A108951 -id:A108951 - cp's OEIS frontend

A329617 Product of distinct exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 6, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 6, 1, 5, 2, 2, 2, 8, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 6, 2, 3, 1, 12, 2, 4, 2, 2, 1, 8, 1, 2, 3, 6, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 6, 3, 2, 6, 1, 5, 4, 2, 1, 8, 2, 2, 2, 4, 1, 12, 2, 3, 2, 2, 2, 6, 1, 6, 3, 8, 1, 6, 1, 4, 6

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Cf. A034386, A108951, A290107, A329600, A329605.

Differs from related A329378 for the first time at n=36. See also A329382.

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
A290107(n) = factorback(vecsort((factor(n)[, 2]), , 8));
A329617(n) = A290107(A108951(n));

a(n) = A290107(A108951(n)) = A290107(A329600(n)).

A329378(n) <= a(n) <= A329382(n) <= A329605(n).

A329618 a(n) = gcd(A001222(n), A324888(n)), where A324888(n) is the minimal number of primorials (A002110) that add to A108951(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A001222, A034386, A108951, A276086, A324886, A324888, A329619, A329620, A329621.

With[{b = Reverse@ Prime@ Range@ 120}, Array[GCD[PrimeOmega@ #1, Total@ IntegerDigits[#2, MixedRadix[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); };
A324886(n) = A276086(A108951(n));
A329618(n) = gcd(bigomega(n), bigomega(A324886(n)));

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

A329647 Bitwise-XOR 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, 0, 2, 1, 3, 0, 3, 0, 2, 1, 2, 0, 3, 1, 4, 1, 1, 0, 3, 0, 2, 1, 5, 2, 3, 0, 2, 0, 0, 1, 5, 1, 2, 3, 6, 0, 3, 0, 4, 1, 1, 0, 3, 1, 2, 1, 4, 0, 3, 1, 2, 0, 7, 2, 5, 0, 3, 1, 7, 0, 2, 0, 6, 3, 0, 1, 3, 1, 2, 0, 7, 1, 3, 2, 2, 1, 1, 0, 5, 0, 2, 1, 6, 2, 3, 0, 4, 0, 6, 0, 3, 1, 2, 3, 7, 1, 1, 1, 4, 0, 0, 1, 5, 3

Offset: 1

Antti Karttunen, Nov 18 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. A034386, A108951, A268387, A329615, A329616.

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
A268387(n) = if(n>1, fold(bitxor, factor(n)[, 2]), 0);
A329647(n) = A268387(A108951(n));

a(n) = A268387(A108951(n)).

a(n) <= A329616(n).

A330686 Primorial deflation of (nonzero) K-champion numbers: a(n) is the unique integer x such that A108951(x) = A307866(1+n).

Original entry on oeis.org

1, 4, 3, 8, 6, 12, 9, 24, 18, 48, 20, 36, 96, 40, 72, 30, 54, 80, 144, 60, 108, 160, 288, 120, 216, 320, 90, 576, 240, 432, 180, 480, 864, 360, 960, 720, 1920, 540, 252, 1440, 3840, 1080, 504, 2880, 1200, 2160, 1008, 5760, 2688, 2400, 4320, 2016, 11520, 3240, 4800, 1512, 8640, 4032, 23040, 1680, 6480, 9600, 3024, 17280, 8064, 7200

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..761 (computed from the b-file of A307866)

Cf. A074206, A181815, A307866, A329900, A353568.

a(n) = A329900(A307866(1+n)).

a(n) = A122111(A353568(n)). - Antti Karttunen, May 20 2022

A330689 Primorial deflation of A330687 (record positions in A050377): a(n) is the unique integer x such that A108951(x) = A330687(n).

Original entry on oeis.org

1, 4, 16, 64, 36, 256, 144, 1024, 81, 576, 324, 2304, 1296, 5184, 2916, 20736, 11664, 82944, 14400, 46656, 331776, 57600, 32400, 1327104, 104976, 230400, 746496, 40000, 129600, 419904, 921600, 160000, 1679616, 3686400, 291600, 640000, 2073600, 6718464, 360000, 1166400, 2560000, 26873856, 1440000, 313600, 4665600, 10240000, 810000

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..294

Cf. A050376, A050377, A108951, A329900, A330687.

PARI
```
A330689(n) = A329900(A330687(n));
```

a(n) = A329900(A330687(n)).

A331292 The next more significant digit after A329348(n) in the primorial base expansion of A108951(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 6, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 3, 0, 1, 0, 0, 0, 0, 5, 0, 2, 0, 0, 3, 0, 16, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 0, 10, 12, 0, 0, 0, 1, 6, 0, 12, 2, 6, 0, 0, 0, 0, 3

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Cf. A007949, A061395, A108951, A117366, A151800, A246277, A324886, A329348, A331188,

A331293.

A331292(n) = { my(h=A061395(n), u = A108951(n) - (A002110(h)*((A331188(n)%prime(1+h))))); ((u/A002110(1+h))%prime(2+h)); };

a(n) = A007949(A246277(A324886(n))).

a(n) = A331293(n) modulo A000040(2+A061395(n)).

