A276086 -id:A276086 - cp's OEIS frontend

A328578 Index of the least prime not dividing A276086(A276086(n)): a(n) = A257993(A276087(n)).

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

Offset: 0

Antti Karttunen, Oct 20 2019

Index of the least significant zero digit in the primorial base expansion of A276086(n), when the rightmost digit is in the position 1.

The scatter plot shows both regular looking as well as more chaotic regions. This can be more clearly seen in related A328579. See also A328839.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Rémy Sigrist, Colored logarithmic scatterplot of the ordinal transform of the sequence for n = 0..510510
Rémy Sigrist, Density plot of the sequence for n = 0..510510
Index entries for sequences related to primorial base

Cf. A000720, A055396, A276086, A276087, A328403, A328570, A328579 (the corresponding prime), A328590, A328763, A328766, A328839.
Cf. A328585 (where equal with A257993), A328587 (less than), A328588 (greater than).
Cf. A328761 (the first occurrence of each n).
Cf. also array A328631 and its rows A005408, A328632, A328633, A328634, A328635, A328636 (positions of terms 1 .. 6 in this sequence).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };
A328578(n) = A328570(A276086(n));

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A328578(n) = A257993(A276086(A276086(n)));

a(n) = A328570(A276086(n)) = A257993(A276087(n)) = A055396(A328403(n)).
a(n) = A000720(A328579(n)).
a(n) = A257993(n) + A328590(n).
a(n) = A055396(A328763(n)).
For all n >= 0, a(A328761(n)) = n.

A324644 a(n) = gcd(sigma(n), A276086(n)).

Original entry on oeis.org

1, 3, 2, 1, 6, 1, 2, 15, 1, 9, 6, 1, 2, 3, 6, 1, 18, 1, 10, 3, 2, 9, 6, 5, 1, 3, 10, 1, 30, 1, 2, 21, 6, 9, 6, 7, 2, 15, 14, 45, 42, 1, 2, 21, 6, 9, 6, 1, 1, 3, 6, 7, 18, 5, 2, 15, 10, 45, 30, 7, 2, 3, 2, 1, 42, 1, 2, 21, 6, 9, 18, 5, 2, 3, 2, 35, 6, 7, 10, 3, 1, 63, 42, 7, 2, 3, 30, 45, 90, 1, 14, 21, 2, 9, 6, 7, 98, 3, 6, 7, 6, 1, 2, 105, 6

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A000203, A009194, A276086, A324198, A324534, A324645, A324646, A364286.

Cf. A088828 (positions of even terms), A176693 (of odd terms).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324644(n) = gcd(sigma(n),A276086(n));

a(n) = gcd(A000203(n), A276086(n)).

A324580 a(n) = n * A276086(n).

Original entry on oeis.org

0, 2, 6, 18, 36, 90, 30, 70, 120, 270, 450, 990, 300, 650, 1050, 2250, 3600, 7650, 2250, 4750, 7500, 15750, 24750, 51750, 15000, 31250, 48750, 101250, 157500, 326250, 210, 434, 672, 1386, 2142, 4410, 1260, 2590, 3990, 8190, 12600, 25830, 7350, 15050, 23100, 47250, 72450, 148050, 42000, 85750, 131250, 267750, 409500

Offset: 0

Antti Karttunen, Mar 09 2019

nonn

Cf. A002110, A276086, A324198, A324577, A324578, A324579, A324582.

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; };
A324580(n) = n*A276086(n);

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

For n >= 0, a(A002110(n)) = A002110(1+n).

A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 3, 1, 2, 5, 1, 1, 4, 5, 5, 1, 2, 1, 1, 1, 10, 1, 1, 3, 12, 1, 1, 1, 2, 1, 1, 1, 4, 1, 5, 1, 2, 1, 5, 5, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 12, 3, 1, 1, 2, 1, 1, 1, 12, 1, 1, 55, 10, 3, 1, 1, 16, 1, 1, 1, 2, 1, 5, 1, 140, 1, 3, 1, 16, 1, 49, 3, 2, 1, 7, 1, 28, 1, 7, 1, 2, 1

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327860, A347958 (inverse Möbius transform), A347959, A351083, A351085, A351086, A351235, A351236.
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A324198, A327858.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));

a(n) = gcd(A003415(n), A327860(n)) = gcd(A003415(n), A003415(A276086(n))).

