A108951 -id:A108951 - cp's OEIS frontend

A373986 Numerator of A373158(n) / A108951(n), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

0, 1, 1, 1, 1, 2, 1, 3, 1, 8, 1, 5, 1, 53, 1, 1, 1, 7, 1, 17, 6, 578, 1, 1, 1, 7508, 1, 107, 1, 19, 1, 5, 193, 127628, 4, 1, 1, 2424923, 2503, 3, 1, 109, 1, 1157, 7, 55773218, 1, 7, 1, 31, 14181, 15017, 1, 5, 13, 9, 269436, 1617423308, 1, 1, 1, 50140122533, 37, 3, 167, 1159, 1, 255257, 18591073, 121, 1, 1, 1, 1855184533703

Offset: 1

Antti Karttunen, Jun 25 2024

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to primorial numbers

Cf. A108951, A373158, A373985, A373987 (denominators), A373988 (rgs-transform).

A373986(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); s/gcd(m,s); };

A373986(n) = numerator(A373158(n)/A108951(n));

a(n) = A373158(n) / A373985(n) = A373158(n) / gcd(A108951(n), A373158(n)).

A283478 a(n) = A097248(A108951(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 5, 210, 6, 15, 7, 2310, 10, 30030, 11, 21, 5, 510510, 30, 9699690, 14, 33, 13, 223092870, 15, 105, 17, 14, 22, 6469693230, 42, 200560490130, 10, 39, 19, 165, 7, 7420738134810, 23, 51, 21, 304250263527210, 66, 13082761331670030, 26, 70, 29, 614889782588491410, 30, 1155, 210, 57, 34, 32589158477190044730, 21, 195, 33, 69, 31

Offset: 1

Antti Karttunen, Mar 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A019565, A097248, A108951, A283475.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, #] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], {n, 58}] (* Michael De Vlieger, Mar 18 2017 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
A283478(n) = A097248(A108951(n));

from sympy import primerange, factorint, nextprime
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a097248(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k
def a(n): return a097248(a108951(n)) # Indranil Ghosh, May 15 2017

(define (A283478 n) (A097248 (A108951 n)))

a(n) = A097248(A108951(n)).
Other identities:
For all n >= 0, a(A019565(n)) = A283475(n).

A329602 Square root of largest square dividing A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 6, 2, 1, 2, 1, 2, 6, 4, 1, 6, 1, 2, 6, 2, 1, 4, 30, 2, 6, 2, 1, 6, 1, 4, 6, 2, 30, 12, 1, 2, 6, 4, 1, 6, 1, 2, 6, 2, 1, 4, 210, 30, 6, 2, 1, 12, 30, 4, 6, 2, 1, 12, 1, 2, 6, 8, 30, 6, 1, 2, 6, 30, 1, 12, 1, 2, 30, 2, 210, 6, 1, 4, 36, 2, 1, 12, 30, 2, 6, 4, 1, 12, 210, 2, 6, 2, 30, 8, 1, 210, 6, 60, 1, 6, 1, 4, 30

Offset: 1

Antti Karttunen, Nov 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10201

Cf. A000188, A108951.

A000188(n) = core(n, 1)[2]; \\ From A000188
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A329602(n) = A000188(A108951(n));

a(n) = A000188(A108951(n)).

A329615 Bitwise-AND of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Cf. A034386, A108951, A267115, A329600, A329616, A329647.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A267115(n) = if(n>1, fold(bitand, factor(n)[, 2]), 0); \\ From A267115
A329615(n) = A267115(A108951(n));

a(n) = A267115(A108951(n)) = A267115(A329600(n)).

a(n) <= A329616(n).

A329616 Bitwise-OR of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 3, 1, 3, 3, 4, 1, 3, 1, 3, 3, 3, 1, 5, 2, 3, 3, 3, 1, 3, 1, 5, 3, 3, 3, 6, 1, 3, 3, 5, 1, 3, 1, 3, 3, 3, 1, 5, 2, 3, 3, 3, 1, 7, 3, 5, 3, 3, 1, 7, 1, 3, 3, 6, 3, 3, 1, 3, 3, 3, 1, 7, 1, 3, 3, 3, 3, 3, 1, 5, 4, 3, 1, 7, 3, 3, 3, 5, 1, 7, 3, 3, 3, 3, 3, 7, 1, 3, 3, 6, 1, 3, 1, 5, 3

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Positions of records are: 1, 2, 4, 6, 16, 24, 36, 54, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 65536, ..., conjectured also to be the positions of the first occurrence of each n.

Cf. A007814, A034386, A108951, A267116, A329600, A329615, A329647.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A267116(n) = if(n>1, fold(bitor, factor(n)[, 2]), 0); \\ From A267116
A329616(n) = A267116(A108951(n));

a(n) = A267116(A108951(n)) = A267116(A329600(n)).
a(n) >= A007814(n).
a(n) >= A329615(n).
a(n) >= A329647(n).

