A143293 -id:A143293 - cp's OEIS frontend

A328836 Numbers k such that A276086(k) is a sum of distinct primorial numbers.

0, 1, 2, 3, 4, 9, 30, 39, 212, 249, 421, 2312, 2559, 30045, 32589, 510511, 512820, 543099, 1021050, 9729723, 10242789, 233335659, 446185742

Offset: 1

Antti Karttunen, Oct 29 2019

Numbers k such that A276086(k) is in A276156, i.e., numbers k for which A328828(A276086(k)) is zero, i.e., numbers k such that in the primorial base expansion of A276086(k) there are no digits larger than 1.
Numbers k for which A276087(k) is squarefree.
No more terms below 2^31.

Index entries for sequences related to primorial base

Sequence A328833 sorted into ascending order.
Cf. A008966, A276086, A276087, A276156, A328828, A328832, A328833.
Positions of zeros in A328829 and in A328844, positions of ones in A328389.
Cf. A143293 (a subsequence).
All the terms of A328313 are included in this sequence, like also in A328837.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328836(n) = !A328828(A276086(n));

A346105 a(n) = A276085(A108951(n)).

Original entry on oeis.org

0, 1, 3, 2, 9, 4, 39, 3, 6, 10, 249, 5, 2559, 40, 12, 4, 32589, 7, 543099, 11, 42, 250, 10242789, 6, 18, 2560, 9, 41, 233335659, 13, 6703028889, 5, 252, 32590, 48, 8, 207263519019, 543100, 2562, 12, 7628001653829, 43, 311878265181039, 251, 15, 10242790, 13394639596851069, 7, 78, 19, 32592, 2561, 628284422185342479, 10, 258, 42

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Additive with a(p^e) = e * A143293(A000720(p)-1), where A143293 is the partial sums of primorials, A002110. (Compare to the formula of A276085).

Cf. A002110, A108951, A143293, A276085, A346108, A346109.

Cf. also A324886.

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)); };
A346105(n) = A276085(A108951(n));

A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; }; \\ This function from A143293
A346105(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A143293(primepi(f[k, 1])-1)); };

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

A351073 Maximal exponent in the prime factorization of A276156(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 5, 1, 1, 3, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 4, 3, 1, 1, 2, 1, 2, 5, 2, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 5, 1, 1, 1, 2, 1, 3, 2, 1, 1, 6, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Feb 03 2022

A328462 Numbers obtained by reinterpreting base-2 representation of odd numbers in primorial base.

Original entry on oeis.org

1, 3, 7, 9, 31, 33, 37, 39, 211, 213, 217, 219, 241, 243, 247, 249, 2311, 2313, 2317, 2319, 2341, 2343, 2347, 2349, 2521, 2523, 2527, 2529, 2551, 2553, 2557, 2559, 30031, 30033, 30037, 30039, 30061, 30063, 30067, 30069, 30241, 30243, 30247, 30249, 30271, 30273, 30277, 30279, 32341, 32343, 32347, 32349, 32371, 32373

Offset: 1

Antti Karttunen, Oct 16 2019

Cf. A002110, A007814.
Row 1 of A328464, odd bisection of A276156 and of A328461.
Cf. A143293 (subsequence).

A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328462(n) = A276156(n+n-1);

A370132 Numbers with no digit larger than 2 in primorial base, A049345.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 240, 241, 242, 243

Offset: 1

Antti Karttunen, Feb 10 2024

Numbers k for which A328114(k) <= 2.

Numbers k such that A276086(k) is cubefree (in A004709).

Antti Karttunen, Table of n, a(n) for n = 1..13122
Index entries for sequences related to primorial base.

Cf. A004709, A049345, A276086, A328114.
Subsequence of A370133.
Subsequences: A328242, A276156 and its subsequences: A002110, A143293.

q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Count[s, ?(# > 2 &)] == 0]; Select[Range[0, 250], q] (* _Amiram Eldar, Mar 06 2024 *)

ismaxprimobasedigit_at_most(n,k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };
isA370132(n) = ismaxprimobasedigit_at_most(n,2);

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).

A380527 Numbers k such that k is a multiple of A327860(k), where A327860 is the arithmetic derivative of the primorial base exp-function.

Original entry on oeis.org

1, 2, 6, 7, 8, 30, 36, 210, 2310, 2340, 2520, 2556, 30030, 30240, 32340, 510510, 510720, 540540, 9699690, 9699720, 9702000, 9729720, 10210200, 223092870, 223092900, 223093080, 223095180, 232792560, 6469693230, 6469693236, 6469693440, 6469695540, 6692786100

Offset: 1

Antti Karttunen, Feb 11 2025

It is conjectured that only terms of A276156 occur here. If any term of A177711 is included, then it must be one of the terms of A381037.

a(34) > A143293(10).

