A143293 -id:A143293 - cp's OEIS frontend

A376401 a(n) = A276075(A376400(n)); Partial sums of A376400.

0, 1, 3, 9, 39, 1089, 70814494839, 7568077812763134673885891483463343434987134201379042046671543939118568810481776089

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(8) has 2129 (decimal) digits.
From the second term onward also the partial sums of A376400.
By induction, it is easy to see that formula a(n) = A276075(A376400(n)) implies that from the second term onward, this sequence gives the partial sums of A376400, as A276075 is fully additive.

Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A376400, A376403.

Cf. also A143293 (when prepended with 0, an analogous sequence for A276085).

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); };
A376400(n) = if(!n,1,my(x=A376400(n-1)); x*A276076(x));
A376401(n) = A276075(A376400(n));
\\ Or alternatively as:
A376401(n) = if(!n,0,A376401(n-1)+A376400(n-1));

a(0) = 0; and for n >= 1, a(n) = a(n-1) + A376400(n-1) = Sum_{i=0..n-1} A376400(i).

A290249 Numbers that are the sum of distinct primorial numbers (A002110) (not including 1).

Original entry on oeis.org

2, 6, 8, 30, 32, 36, 38, 210, 212, 216, 218, 240, 242, 246, 248, 2310, 2312, 2316, 2318, 2340, 2342, 2346, 2348, 2520, 2522, 2526, 2528, 2550, 2552, 2556, 2558, 30030, 30032, 30036, 30038, 30060, 30062, 30066, 30068, 30240, 30242, 30246, 30248, 30270, 30272, 30276, 30278, 32340, 32342, 32346, 32348, 32370

Offset: 1

Ilya Gutkovskiy, Jul 24 2017

nonn

			38 is in the sequence because 38 = 2 + 6 + 30 = 2 + 2*3 + 2*3*5.

Index entries for sequences related to primorial numbers

Cf. A002110, A059589, A059590, A143293, A177711, A276156.

Rest[f[x_] := Product[1 + x^Product[Prime[m], {m, 1, k}], {k, 1, 6}]; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, 32370}]]]

Nonzero exponents in expansion of Product_{k>=1} (1 + x^A002110(k)).

A328399 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328475(i) = A328475(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 4, 5, 1, 6, 2, 7, 7, 8, 4, 9, 5, 10, 10, 11, 7, 12, 8, 13, 13, 14, 10, 15, 11, 16, 16, 17, 16, 18, 17, 19, 19, 20, 1, 21, 2, 22, 22, 23, 4, 24, 5, 25, 25, 26, 7, 27, 8, 28, 28, 29, 10, 30, 11, 31, 31, 32, 31, 33, 32, 34, 34, 35, 16, 36, 17, 37, 37, 38, 19, 39, 20, 40, 40, 41, 22, 42, 23, 43, 43, 44, 25, 45, 26, 46, 46, 47, 46, 48, 47

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Restricted growth sequence transform of A328475, defined as A328475(n) = A111701(A276086(n)).

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A002110, A053589, A111701, A276086, A143293 (indices of 1's after a(0)=1).

Cf. also A328477.

up_to = 32768;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328475(n) = A111701(A276086(n));
v328399 = rgs_transform(vector(up_to+1, n, A328475(n-1)));
A328399(n) = v328399[1+n];

A343047 a(n) = A343046(n, n).

Original entry on oeis.org

0, 1, 6, 9, 12, 15, 210, 217, 246, 249, 252, 255, 420, 427, 456, 459, 492, 495, 630, 637, 666, 669, 702, 705, 840, 847, 876, 879, 912, 915, 30030, 30061, 30246, 30279, 30252, 30285, 32550, 32587, 32586, 32589, 32592, 32595, 32760, 32797, 32796, 32799, 32832

Offset: 0

Rémy Sigrist, Apr 05 2021

This sequence has similarities with A087019 and A343043.

			For n = 2:
- the primorial base representation of 2 is "10", so:
         1 0
       x 1 0
       -----
         0 0
     + 1 0
     -------
       1 0 0
- hence a(2) = 2*3 = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A343047
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to primorial base

Cf. A002110, A143293, A343046.

Cf. A087019, A343043.

PARI
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See Links section.
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a(A002110(k)) = A002110(2*k) for any k >= 0.

a(A143293(k)) = A143293(2*k) for any k >= 0.

A343404 For any number n with representation (d_w, ..., d_1) in primorial base, a(n) is the least number m such that m mod prime(k) = d_k for k = 1..w (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

0, 1, 4, 1, 2, 5, 6, 21, 16, 1, 26, 11, 12, 27, 22, 7, 2, 17, 18, 3, 28, 13, 8, 23, 24, 9, 4, 19, 14, 29, 120, 15, 190, 85, 50, 155, 36, 141, 106, 1, 176, 71, 162, 57, 22, 127, 92, 197, 78, 183, 148, 43, 8, 113, 204, 99, 64, 169, 134, 29, 30, 135, 100, 205

Offset: 0

Rémy Sigrist, Apr 14 2021

Leading zeros in primorial base expansions are ignored.

The Chinese remainder theorem ensures that this sequence is well defined and provides a way to compute it.

			For n = 42 :
- the expansion of 42 in primary base is "1200",
- so a(42) mod 2 = 0 => a(42) = 2*t for some t >= 0,
     a(42) mod 3 = 0 => a(42) = 6*u for some u >= 0,
     a(42) mod 5 = 2 => a(42) = 12 + 30*v for some v >= 0,
     a(42) mod 7 = 1 => a(42) = 162 + 210*w for some w >= 0,
- we choose w=0 so as to minimize the value,
- hence a(42) = 162.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Chinese remainder theorem
Index entries for sequences related to primorial base

Cf. A002110, A079276, A143293, A235168, A343405 (fixed points).

a(n) = { my (v=Mod(0,1)); forprime (p=2, oo, if (n==0, return (lift(v)), v=chinese(v, Mod(n, p)); n\=p)) }

a(n) = 1 iff n belongs to A143293.
a(n) = n iff n belongs to A343405.
a(n) < A002110(k) for any n < A002110(k) and k >= 0.
a(A002110(k)) = A079276(k+1) * A002110(k) for any k >= 0.

