A351954 -id:A351954 - cp's OEIS frontend

A276076 Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450, 3675, 7350, 11025, 22050, 6125, 12250, 18375, 36750, 55125, 110250, 343

Offset: 0

Antti Karttunen, Aug 18 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the digit in one-based position k of the factorial base representation of n. See the examples.

			   n  A007623   polynomial     encoded as             a(n)
   -------------------------------------------------------
   0       0    0-polynomial   (empty product)        = 1
   1       1    1*x^0          prime(1)^1             = 2
   2      10    1*x^1          prime(2)^1             = 3
   3      11    1*x^1 + 1*x^0  prime(2) * prime(1)    = 6
   4      20    2*x^1          prime(2)^2             = 9
   5      21    2*x^1 + 1*x^0  prime(2)^2 * prime(1)  = 18
   6     100    1*x^2          prime(3)^1             = 5
   7     101    1*x^2 + 1*x^0  prime(3) * prime(1)    = 10
and:
  23     321  3*x^2 + 2*x + 1  prime(3)^3 * prime(2)^2 * prime(1)
                                      = 5^3 * 3^2 * 2 = 2250.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Indranil Ghosh, Python program for computing this sequence.
Index entries for sequences related to factorial base representation.

Cf. A000040, A007623, A084558, A099563, A257687, A276009.
Cf. A276075 (a left inverse).
Cf. A276078 (same terms in ascending order).
Cf. also A000142, A001221, A001222, A002110, A007489, A008836, A019565, A033312, A034968, A048675, A051903, A059590, A060130, A076954, A246359, A248663, A262725, A276073, A276074, A351576, A351577, A351950, A351951, A351952, A351954.
Cf. also A275733, A275734, A275735, A275725 for other such encodings of factorial base related polynomials, and A276086 for a primorial base analog.

a[n_] := Module[{k = n, m = 2, r, p = 2, q = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, q *= p^r; p = NextPrime[p]; m++]; q]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

a(0) = 1, for n >= 1, a(n) = A275733(n) * a(A276009(n)).
Or: for n >= 1, a(n) = a(A257687(n)) * A000040(A084558(n))^A099563(n).
Other identities.
For all n >= 0:
A276075(a(n)) = n.
A001221(a(n)) = A060130(n).
A001222(a(n)) = A034968(n).
A051903(a(n)) = A246359(n).
A048675(a(n)) = A276073(n).
A248663(a(n)) = A276074(n).
a(A007489(n)) = A002110(n).
a(A059590(n)) = A019565(n).
For all n >= 1:
a(A000142(n)) = A000040(n).
a(A033312(n)) = A076954(n-1).
From Antti Karttunen, Apr 18 2022: (Start)
a(n) = A276086(A351576(n)).
A276085(a(n)) = A351576(n)
A003557(a(n)) = A351577(n).
A003415(a(n)) = A351950(n).
A069359(a(n)) = A351951(n).
A083345(a(n)) = A342001(a(n)) = A351952(n).
A351945(a(n)) = A351954(n).
A181819(a(n)) = A275735(n).
(End)
lambda(a(n)) = A262725(n+1), where lambda is Liouville's function, A008836. - Antti Karttunen and Peter Munn, Aug 09 2024

Name changed by Antti Karttunen, Apr 18 2022

A275735 Prime-factorization representations of "factorial base level polynomials": a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 5, 10, 10, 20, 15, 30, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 10, 20, 20, 40, 30, 60, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 15, 30, 30, 60, 45, 90, 5, 10, 10, 20, 15, 30, 10, 20, 20, 40, 30, 60, 15, 30, 30, 60, 45, 90, 25, 50, 50, 100, 75

Offset: 0

Antti Karttunen, Aug 09 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of times a nonzero digit k occurs in the factorial base representation of n. See the examples.

			For n = 0 whose factorial base representation (A007623) is also 0, there are no nonzero digits at all, thus there cannot be any prime present in the encoding, and a(0) = 1.
For n = 1 there is just one 1, thus a(1) = prime(1) = 2.
For n = 2 ("10"), there is just one 1-digit, thus a(2) = prime(1) = 2.
For n = 3 ("11") there are two 1-digits, thus a(3) = prime(1)^2 = 4.
For n = 18 ("300") there is just one 3, thus a(18) = prime(3) = 5.
For n = 19 ("301") there is one 1 and one 3, thus a(19) = prime(1)*prime(3) = 2*5 = 10.
For n = 141 ("10311") there are three 1's and one 3, thus a(141) = prime(1)^3 * prime(3) = 2^3 * 5^1 = 40.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A000079, A001221, A001222, A003961, A007623, A008683, A181819, A225901, A257511, A257684, A265349, A265350, A264990, A275729, A275806, A351954.
Cf. also A275725, A275733, A275734 for other such prime factorization encodings of A060117/A060118-related polynomials, and also A276076.
Differs from A227154 for the first time at n=18, where a(18) = 5, while A227154(18) = 4.