Index entries for sequences related to primorial base.

Cf. A003415, A053669, A143293, A276086, A327860.
Subsequence of A381035. Conjectured to be a subsequence of A276156.
Subsequences: A002110, A328110.
Cf. also A177711, A351087, A381037.

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
is_A380527(n) = !(n%A327860(n));

A328395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A276087(i)) = A046523(A276087(j)) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 5, 6, 1, 7, 4, 8, 9, 10, 11, 7, 5, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 2, 5, 9, 5, 6, 8, 6, 23, 24, 1, 25, 4, 26, 27, 28, 11, 29, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 9, 9, 42, 12, 24, 12, 7, 43, 44, 45, 46, 47, 48, 27, 49, 33, 50, 35, 51, 52, 53, 54, 55, 56, 57, 58, 59, 20, 60, 61, 30, 27, 62, 63, 64, 65, 66, 15, 67, 68, 69, 8

Offset: 0

Antti Karttunen, Oct 15 2019

nonn

Restricted growth sequence transform of function f(n) = A278226(A276086(n)) = A046523(A276086(A276086(n))).
For all i, j:
  a(i) = a(j) => A328397(i) = A328397(j) => A328389(i) = A328389(j).

Antti Karttunen, Table of n, a(n) for n = 0..32589
Index entries for sequences related to primorial base

Cf. A046523, A276086, A276087, A328389.
Cf. A143293 (positions of 1's after the initial one).
Cf. also A278226, A286626, A328388, A328396, A328397.

up_to = 32589;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
v328395 = rgs_transform(vector(1+up_to, n, A046523(A276087(n-1))));
A328395(n) = v328395[1+n];

A328766 Number of nonleading zeros in primorial base expansion of A276086(n).

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 3, 1, 1, 0, 3, 1, 3, 1, 1, 0, 3, 0, 3, 1, 1, 0, 3, 0, 3, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 4, 1, 1, 0, 4, 0, 4, 1, 1, 0, 4, 0, 4, 1, 1, 0, 4, 1, 4, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 4, 1, 2, 0, 4, 0, 4, 1, 1, 0, 4, 0, 4, 2, 2, 1, 4, 0, 4, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 5, 1, 1, 0, 4

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

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

Cf. A079067, A143293, A276086, A276087, A328620, A328763.

Cf. also A328578.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328620(n) = { my(s=0, p=2); while(n, s += (0==(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328766(n) = A328620(A276086(n));

a(n) = A328620(A276086(n)) = A079067(A276087(n)).
a(n) = A001221(A328763(n)) - 1.
For all n >= 1, a(A143293(n-1)) = n. [Note however that these are not the first occurrences of each n, that is, A143293 does not give the indices of records]

A370130 a(n) = A369669(A276086(n)), where A369669 is the greatest common divisor of the first and second arithmetic derivative of n, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 16, 1, 5, 5, 5, 5, 5, 5, 100, 25, 25, 175, 25, 25, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 17, 1, 1, 5, 5, 5, 5, 20, 5, 25, 25, 25, 25, 325, 25, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 11, 1, 4, 1, 1, 1, 1, 1, 5, 5, 320, 95, 5, 5, 25, 25, 25, 25, 100, 25, 7, 7, 112, 7, 7, 7, 7, 7, 7, 7, 28

Offset: 0

Antti Karttunen, Feb 10 2024

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to primorial base

Cf. A003415, A068346, A085731, A276086, A327860, A328242 (positions of 1's), A370131.

Cf. A000012, A024451, A143293.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369669(n) = { my(d=A003415(n)); gcd(d, A003415(d)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A370130(n) = A369669(A276086(n));

a(n) = A369669(A276086(n)).
a(n) = gcd(A327860(n), A370131(n)).
For n >= 1, a(n) = A085731(A327860(n)).

cp's OEIS Frontend

A328836 Numbers k such that A276086(k) is a sum of distinct primorial numbers.

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A346105 a(n) = A276085(A108951(n)).

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A351073 Maximal exponent in the prime factorization of A276156(n).

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A328462 Numbers obtained by reinterpreting base-2 representation of odd numbers in primorial base.

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A370132 Numbers with no digit larger than 2 in primorial base, A049345.

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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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A380527 Numbers k such that k is a multiple of A327860(k), where A327860 is the arithmetic derivative of the primorial base exp-function.

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A328395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A276087(i)) = A046523(A276087(j)) for all i, j.

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A328766 Number of nonleading zeros in primorial base expansion of A276086(n).