A331293 Let h = A061395(n) and u = A108951(n) - (A002110(h)*A329348(n)). Then a(n) = u/A002110(1+h).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 4, 0, 7, 0, 0, 1, 0, 5, 0, 0, 2, 4, 0, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 19, 8, 0, 0, 0, 14, 2, 0, 0, 0, 0, 3, 0, 0, 3, 10, 1, 0, 0, 0, 0, 5, 0, 9, 0, 0, 25, 0, 16, 0, 0, 2, 43, 0, 0, 2, 1, 0, 0, 0, 0, 10, 12, 0, 0, 0, 1, 6, 0, 38, 2, 17, 0, 0, 0, 0, 16

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A002110, A061395, A108951, A329348, A331188, A331292.

Cf. A331294 (positions of records).

A331293(n) = { my(h=A061395(n), u = A108951(n) - (A002110(h)*(A331188(n)%prime(1+h)))); (u/A002110(1+h)); };

a(n) modulo A151800(A117366(n)) = A331292(n).

A344698 a(n) = A344696(A108951(n)).

Original entry on oeis.org

1, 1, 1, 7, 1, 7, 1, 5, 91, 7, 1, 5, 1, 7, 91, 31, 1, 65, 1, 5, 91, 7, 1, 31, 2821, 7, 25, 5, 1, 65, 1, 21, 91, 7, 2821, 403, 1, 7, 91, 31, 1, 65, 1, 5, 25, 7, 1, 21, 7657, 403, 91, 5, 1, 155, 2821, 31, 91, 7, 1, 403, 1, 7, 25, 127, 2821, 65, 1, 5, 91, 403, 1, 91, 1, 7, 155, 5, 7657, 65, 1, 21, 3751, 7, 1, 403, 2821

Offset: 1

Antti Karttunen, May 26 2021

nonn

Records seem to occur on certain squares: 1, 4, 9, 25, 49, 100, 121, 289, 361, 529, 841, 961, etc. See A344701.

Cf. A000203, A001615, A108951, A337203, A344696, A344699, A344701 (positions of records).

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
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344696(n) = { my(u=sigma(n)); (u/gcd(u,A001615(n))); };
A344698(n) = A344696(A108951(n));

A344699 a(n) = A344697(A108951(n)).

Original entry on oeis.org

1, 1, 1, 6, 1, 6, 1, 4, 72, 6, 1, 4, 1, 6, 72, 24, 1, 48, 1, 4, 72, 6, 1, 24, 2160, 6, 18, 4, 1, 48, 1, 16, 72, 6, 2160, 288, 1, 6, 72, 24, 1, 48, 1, 4, 18, 6, 1, 16, 5760, 288, 72, 4, 1, 108, 2160, 24, 72, 6, 1, 288, 1, 6, 18, 96, 2160, 48, 1, 4, 72, 288, 1, 64, 1, 6, 108, 4, 5760, 48, 1, 16, 2592, 6, 1, 288, 2160

Offset: 1

Antti Karttunen, May 26 2021

nonn

Cf. A000203, A001615, A108951, A337203, A344697, A344698, A344701 (apparently positions of records).

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]) };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344697(n) = { my(u=A001615(n)); (u/gcd(u,sigma(n))); };
A344699(n) = A344697(A108951(n));

A346092 a(n) = A276085(A328572(A108951(n))).

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 0, 0, 30, 0, 18, 0, 210, 150, 8, 0, 36, 0, 90, 1050, 2310, 0, 12, 660, 30030, 0, 630, 0, 120, 0, 0, 11550, 510510, 3780, 108, 0, 9699690, 150150, 0, 0, 0, 0, 6930, 840, 223092870, 0, 60, 11550, 1560, 2552550, 90090, 0, 216, 36960, 1470, 48498450, 6469693230, 0, 480, 0, 200560490130, 5040, 32, 360360

Offset: 1

Antti Karttunen, Jul 09 2021

nonn

Cf. A002110, A108951, A276085, A328572, A344592, A346093.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A346092(n) = A276085(A328572(A108951(n)));

a(n) = A276085(A344592(n)) = A276085(A328572(A108951(n))).

a(n) = A108951(n) - A346093(n).

cp's OEIS Frontend

A329617 Product of distinct 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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PARI

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A329618 a(n) = gcd(A001222(n), A324888(n)), where A324888(n) is the minimal number of primorials (A002110) that add to A108951(n).

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A329647 Bitwise-XOR 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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A330686 Primorial deflation of (nonzero) K-champion numbers: a(n) is the unique integer x such that A108951(x) = A307866(1+n).

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A330689 Primorial deflation of A330687 (record positions in A050377): a(n) is the unique integer x such that A108951(x) = A330687(n).

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A331292 The next more significant digit after A329348(n) in the primorial base expansion of A108951(n).

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A331293 Let h = A061395(n) and u = A108951(n) - (A002110(h)*A329348(n)). Then a(n) = u/A002110(1+h).

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A344698 a(n) = A344696(A108951(n)).

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A344699 a(n) = A344697(A108951(n)).

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