A329621 a(n) = gcd(A056239(n), A324888(n)) = gcd(A001222(A108951(n)), A001222(A324886(n))).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 6, 2, 1, 1, 6, 1, 2, 2, 1, 6, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 8, 1, 3, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 9, 3, 1, 1, 8, 2, 1, 4, 2, 1, 6, 8, 1, 4, 2, 1, 1, 2, 1, 1, 1, 1, 3, 8, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A001222, A034386, A056239, A108951, A276086, A324886, A324888, A329618, A329622.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 500]}, Array[GCD @@ PrimeOmega@ {#, Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[#, b]} &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105]] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329621(n) = { my(u=A108951(n)); gcd(bigomega(u), bigomega(A276086(u))); };

a(n) = gcd(A056239(n), A324888(n)) = gcd(A001222(A108951(n)), A001222(A324886(n))).

A329622 a(n) = A056239(n) - A324888(n) = A001222(A108951(n)) - A001222(A324886(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2019

sign

Cf. A001222, A034386, A056239, A108951, A276086, A324886, A324888, A329619, A329621.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 500]}, Array[Subtract @@ PrimeOmega@ {#, Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[#, b]} &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 95]] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329622(n) = { my(u=A108951(n)); (bigomega(u) - bigomega(A276086(u))); };

a(n) = A056239(n) - A324888(n) = A001222(A108951(n)) - A001222(A324886(n)).

A373987 Denominator of A373158(n) / A108951(n), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 4, 3, 15, 1, 12, 1, 105, 5, 2, 1, 36, 1, 60, 35, 1155, 1, 4, 15, 15015, 12, 420, 1, 180, 1, 16, 1155, 255255, 105, 9, 1, 4849845, 15015, 20, 1, 1260, 1, 4620, 180, 111546435, 1, 48, 105, 900, 85085, 60060, 1, 108, 385, 70, 1616615, 3234846615, 1, 18, 1, 100280245065, 1260, 16, 5005, 13860, 1, 1021020

Offset: 1

Antti Karttunen, Jun 25 2024

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to primorial numbers

Cf. A108951, A373158, A373985, A373986 (numerators).

A373987(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); m/gcd(m,s); };

A373987(n) = denominator(A373158(n)/A108951(n));

a(n) = A108951(n) / A373985(n) = A108951(n) / gcd(A108951(n), A373158(n)).

A324887 a(n) = A108951(n) * A276086(A108951(n)).

Original entry on oeis.org

2, 6, 30, 36, 210, 300, 2310, 120, 1260, 2940, 30030, 15000, 510510, 50820, 21176820, 3600, 9699690, 88200, 223092870, 288120, 2232166860, 780780, 6469693230, 42000, 645668100, 17357340, 11880, 12298440, 200560490130, 66555720, 7420738134810, 672, 66899572740, 368588220, 228227900600700, 216090000, 304250263527210

Offset: 1

Antti Karttunen, Mar 30 2019

nonn

Cf. A034386, A108951, A276086, A324580, A324886, A324888.

Permutation of A324577.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324886(n) = A276086(A108951(n));
A324887(n) = (A324886(n)*A108951(n));

a(n) = A324580(A108951(n)) = A108951(n) * A324886(n).

A329612 a(n) = gcd(d(n), d(A108951(n))), where d(n) gives the number of divisors of n, A000005(n), and A108951 is fully multiplicative with a(prime(i)) = prime(i)# = prime(1) * ... * prime(i).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 4, 2, 2, 2, 4, 2, 5, 2, 6, 2, 2, 4, 4, 2, 2, 3, 4, 4, 2, 2, 8, 2, 6, 4, 4, 2, 3, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 2, 3, 6, 4, 2, 2, 4, 4, 8, 4, 4, 2, 6, 2, 4, 2, 7, 4, 8, 2, 2, 4, 8, 2, 6, 2, 4, 6, 2, 2, 8, 2, 2, 5, 4, 2, 12, 4, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 2, 2, 6, 2, 9, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A000005, A034386, A108951, A329614.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329612(n) = gcd(numdiv(n),numdiv(A108951(n)));

a(n) = gcd(A000005(n),A329605(n)) = gcd(A000005(n),A000005(A108951(n))).

cp's OEIS Frontend

A373986 Numerator of A373158(n) / A108951(n), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A283478 a(n) = A097248(A108951(n)).

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A329602 Square root of largest square dividing A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A329615 Bitwise-AND of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A329616 Bitwise-OR of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A329621 a(n) = gcd(A056239(n), A324888(n)) = gcd(A001222(A108951(n)), A001222(A324886(n))).

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A329622 a(n) = A056239(n) - A324888(n) = A001222(A108951(n)) - A001222(A324886(n)).

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A373987 Denominator of A373158(n) / A108951(n), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

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