A324202 -id:A324202 - cp's OEIS frontend

A324203 Lexicographically earliest sequence such that a(i) = a(j) => A324202(i) = A324202(j) for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Restricted growth sequence transform of A324202.
For all i, j:
  a(i) = a(j) => A324190(i) = A324190(j),
  a(i) = a(j) => A324191(i) = A324191(j).

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

Cf. A300827, A324181, A324190, A324191, A324202.

up_to = 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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324202(n) = A046523(factorback(apply(x -> prime(1+x),apply(A297167, select(d -> d>1,divisors(n))))));
v324203 = rgs_transform(vector(up_to, n, A324202(n)));
A324203(n) = v324203[n];

A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Number of distinct values of the sum {excess of d} + {the index of the largest prime factor of d} (that is, A046660(d) + A061395(d)) that occurs over all divisors d > 1 of n.

Number of distinct values A297112 obtains over the divisors > 1 of n; a(1) = 0.

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

Cf. A000005, A001221, A001222, A046660, A061395, A324120, A324179, A324191, A324192, A324202, A324203.

Cf. also A156552, A297112.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));

a(n) = A001221(A324202(n)).
a(n) >= A324120(n).
a(n) >= A001222(n) >= A001221(n). [See A324179 and A324192 for differences]
a(n) <= A000005(n)-1. [See A324191 for differences]
For all primes p, a(p^k) = k.

A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

Original entry on oeis.org

1, 2, 3, 6, 5, 18, 7, 30, 15, 50, 11, 270, 13, 98, 75, 210, 17, 450, 19, 1050, 147, 242, 23, 9450, 35, 338, 105, 3234, 29, 11250, 31, 2310, 363, 578, 245, 47250, 37, 722, 507, 57750, 41, 43218, 43, 9438, 2625, 1058, 47, 727650, 77, 2450, 867, 17238, 53, 22050, 605, 210210, 1083, 1682, 59, 8268750, 61, 1922, 8085, 30030

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A046660, A046523, A048675, A061395, A097248, A156552, A297113, A324202, A332462.

A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A332461(n) = if(1==n,1, my(m=1); fordiv(n,d,if(d>1, m *= prime(A297113(d)))); (m));

a(n) = Product_{d|n, d>1} A000040(A297113(d)).
a(p) = p for all primes p.
For all n >= 0, a(2^n) = A002110(n).
For all n >= 1:
A046523(a(n)) = A324202(n).
A048675(a(n)) = A156552(n).
A097248(a(n)) = A332462(n).

A324193 a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 54, 1, 14, 15, 30, 1, 90, 1, 150, 21, 22, 1, 1350, 5, 26, 15, 294, 1, 2250, 1, 210, 33, 34, 35, 6750, 1, 38, 39, 5250, 1, 6174, 1, 726, 375, 46, 1, 66150, 7, 350, 51, 1014, 1, 3150, 55, 16170, 57, 58, 1, 1181250, 1, 62, 735, 2310, 65, 23958, 1, 1734, 69, 17150, 1, 1653750, 1, 74, 525, 2166, 77, 39546, 1, 404250, 105

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

An auxiliary sequence for defining A300827, which is the restricted growth sequence transform of this sequence. A324202 is a similar sequence, but is not limited to the proper divisors of n, and in contrast to this, also finds the least prime signature representative (A046523) of the product formed.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences computed from indices in prime factorization

Cf. A001221, A001222, A007913, A061395, A087207, A297167, A300827 (rgs-transform), A324120, A324180.

Cf. also A324202.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324193(n) = { my(m=1); if(n<=2, n-1, fordiv(n, d, if((d>1)&(dA297167(d)))); (m)); };

a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).
For all n > 0:
A001222(a(n)) = A000005(n)-2.
A001221(A007913(a(n))) = A324120(n).
A087207(A007913(a(n))) = A324180(n).

A324537 a(n) = A003557(k), where k = Product_{d|n, d>2} prime(A297167(d)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 5, 3, 1, 1, 6, 1, 3, 5, 7, 1, 12, 1, 11, 1, 5, 1, 54, 1, 1, 7, 13, 5, 36, 1, 17, 11, 9, 1, 250, 1, 7, 9, 19, 1, 60, 1, 15, 13, 11, 1, 30, 7, 5, 17, 23, 1, 1620, 1, 29, 5, 1, 11, 686, 1, 13, 19, 375, 1, 540, 1, 31, 15, 17, 7, 2662, 1, 45, 1, 37, 1, 3500, 13, 41, 23, 7, 1, 2430, 11, 19, 29, 43, 17, 420, 1, 35, 7, 75, 1

Offset: 1

Antti Karttunen, Mar 07 2019

nonn

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

Cf. A000961 (positions of ones), A003557, A297167, A300827, A324191, A324193, A324202, A324538.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A324537(n) = { my(m=1); fordiv(n, d, if(d>2, m *= prime(A297167(d)))); A003557(m); };

A001222(a(n)) = A324191(n) - 1.

cp's OEIS Frontend

A324203 Lexicographically earliest sequence such that a(i) = a(j) => A324202(i) = A324202(j) for all i, j >= 1.

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A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

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A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

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A324193 a(1) = 0; for n > 1, a(n) = Product_{d|n, d>1, dA297167(d)).

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A324537 a(n) = A003557(k), where k = Product_{d|n, d>2} prime(A297167(d)).

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