A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

Original entry on oeis.org

1, -1, -3, 0, -1, 5, -1, 0, 6, -3, -1, -2, -1, 1, -9, 0, -1, -16, -1, 4, 3, 1, -1, 0, -24, 1, -12, 0, -1, 43, -1, 0, 3, 1, -5, 14, -1, 1, 3, 0, -1, -11, -1, 0, 54, 1, -1, 0, -6, 32, 3, 0, -1, 44, -3, -6, 3, 1, -1, -50, -1, 1, -24, 0, 1, -5, -1, 0, 3, -15, -1, -4, -1, 1, 96, 0, -5, -5, -1, 0, 24, 1, -1, 8, -3, 1, 3, 0, -1

Offset: 1

Antti Karttunen, Jul 13 2021

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A276086, A324198, A346243.
Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).
Cf. also A354347, A354348, A354823, A354824.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
v346242 = DirInverseCorrect(vector(up_to,n,A324198(n)));
A346242(n) = v346242[n];

a(n) = A346243(n) - A324198(n).
From Antti Karttunen, Jun 09 2022: (Start)
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA324198(n/d) * a(d).
For all n >= 1, A000035(a(n)) = A008966(n).
For all n >= 1, a(A045344(n)) = -1.
(End)

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

6, 30, 32, 36, 60, 210, 212, 213, 214, 216, 240, 420, 2310, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2322, 2324, 2328, 2340, 2342, 2343, 2344, 2346, 2348, 2349, 2352, 2370, 2372, 2376, 2400, 2520, 2522, 2523, 2524, 2526, 2528, 2550, 2552, 2730, 4620, 4622, 4623, 4624, 4626, 4628, 4632, 4650, 4652, 4656

Offset: 1

Antti Karttunen, Feb 05 2022

Conjecture: Apart from the initial 6, the rest of terms are the numbers k for which A003415(k) > A276086(k), thus giving the positions of zeros in A351232. In other words, it seems that only k=6 satisfies A003415(k) = A276086(k). See also comments in A351088.

Antti Karttunen, Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to primorial base

Cf. A003415, A024451, A048103, A060735, A235991, A276085, A276086, A327862, A351088, A351089, A351232, A369656, A369663, A369664, A371104.
Union of A370127 and A370128.
Subsequence of A328118.
Subsequences: A351229, A369959, A369960, A369970 (after its two initial terms).
Cf. also A369650.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351228(n) = (A003415(n)>=A276086(n));

A328316 Iterates of A276086 starting from 0.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 18, 125, 43218, 258413198822535882125

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

The unique infinite sequence such that a(0) = 0, a(n) = A276085(a(n+1)) for n >= 0, and A129251(a(n)) = 0 for n >= 1, i.e., all nonzero terms must be in A048103.

a(10) is 240 decimal digits long (can be found in b-file), and a(11) is too big to fit even into a b-file as it is 32700 decimal digits long, but it can be found in the given a-file.

Antti Karttunen, Table of n, a(n) for n = 0..10
Antti Karttunen, Terms a(0) .. a(11), without running index
Index entries for sequences related to primorial base

Cf. A002110, A048103, A129251, A276085, A276086, A328317 (the smallest prime not dividing a(n)), A328318, A328319 (digit sum in primorial base), A328322 (max. digit), A328323.
Cf. A153013, and also A109162, A179016, A219666, A259934 for more or less analogous sequences.
Cf. also A328313.

A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
NestList[A276086, 0, 10] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));

a(0) = 0; and for n > 0, a(n) = A276086(a(n-1)).

A380459 a(n) = Product_{d|n} A276086(d)^A349394(n/d).