A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A275735(n) = A181819(A276076(n)); \\ Antti Karttunen, Apr 03 2022

from sympy import prime
from operator import mul
import collections
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(n) = A275734(A225901(n)).
A001221(a(n)) = A275806(n).
A001222(a(n)) = A060130(n).
A048675(a(n)) = A275729(n).
A051903(a(n)) = A264990(n).
A008683(a(A265349(n))) = -1 or +1 for all n >= 0.
A008683(a(A265350(n))) = 0 for all n >= 1.
From Antti Karttunen, Apr 03 2022: (Start)
A342001(a(n)) = A351954(n).
a(n) = A181819(A276076(n)). (End)

A351945 a(n) = A342001(A181819(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 5, 1, 5, 2, 2, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 7, 1, 3, 1, 5, 5, 2, 1, 9, 1, 5, 2, 5, 1, 7, 2, 7, 2, 2, 1, 8, 1, 2, 5, 1, 2, 3, 1, 5, 2, 3, 1, 8, 1, 2, 5, 5, 2, 3, 1, 9, 1, 2, 1, 8, 2, 2, 2, 7, 1, 8, 2, 5, 2, 2, 2, 13, 1, 5, 5, 2, 1, 3, 1, 7, 3

Offset: 1

Antti Karttunen, Apr 02 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Cf. A003415, A003557, A181819, A342001, A351942, A351954 [= a(A276076(n))].

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A342001(n) = (A003415(n) / A003557(n));
A351945(n) = A342001(A181819(n));

a(n) = A342001(A181819(n)) = A351942(n) / A351944(n).

A351952 a(n) = A351950(n) / A351577(n).

Original entry on oeis.org

0, 1, 1, 5, 2, 7, 1, 7, 8, 31, 13, 41, 2, 9, 11, 37, 16, 47, 3, 11, 14, 43, 19, 53, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 19, 73, 92, 289, 127, 359, 26, 87, 113, 331, 148, 401, 2, 11, 13, 47, 20, 61, 17, 69, 86, 277, 121, 347, 24, 83, 107, 319, 142, 389, 31, 97, 128, 361, 163, 431, 3, 13, 16, 53, 23, 67

Offset: 0

Antti Karttunen, Apr 01 2022

Compare how different the scatter plot is to that of A342002, albeit with a very similar definition.

Note: this is at least partly because the other uses linear and the other uses logarithmic scatter plot. - Antti Karttunen, Oct 23 2024

Antti Karttunen, Table of n, a(n) for n = 0..20160
Index entries for sequences related to factorial base representation

Cf. A000142 (positions of 1's), A003415, A003557, A083345, A225901, A276076, A342001, A342002, A351576, A351577, A351950, A351953, A351954.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A351952(n) = { my(u=A276076(n)); (A003415(u) / A003557(u)); };

a(n) = A351950(n) / A351577(n).
a(n) = A342001(A276076(n)) = A083345(A276076(n)).
a(n) = A342002(A351576(n)).
a(n) = A351953(A225901(n)).

A353577 Arithmetic derivative without its inherited divisor applied to the prime shadow of the primorial base exp-function: a(n) = A342001(A181819(A276086(n))).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 2, 2, 3, 5, 8, 1, 5, 5, 8, 2, 7, 1, 7, 7, 12, 8, 31, 1, 9, 9, 16, 10, 41, 1, 2, 2, 3, 5, 8, 2, 3, 3, 4, 8, 11, 5, 8, 8, 11, 7, 10, 7, 12, 12, 17, 31, 46, 9, 16, 16, 23, 41, 62, 1, 5, 5, 8, 2, 7, 5, 8, 8, 11, 7, 10, 2, 7, 7, 10, 3, 9, 8, 31, 31, 46, 13, 41, 10, 41, 41, 62, 17, 55, 1, 7, 7, 12, 8

Offset: 0

Antti Karttunen, Apr 30 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A181819, A276086, A328835, A342001, A351942, A351945, A353524, A353576.
Cf. A060735 (positions of 1's).
Cf. also A342002, A351954 (similar or analogous definitions).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A353577(n) = A342001(A181819(A276086(n)));

a(n) = A342001(A328835(n)) = A351945(A276086(n)) = A342001(A181819(A276086(n))).

a(n) = A353576(n) / A353524(n).

cp's OEIS Frontend

A276076 Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.

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A275735 Prime-factorization representations of "factorial base level polynomials": a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).

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A351945 a(n) = A342001(A181819(n)).

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A351952 a(n) = A351950(n) / A351577(n).

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A353577 Arithmetic derivative without its inherited divisor applied to the prime shadow of the primorial base exp-function: a(n) = A342001(A181819(A276086(n))).

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