A276076 -id:A276076 - cp's OEIS frontend

A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

1, 2, 2, 6, 4, 12, 2, 6, 6, 30, 12, 60, 4, 12, 12, 60, 36, 180, 8, 24, 24, 120, 72, 360, 2, 6, 6, 30, 12, 60, 6, 30, 30, 210, 60, 420, 12, 60, 60, 420, 180, 1260, 24, 120, 120, 840, 360, 2520, 4, 12, 12, 60, 36, 180, 12, 60, 60, 420, 180, 1260, 36, 180, 180, 1260, 900, 6300, 72, 360, 360, 2520, 1800, 12600, 8, 24, 24, 120, 72, 360, 24, 120, 120, 840, 360, 2520

Offset: 0

Antti Karttunen, Nov 16 2016

This sequence can be used for filtering certain factorial base related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A276076(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Any such sequence should match where the result is computed from the nonzero digits (that may also be > 9) in the factorial base representation of n, but does not depend on their order. Some of these are listed on the last line of the Crossrefs section.
Note that as A275735 is present in that list it means that the sequences matching to its filter-sequence A278235 form a subset of the sequences matching to this sequence. Also, for A275735 there is a stronger condition that for any i, j: a(i) = a(j) <=> A275735(i) = A275735(j), which if true, would imply that there is an injective function f such that f(A275735(n)) = A278236(n), and indeed, this function seems to be A181821.

Cf. A007623, A046523, A181821, A276076.
Similar sequences: A278222 (base-2 related), A069877 (base-10), A278226 (primorial base), A278225, A278234, A278235 (other variants for factorial base),
Differs from A278226 for the first time at n=24, where a(24)=2, while A278226(24)=16.
Sequences that partition N into same or coarser equivalence classes: A275735 (<=>), A034968, A060130, A227153, A227154, A246359, A257079, A257511, A257679, A257694, A257695, A257696, A264990, A275729, A275806, A275948, A275964, A278235.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; s = ReverseSort[s]; Times @@ (Prime[Range[Length[s]]] ^ s)]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

(define (A278236 n) (A046523 (A276076 n)))

a(n) = A046523(A276076(n)).

a(n) = A181821(A275735(n)). [Empirical formula found with the help of equivalence class matching. Not yet proved.]

A351950 Arithmetic derivative of the factorial base exp-function: a(n) = A003415(A276076(n)).

Original entry on oeis.org

0, 1, 1, 5, 6, 21, 1, 7, 8, 31, 39, 123, 10, 45, 55, 185, 240, 705, 75, 275, 350, 1075, 1425, 3975, 1, 9, 10, 41, 51, 165, 12, 59, 71, 247, 318, 951, 95, 365, 460, 1445, 1905, 5385, 650, 2175, 2825, 8275, 11100, 30075, 14, 77, 91, 329, 420, 1281, 119, 483, 602, 1939, 2541, 7287, 840, 2905, 3745, 11165, 14910, 40845

Offset: 0

Antti Karttunen, Apr 01 2022

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A003415, A276076, A351576, A351952.
Differs from a similarly defined A327860 for the first time at n=24.
Coincides with A351951 on n given by A059590.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 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; };
A351950(n) = A003415(A276076(n));

a(n) = A003415(A276076(n)).

a(n) = A327860(A351576(n)).

A351954 Arithmetic derivative without its inherited divisor applied to the prime shadow of the factorial base exp-function: a(n) = A342001(A181819(A276076(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, 2, 2, 3, 5, 8, 2, 3, 3, 4, 8, 11, 5, 8, 8, 11, 7, 10, 7, 12, 12, 17, 31, 46, 1, 5, 5, 8, 2, 7, 5, 8, 8, 11, 7, 10, 2, 7, 7, 10, 3, 9, 8, 31, 31, 46, 13, 41, 1, 7, 7, 12, 8, 31, 7, 12, 12, 17, 31, 46, 8, 31, 31, 46, 13, 41, 2, 9, 9, 14, 11

Offset: 0

Antti Karttunen, Apr 02 2022

Compare the scatter plot to those of A275735, A353575 and of A353577. - Antti Karttunen, Apr 30 2022

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

Cf. A003415, A003557, A181819, A275735, A276076, A342001, A351576, A351945, A351952, A353575, A353577.

Cf. A051683 (positions of 1's).

