A276156 -id:A276156 - cp's OEIS frontend

A177711 Natural numbers which are not sums of one or more distinct primorials.

4, 5, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Jonathan Vos Post, May 11 2010

Numbers with a digit larger than one in primorial base representation, A049345. Numbers k for which A276086(k) is not squarefree. - Antti Karttunen, Feb 17 2025

			1 and 2 are not in the sequence, as they are the first and second primorials, 0# and 1#. 3 is not in the sequence, as 3 = 1+2. Neither 4 nor 5 can be the sum of distinct primorials (i.e. 4=2+2 or 5 = 2+2+1 repeat a primorial). 6 is not in the sequence, as it is 3#. 7 and 8 are not in the sequence as 7 = 6+1 and 8 = 6+2. 9 is not in the sequence, as 9 = 6+2+1.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A002110, A049345, A177689, A177697, A177708, A177709, A276086.
Complement of A276156.
Positions of terms > 1 in A328114.
Subsequences: A380535, A381034.

is_A177711(n) = { my(p=2); while(n, if(n%p > 1, return(1)); n = n\p; p = nextprime(1+p)); (0); }; \\ Antti Karttunen, Feb 17 2025

COMPLEMENT of {Primorial numbers A002110 UNION A177689 Sums of 2 distinct primorials UNION Sums of 3 distinct primorials A177697 UNION Sums of 4 distinct primorials A177709 UNION ...}.

{k such that A328114(k) > 1}. - Antti Karttunen, Feb 17 2025

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

Original entry on oeis.org

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));

A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 40, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 115, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 228, 229, 233, 239, 241, 251, 252, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Antti Karttunen, May 26 2021

nonn

Numbers k such that A108951(k) is in A276156.

			A108951(40) = 240 and 240 is in A276156 because 240 = A002110(4) + A002110(3) = 210 + 30, therefore 40 is included in this sequence.

Positions of ones in A329344, in A344592 and in A344593.
Positions of squarefree terms in A324886.
Union of A008578 and A351959.
Cf. A002110, A108951, A276156, A351957 (characteristic function).
Cf. also A351958.

Name changed by Antti Karttunen, Apr 04 2022

A283985 Sums of distinct terms of A143293: a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 39, 40, 42, 43, 48, 49, 51, 52, 249, 250, 252, 253, 258, 259, 261, 262, 288, 289, 291, 292, 297, 298, 300, 301, 2559, 2560, 2562, 2563, 2568, 2569, 2571, 2572, 2598, 2599, 2601, 2602, 2607, 2608, 2610, 2611, 2808, 2809, 2811, 2812, 2817, 2818, 2820, 2821, 2847, 2848, 2850, 2851, 2856, 2857, 2859, 2860, 32589

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A002110, A030308, A143293, A276085, A276156, A283477, A283984.

A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; }; \\ This function from Charles R Greathouse IV, Feb 05 2014
A030308(n,k) = bittest(n,k);
A283985(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A143293(i));

from sympy import primorial, primepi, prime, primerange, factorint
from operator import mul
from functools import reduce
def a002110(n): return 1 if n<1 else primorial(n)
def a276085(n):
    f=factorint(n)
    return sum([f[i]*a002110(primepi(i) - 1) for i in f])
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a276085(a108951(a019565(n)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

(define (A283985 n) (A276085 (A283477 n)))

a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).
a(n) = A276085(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A143293(n).

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);

A328581 Product of nonzero digits in primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 3, 3, 3, 3, 6, 6, 4, 4, 4, 4, 8, 8, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 3, 3, 3, 3, 6, 6, 4, 4, 4, 4, 8, 8, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 6, 6, 6, 6, 12, 12, 8, 8, 8, 8, 16, 16, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Oct 21 2019

a(0) = 1 as an empty product.

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

Cf. A005361, A049345, A276086, A324655, A328582.
Cf. A276156 (positions of 1's).
Cf. also A227153 (an analogous sequence).

a[n_] := Module[{k = n, p = 2, s = 1, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, If[r > 0, s *= r]; p = NextPrime[p]]; s]; Array[a, 100, 0] (* Amiram Eldar, Mar 06 2024 *)

A328581(n) = { my(m=1, p=2); while(n, if(n%p, m *= (n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A005361(A276086(n)).

A328582 Least common multiple of nonzero digits in primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 6, 6, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Oct 21 2019

nonn

a(0) = 1 as an empty product.

Cf. A072411, A276086, A328581.

Cf. A276156 (positions of 1's).

A328582(n) = { my(m=1, p=2); while(n, if(n%p, m = lcm(m,n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A072411(A276086(n)).

A369642 Composite numbers k, not squarefree semiprimes, such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

9, 16, 28, 30, 45, 108, 112, 136, 189, 198, 210, 212, 225, 236, 244, 246, 282, 290, 361, 374, 399, 435, 507, 1480, 1940, 2132, 2212, 2308, 2356, 2524, 2655, 2766, 2802, 3018, 3054, 3501, 3590, 3771, 3938, 4225, 4454, 4755, 4809, 5005, 5763, 6123, 6771, 9024, 9936, 10295, 11881, 12221, 16296, 22491, 24389, 26865

Offset: 1

Antti Karttunen, Jan 31 2024

nonn

Composite numbers k, not squarefree semiprimes, such that A327859(k) = A276086(A003415(k)) is squarefree number, or equally, k' is in A276156.

Squares that appear in this sequence: 9, 16, 225, 361, 4225, 11881, 1371241, 1635841, 225930961, 228644641, 229189321, 262083721, ...

Antti Karttunen, Table of n, a(n) for n = 1..2450
Index entries for sequences related to primorial base

Sequence A369641 without any terms of A006881.
Cf. A003415, A276086, A276156, A327859, A369647 (subsequence after its two initial terms).
Nonsquarefree terms all occur in A369639.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
A369640(n) = if(n<2 || isprime(n), 0, ismaxprimobasedigit_at_most(A003415(n),1));
isA369642(n) = (((bigomega(n)>2)||(bigomega(n)>omega(n))) && A369640(n));

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);

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));

cp's OEIS Frontend

A177711 Natural numbers which are not sums of one or more distinct primorials.

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

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A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

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A283985 Sums of distinct terms of A143293: a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

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

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A328581 Product of nonzero digits in primorial base expansion of n.

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A328582 Least common multiple of nonzero digits in primorial base expansion of n.

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A369642 Composite numbers k, not squarefree semiprimes, such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

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