A369243 a(n) is the least integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.

Original entry on oeis.org

2, 0, 14, 45, 198, 5114, 65174, 1086194, 20485574, 465779078, 0, 293420849770, 318745032938881

Offset: 0

Antti Karttunen, Jan 19 2024

a(n) = the smallest integer k for which A003415(k) = A143293(n), and 0 if no such k exists.

			a(0) = 2 as the least number k such that A003415(k) = A143293(0) = 1 is 2.
a(1) = 0 as there is no number k such that A003415(k) = A143293(1) = 3.

Antti Karttunen, PARI program for computing terms of this and related sequences.

Cf. A003415, A143293, A369244.

Cf also A368703.

a(n) <= A369244(n).

A369244 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.

Original entry on oeis.org

0, 14, 74, 198, 10295, 65174, 40354813, 20485574, 680909375411, 0, 17866904665985941, 318745032938881

Offset: 1

Antti Karttunen, Jan 19 2024

a(n) = the largest integer k for which A003415(k) = A143293(n), and 0 if no such k exists.

Antti Karttunen, PARI program for computing terms of this and related sequences.

Cf. A003415, A143293, A369243.

Cf. also A368704.

a(n) >= A369243(n).

A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

Original entry on oeis.org

2, 9, 26, 122, 1382, 21446, 204566, 9699686, 90387605

Offset: 1

Antti Karttunen, Jun 22 2024

For n > 1, the index of the next term in A373849, after its sixth term 0, that is a multiple of A002110(n), as for n >= 1, the smallest k such that A003415(k) == A276086(k) mod A002110(n) gives the sequence 1, 6, 6, 6, 6, 6, 6, 6, ..., because A003415(6) = A276086(6).
Provided that such k exists for every n (and the escape clause is not needed), then the sequence is by necessity monotonic. If it is strictly monotonic, then it implies that k=6 is the only k such that A003415(k) = A276086(k). See also comments in A351228.
Note that if we instead search for the smallest k such that A276086(k) is a multiple of A002110(n) we obtain A143293, partial sums of the primorial numbers. See also A368703.

Cf. A002110, A003415, A143293, A276086, A373849.

Cf. also A351228, A368703, A371104, A373848.

A002110(n) = prod(i=1,n,prime(i));
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); };
A373603(n) = { my(m=A002110(n), c=2); for(i=1,oo,if(0==((A276086(i)-A003415(i))%m), c--; if(0==c, return(i)))); };

A376398 a(n) = A276085(n!), where A276085 is the primorial base log-function.

Original entry on oeis.org

0, 1, 3, 5, 11, 14, 44, 47, 51, 58, 268, 272, 2582, 2613, 2621, 2625, 32655, 32660, 543170, 543178, 543210, 543421, 10243111, 10243116, 10243128, 10245439, 10245445, 10245477, 233338347, 233338356, 6703031586, 6703031591, 6703031803, 6703061834, 6703061870, 6703061876, 207263552006, 207264062517, 207264064829, 207264064838

Offset: 1

Antti Karttunen, Oct 28 2024

nonn

Partial sums of A276085.
Cf. A000142, A002110.
Cf. also A143293, A336415, A376397.

A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
A376398(n) = A276085(n!);

a(n) = A276085(A000142(n)).

a(n) = Sum_{i=1..n} A276085(i).

A341433 Numbers that are divisible by the product of their digits in primorial base representation.

Original entry on oeis.org

1, 3, 9, 21, 39, 51, 99, 249, 261, 309, 669, 729, 2559, 2571, 2619, 2979, 3051, 4239, 7179, 7191, 32589, 32601, 32649, 32661, 33009, 33021, 37209, 37269, 37629, 51489, 92649, 92709, 93069, 97281, 272889, 274509, 543099, 543111, 543159, 543519, 543591, 544779

Offset: 1

Amiram Eldar, Feb 11 2021

The primorial base repunits (A143293) are all terms since their product of digits in primorial base is 1.

All the terms are zeroless in primorial base, and therefore they are terms of A328574. In particular, since the last digit of even numbers in primorial base is 0, all the terms are odd numbers.

			9 is a term since 9 in primorial base is 111 (9 = 3! + 2! + 1!) and 9 is divisible by 1*1*1 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..150
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

A143293 is a subsequence.
Subsequence of A328574.
Cf. A007602, A049345, A235168, A286590, A333426.

max = 12; bases = Prime@Range[max, 1, -1]; nmax = Times @@ bases - 1; q[n_] := FreeQ[(d = IntegerDigits[n, MixedRadix[bases]]), 0] && Divisible[n, Times @@ d]; Select[Range[1, 10^5, 2], q]

cp's OEIS Frontend

A376401 a(n) = A276075(A376400(n)); Partial sums of A376400.

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A290249 Numbers that are the sum of distinct primorial numbers (A002110) (not including 1).

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A328399 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328475(i) = A328475(j) for all i, j.

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A343047 a(n) = A343046(n, n).

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A343404 For any number n with representation (d_w, ..., d_1) in primorial base, a(n) is the least number m such that m mod prime(k) = d_k for k = 1..w (where prime(k) denotes the k-th prime number).

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A369243 a(n) is the least integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.

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A369244 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.

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A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

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A376398 a(n) = A276085(n!), where A276085 is the primorial base log-function.

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