Original entry on oeis.org

1, 2, 2, 12, 2, 18, 2, 1296, 48, 54, 2, 1620, 2, 30, 108, 25194240, 2, 4050, 2, 131220, 60, 270, 2, 12150000, 576, 150, 3317760, 67500, 2, 33750, 2, 142818689064960000, 540, 1350, 180, 2050312500, 2, 750, 300, 1195742250000, 2, 281250, 2, 82012500, 26244000, 6750, 2, 92264062500000000, 1280, 13668750, 2700, 42187500, 2

Offset: 1

Antti Karttunen, Jan 31 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2047
Index entries for sequences related to primorial base.

Cf. A003415, A276085, A276086, A349394, A380460 (rgs-transform).

Cf. also A329350, A329380.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A349394(n) = { my(p=0, e); if((e=isprimepower(n, &p)), p^(e-1), 0); };
A380459(n) = { my(m=1); fordiv(n, d, m *= A276086(d)^A349394(n/d)); (m); };

For n >= 1, A276085(a(n)) = A003415(n).

A324583 Numbers k such that k and A276086(k) are coprime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 41, 43, 44, 46, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 71, 72, 73, 74, 76, 78, 79, 82, 83, 86, 88, 89, 90, 92, 94, 95, 96, 97, 101, 102, 103, 104, 106, 107, 108, 109, 113, 114, 116, 118, 120, 121

Offset: 1

Antti Karttunen, Mar 10 2019

nonn

Numbers k for which A324198(k) = 1.
For terms k > 0 it holds that:
  A000005(A324580(k)) = A000005(k) * A324655(k),
  A000010(A324580(k)) = A000010(k) * A324650(k),
  A000203(A324580(k)) = A000203(k) * A324653(k),
and similarly for any multiplicative function.

Cf. A324584 (complement), A356162 (characteristic function).
Cf. A276086, A324580, A324650, A324653, A324655.
Some subsequences are: A055932 ⊃ A025487 ⊃ A002182, and also A002110.
Subsequence of A356316.
Positions of 1's in A324198, positions 0's in A351254, A356302 and A356303, positions of fixed points in A351250 and in A356309.
Cf. also A355821, A356311.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324198(n) = gcd(n,A276086(n));
for(n=0,oo,if(1==A324198(n),print1(n,", ")));

Initial 0 prepended by Antti Karttunen, Nov 03 2022

A328382 a(n) = A276086(n) mod A003415(n), where A276086 is the primorial base exp-function and A003415 is the arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 3, 0, 3, 0, 9, 0, 3, 6, 1, 0, 20, 0, 15, 0, 7, 0, 9, 0, 0, 24, 25, 0, 7, 0, 21, 0, 6, 6, 35, 0, 0, 2, 43, 0, 11, 0, 45, 36, 0, 0, 91, 0, 15, 10, 35, 0, 1, 14, 61, 4, 5, 0, 49, 0, 15, 39, 57, 0, 1, 0, 15, 14, 22, 0, 133, 0, 9, 35, 65, 0, 19, 0, 71, 30, 42, 0, 121, 2, 30, 6, 105, 0, 97, 6, 69, 18, 0, 6, 83, 0, 63, 15, 35, 0, 21

Offset: 2

Antti Karttunen, Oct 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 2..30030
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327858, A358220, A358221 (positions of 0's), A358232 (of 1's), A358228 (of odd terms), A358229 (of even terms), A358227 (parity of terms).

Cf. also A328386.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328382(n) = (A276086(n)%A003415(n));

a(n) = A276086(n) mod A003415(n).

For n >= 2, gcd(a(n), A003415(n)) = A327858(n).

cp's OEIS Frontend

A328578 Index of the least prime not dividing A276086(A276086(n)): a(n) = A257993(A276087(n)).

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A324644 a(n) = gcd(sigma(n), A276086(n)).

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A324580 a(n) = n * A276086(n).

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A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

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A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

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A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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A328316 Iterates of A276086 starting from 0.

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A380459 a(n) = Product_{d|n} A276086(d)^A349394(n/d).

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A324583 Numbers k such that k and A276086(k) are coprime, where A276086 is the primorial base exp-function.

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