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])));
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; };
A342001(n) = (A003415(n) / A003557(n));
A351945(n) = A342001(A181819(n));
A351954(n) = A351945(A276076(n));

a(n) = A342001(A275735(n)) = A351945(A276076(n)).

a(n) = A353577(A351576(n)). - Antti Karttunen, Apr 30 2022

Verbal description added to the definition by Antti Karttunen, Apr 30 2022

A276073 A276076-polynomials evaluated at x=2: a(n) = A048675(A276076(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 12, 13, 14, 15, 16, 17, 8, 9, 10, 11, 12, 13, 12, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 24, 25, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 24, 25, 24, 25, 26, 27, 28, 29, 28, 29, 30, 31, 32, 33, 24, 25, 26, 27, 28, 29, 28, 29, 30, 31, 32, 33, 32, 33, 34, 35, 36, 37, 36

Offset: 0

Antti Karttunen, Aug 18 2016

Antti Karttunen, Table of n, a(n) for n = 0..5039
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A048675, A276076.

Cf. also A276074.

(define (A276073 n) (A048675 (A276076 n)))

a(n) = A048675(A276076(n)).

A376403 a(0) = 0, and for n > 0, a(n) = a(n-1) + A276076(a(n-1)), where A276076 is the factorial base exp-function.

Original entry on oeis.org

0, 1, 3, 9, 39, 1089, 520179, 1466909163669354042297

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(8) has 212 digits, a(9) has 10654 digits.

By induction, it is easy to see that formula a(n) = A276075(A376399(n)) implies that from the second term onward, this sequence gives the partial sums of A376399. See more comments in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..8
Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A376399, A376401.

Cf. also A143293 (when prepended with 0, an analogous sequence for A276086).

A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376403(n) = if(!n,0,A376403(n-1)+A276076(A376403(n-1)));

a(n) = A276075(A376399(n)).

a(0) = 0; and for n > 0, a(n) = a(n-1) + A376399(n-1) = Sum_{i=0..n-1} A376399(i).

A276074 A276076-polynomials evaluated at X=2 over the field GF(2): a(n) = A248663(A276076(n)).

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1

Offset: 0

Antti Karttunen, Aug 18 2016

Antti Karttunen, Table of n, a(n) for n = 0..5039
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A248663, A276076.

Cf. also A276073, A275808

(define (A276074 n) (A248663 (A276076 n)))

a(n) = A248663(A276076(n)).

A376400 a(0) = 1, and for n > 0, a(n) = a(n-1) * A276076(a(n-1)), where A276076 is the factorial base exp-function.

Original entry on oeis.org

1, 2, 6, 30, 1050, 70814493750, 7568077812763134673885891483463343434987134201379042046671543939118568739667281250

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(7) has 2129 (decimal) digits.
Like A376399, this satisfies A276075(a(n)) = a(n-1) + A276075(a(n-1)), for all n >= 1, so also here, applying A276075 to the terms gives the partial sums shifted right once, A376401.
However, unlike A376399, this is not a subsequence of A276078: a(5) = 70814493750 is the first term that is in A276079.

Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A276078, A276079, A376399, A376403.
Cf. A376401 (= A276075(a(n)), also gives the partial sums from its second term onward).
Cf. also analogous sequences A002110 (for A276086) and A376408 (for A019565).

A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376400(n) = if(!n,1,my(x=A376400(n-1)); x*A276076(x));

A376417 a(n) = n - A276076(A276075(n)), where A276075 and A276076 are factorial base log and exp-functions.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 22, 7, 0, 0, 0, 14, 0, 0, 0, 31, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 43, 0, 0, 0, 13, 0, 44, 0, 14, 0, 0, 0, 15, 0, 0, 0, 59, 0, 0, 0, 17, 0, 0, 0, 62, 0, 0, 0, 19, 0, 0, 0, 35, 66, 0, 0, 21, 0, 0, 0, 22, 0, 0, 0, 23, 0, 0, 0, 86, 0, 0, 0, 25

