0, 17, 0, 17, 0

Offset: 1

Antti Karttunen, Jul 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A064989, A122111, A252463, A253553, A253566, A292383, A336121, A336124, A336125, A336312.

Cf. A336119 (positions of zeros).

\\ Uses also code given in A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336120(n) = if(1==n,0,(3==A336124(n))+(2*A336120(A253553(n))));

a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + 2*a(A253553(n)).
a(n) = A292383(A122111(n)).
a(n) = A253566(n) - A336125(n).
A000120(a(n)) = A336121(n).

A336125 a(n) = A292385(A122111(n)).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 5, 5, 8, 16, 10, 32, 16, 10, 10, 64, 8, 128, 20, 20, 32, 256, 20, 8, 64, 11, 40, 512, 16, 1024, 20, 40, 128, 16, 21, 2048, 256, 80, 40, 4096, 32, 8192, 80, 22, 512, 16384, 40, 17, 17, 160, 160, 32768, 16, 32, 80, 320, 1024, 65536, 42, 131072, 2048, 44, 41, 64, 64, 262144, 320, 640, 34, 524288, 41, 1048576, 4096, 20

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A064989, A122111, A253553, A253566, A292385, A336120, A336123, A336313.

\\ Uses also code given in A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336125(n) = if(n<=2,n-1,(1==A336124(n))+(2*A336125(A253553(n))));

A064989(n) = {my(f); 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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A252463(n) = if(!(n%2),n/2,A064989(n));
A292385(n) = if(n<=2,n-1,(1==(n%4))+(2*A292385(A252463(n))));
A336125(n) = A292385(A122111(n));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + 2*a(A253553(n)).
a(n) = A292385(A122111(n)).
a(n) = A253566(n) - A336120(n).
A000120(a(n)) = A336123(n).

A336312 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336120(i)) = A278222(A336120(j)) for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 3, 1, 4, 1, 2, 2, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 2, 3, 1, 2, 1, 2, 1, 2, 1, 4, 2, 3, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 4, 1, 2, 1, 2, 2, 2, 1, 3, 1, 4, 1, 2, 1, 2, 2, 4, 1, 3, 1, 3, 1, 3, 1, 3, 2

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

Restricted growth sequence transform of A278222(A336120(n)).
For all i, j:
  a(i) = a(j) => A336121(i) = A336121(j) => A335909(i) = A335909(j).

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

Cf. A336311, A336313.
Cf. A336119 (positions of ones).
Cf. also A292583, A332901, A335909, A336121.

up_to = 1024; \\ 65537;
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; };
\\ Needs also code from A336124:
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336120(n) = if(1==n,0,(3==A336124(n))+(2*A336120(A253553(n))));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v336312 = rgs_transform(vector(up_to,n,A278222(A336120(n))));
A336312(n) = v336312[n];

A356300 Square array read by antidiagonals. A(n,k) is the nearest common ancestor of n and k in the binary tree depicted in A253563.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 3, 4, 3, 4, 7, 4, 3, 4, 3, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Antti Karttunen, Aug 03 2022

Array is symmetric and is read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .

Also the nearest common ancestor of n and k in the tree depicted in A253565 (the mirror image of the A253563-tree).

			The top left 21x21 corner of the array:
n/k  |  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21
-----+----------------------------------------------------------------------------
   1 |  1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
   2 |  1, 2, 2, 2, 2, 2, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,
   3 |  1, 2, 3, 2, 3, 2, 3, 2, 3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3,
   4 |  1, 2, 2, 4, 2, 4, 2, 4, 2,  4,  2,  4,  2,  4,  2,  4,  2,  4,  2,  4,  2,
   5 |  1, 2, 3, 2, 5, 2, 5, 2, 3,  2,  5,  2,  5,  2,  3,  2,  5,  2,  5,  2,  3,
   6 |  1, 2, 2, 4, 2, 6, 2, 4, 2,  6,  2,  4,  2,  6,  2,  4,  2,  6,  2,  4,  2,
   7 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2,  7,  2,  7,  2,  3,  2,  7,  2,  7,  2,  3,
   8 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2,  8,  2,  4,  2,  8,  2,  4,  2,  8,  2,
   9 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2,  9,  2,  3,  2,  3,  2,  9,
  10 |  1, 2, 2, 4, 2, 6, 2, 4, 2, 10,  2,  4,  2, 10,  2,  4,  2,  6,  2,  4,  2,
  11 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 11,  2,  3,  2, 11,  2, 11,  2,  3,
  12 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2, 12,  2,  4,  2,  8,  2,  4,  2, 12,  2,
  13 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 13,  2, 13,  2,  3,
  14 |  1, 2, 2, 4, 2, 6, 2, 4, 2, 10,  2,  4,  2, 14,  2,  4,  2,  6,  2,  4,  2,
  15 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2, 15,  2,  3,  2,  3,  2, 15,
  16 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2,  8,  2,  4,  2, 16,  2,  4,  2,  8,  2,
  17 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 17,  2, 17,  2,  3,
  18 |  1, 2, 2, 4, 2, 6, 2, 4, 2,  6,  2,  4,  2,  6,  2,  4,  2, 18,  2,  4,  2,
  19 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 17,  2, 19,  2,  3,
  20 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2, 12,  2,  4,  2,  8,  2,  4,  2, 20,  2,
  21 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2, 15,  2,  3,  2,  3,  2, 21,
.
A(3,6) = A(6,3) = 2 because the nearest common ancestor of 3 and 6 in the tree described in A253563 (and in A253565) is 2.
A(4,6) = A(6,4) = 4 because 6 occurs as a descendant of 4 in A253563-tree, thus their nearest common ancestor is 4 itself.

Index entries for sequences computed from indices in prime factorization

Cf. A253553, A253563, A253565, A356301.

Cf. also A348041.

up_to = 105;
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356300list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A356300sq(col,(a-(col-1))))); (v); };
v356300 = A356300list(up_to);
A356300(n) = v356300[n];

A356306 The nearest common ancestor of A000265(n) and gcd(A000265(n), sigma(n)) in the A253563-tree.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 9, 7, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3

Offset: 1

Antti Karttunen, Aug 03 2022

nonn

Cf. A000203, A000265, A161942, A355931, A356300, A356301, A356307, A356308.

Cf. also A356156.

A000265(n) = (n>>valuation(n,2));
A161942(n) = A000265(sigma(n));
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356306(n) = A356300sq(A000265(n), gcd(n, A161942(n)));

a(n) = A356300(A000265(n), A355931(n)).

cp's OEIS Frontend

A336121 a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + a(A253553(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A336123 a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + a(A253553(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A253550 Shift one instance of the largest prime one step towards larger primes: a(1) = 1, for n>1: a(n) = (n / prime(g)) * prime(g+1), where g = A061395(n), index of the greatest prime dividing n.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A336120 a(n) = A292383(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A336125 a(n) = A292385(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A336312 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336120(i)) = A278222(A336120(j)) for all i, j >= 1.

Views

Author

Keywords

Comments