Offset: 1

Antti Karttunen, Nov 03 2024

nonn

All terms are nonnegative because for all n, x = A276076(A276075(n)) <= n, as any factor prime(i)^k || n (with k > i) will propagate carries (in the image of fully additive A276075) towards more significant digit positions, which A276076 will convert back to the exponents of larger primes, but for each new instance of such larger prime present in x, enough instances of smaller primes in n have been eliminated (by the carry process) so that the net change of magnitude is negative, unless there are no such factors present at all in n (i.e., when n is a term of A276078), then A276076(A276075(n)) = n, and a(n) = 0.
  This implies also that the least k for which A276075(k) = n is k = A276076(n).
Differs from similar A376418 for the first time at n=625, 1250, 1875, 2500, 3125, 3375, 3750, 4375, 4500, 5000, 5625, ...

			a(625) = 618, as 625 = 5^4 = prime(3)^4, thus A276075(625) = 4 * 3! = 24, but on the other hand, A276076(24) = prime(4) = 7, and 625 - 7 = 618.
a(2500) = 2479, as 2500 = 2^2 * 5^4 = prime(1)^2 * prime(3)^4, thus A276075(2500) = 2 * 1! + 4 * 3! = 26, but on the other hand, A276076(26) = prime(2)*prime(4) = 21 (as A007623(26) = 1010), and 2500 - 21 = 2479.
a(16807) = 16796, as 16807 = prime(4)^5 = 7^5, thus A276075(16807) = 5 * 4! = 120, but on the other hand, A276076(120) = prime(5) = 11, and 16807 - 11 = 16796.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to factorial base representation

Cf. A007623, A276075, A276076, A276078 (indices of 0's), A276079 (of terms > 0), A376418.

A276075(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*(primepi(f[k, 1])!)); };
A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376417(n) = (n - A276076(A276075(n)));

A351577 a(n) = A003557(A276076(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 15, 15, 25, 25, 25, 25, 75, 75, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 15, 15, 25, 25, 25, 25, 75, 75, 7, 7, 7, 7, 21, 21, 7, 7, 7, 7, 21, 21, 35, 35, 35, 35, 105, 105, 175, 175, 175, 175, 525, 525, 49, 49, 49, 49, 147, 147, 49, 49, 49, 49, 147, 147, 245

Offset: 0

Antti Karttunen, Apr 01 2022

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A003557, A085731, A276009, A276076, A328572, A351576, A351950, A351952.

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; };
A351577(n) = A003557(A276076(n));

a(n) = A003557(A276076(n)).
a(n) = A276076(A276009(n)).
a(n) = A328572(A351576(n)).
a(n) = A085731(A276076(n)) = gcd(A276076(n), A351950(n)).

A351951 a(n) = A069359(A276076(n)).

Original entry on oeis.org

0, 1, 1, 5, 3, 15, 1, 7, 8, 31, 24, 93, 5, 35, 40, 155, 120, 465, 25, 175, 200, 775, 600, 2325, 1, 9, 10, 41, 30, 123, 12, 59, 71, 247, 213, 741, 60, 295, 355, 1235, 1065, 3705, 300, 1475, 1775, 6175, 5325, 18525, 7, 63, 70, 287, 210, 861, 84, 413, 497, 1729, 1491, 5187, 420, 2065, 2485, 8645, 7455, 25935, 2100

Offset: 0

Antti Karttunen, Apr 01 2022

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A069359, A276076, A329029, A351576.

Coincides with A351950 on positions given by A059590.

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
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; };
A351951(n) = A069359(A276076(n));

a(n) = A069359(A276076(n)).

a(n) = A329029(A351576(n)).

cp's OEIS Frontend

A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

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A351950 Arithmetic derivative of the factorial base exp-function: a(n) = A003415(A276076(n)).

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A351954 Arithmetic derivative without its inherited divisor applied to the prime shadow of the factorial base exp-function: a(n) = A342001(A181819(A276076(n))).

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A276073 A276076-polynomials evaluated at x=2: a(n) = A048675(A276076(n)).

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A376403 a(0) = 0, and for n > 0, a(n) = a(n-1) + A276076(a(n-1)), where A276076 is the factorial base exp-function.

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A276074 A276076-polynomials evaluated at X=2 over the field GF(2): a(n) = A248663(A276076(n)).

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A376400 a(0) = 1, and for n > 0, a(n) = a(n-1) * A276076(a(n-1)), where A276076 is the factorial base exp-function.

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A376417 a(n) = n - A276076(A276075(n)), where A276075 and A276076 are factorial base log and exp-functions.

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A351577 a(n) = A003557(A276076(n)).