A177711 -id:A177711 - cp's OEIS frontend

A276156 Numbers obtained by reinterpreting base-2 representation of n in primorial base: a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1 + A276154(a(n)).

0, 1, 2, 3, 6, 7, 8, 9, 30, 31, 32, 33, 36, 37, 38, 39, 210, 211, 212, 213, 216, 217, 218, 219, 240, 241, 242, 243, 246, 247, 248, 249, 2310, 2311, 2312, 2313, 2316, 2317, 2318, 2319, 2340, 2341, 2342, 2343, 2346, 2347, 2348, 2349, 2520, 2521, 2522, 2523, 2526, 2527, 2528, 2529, 2550, 2551, 2552, 2553, 2556, 2557, 2558, 2559, 30030, 30031

Offset: 0

Antti Karttunen, Aug 24 2016

Numbers that are sums of distinct primorial numbers, A002110.

Numbers with no digits larger than one in primorial base, A049345.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to primorial base

Cf. A000040, A001511, A002110, A007088, A007814, A019565, A049345, A257993, A276084, A276085, A276154, A351073, A328461, A328473, A328474, A328571, A328831, A328836, A341518, A344591.
Complement of A177711.
Subsequences: A328233, A328832, A328462 (odd bisection).
Conjectured subsequences: A328110, A380527.
Fixed points of A328841, positions of zeros in A328828, A328842, and A329032, positions of ones in A328581, A328582, and A381032.
Positions of terms < 2 in A328114.
Indices where A327860 and A329029 coincide.
Cf. also table A328464 (and its rows).
Cf. also A059590, A283985, A290249, A342921.

nn = 65; b = MixedRadix[Reverse@ Prime@ Range[IntegerLength[nn, 2] - 1]]; Table[FromDigits[IntegerDigits[n, 2], b], {n, 0, 65}] (* Version 10.2, or *)
Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ IntegerDigits[n, 2], {n, 0, 65}] (* Michael De Vlieger, Aug 26 2016 *)

A276156(n) = { my(s=0, p=1, r=1); while(n, if(n%2, s += r); n>>=1; p = nextprime(1+p); r *= p); (s); }; \\ Antti Karttunen, Feb 03 2022

from sympy import prime, primorial, primepi, factorint
from operator import mul
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 a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) # after Chai Wah Wu
def a(n): return 0 if n==0 else a276085(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 23 2017

a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1+A276154(a(n)).
Other identities. For all n >= 0:
a(n) = A276085(A019565(n)).
A049345(a(n)) = A007088(n).
A257993(a(n)) = A001511(n).
A276084(a(n)) = A007814(n).
A051903(a(n)) = A351073(n).

A381034 Numbers that have a digit > 1 in their primorial base expansion, and that are multiples of the corresponding radix prime of the least significant such digit.

Original entry on oeis.org

15, 20, 25, 45, 50, 55, 63, 75, 80, 85, 91, 98, 105, 110, 115, 126, 135, 140, 145, 165, 170, 175, 182, 189, 195, 200, 205, 225, 230, 235, 255, 260, 265, 273, 285, 290, 295, 301, 308, 315, 320, 325, 336, 345, 350, 355, 375, 380, 385, 392, 399, 405, 410, 415, 429, 435, 440, 445, 451, 465, 470, 475, 483, 495, 500, 505

Offset: 1

Antti Karttunen, Feb 17 2025

nonn

			A049345(15) = 211, the least significant digit larger than one is 2, where the corresponding radix prime is 5, and 5 divides 15, therefore 15 is included.

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

Cf. A049345, A381032, A381033 (characteristic function), A381037 (subsequence).

Subsequence of the intersection A177711 and A381035.

PARI
```
is_A381034 = A381033;
```

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));

A381037 Numbers with a digit > 1 in their primorial base expansion that are multiples of the corresponding radix primes of all such digits.

Original entry on oeis.org

15, 20, 25, 45, 50, 55, 63, 91, 98, 105, 126, 140, 175, 182, 189, 225, 230, 235, 255, 260, 265, 273, 301, 308, 315, 336, 350, 385, 392, 399, 429, 440, 451, 638, 660, 693, 770, 847, 1056, 1089, 1100, 1155, 1232, 1298, 1386, 1485, 1507, 1683, 1705, 1716, 1771, 1892, 2079, 2101, 2145, 2325, 2330, 2335, 2355, 2360, 2365

Offset: 1

Antti Karttunen, Feb 17 2025

nonn

Numbers k such that A328572(k) > 1 and A007947(A328572(k)) divides k.

			For n=75, A049345(75) = 2211, with two digits > 1. Their corresponding radix primes are 5 and 7, but as 75 is a multiple of only 5 but not of 7, 75 is NOT included in this sequence.
For n = 105, A049345(105) = 3211, with two digits > 1. Their corresponding radix primes are 5 and 7 that both divide 105, therefore 105 is included.

Antti Karttunen, Table of n, a(n) for n = 1..12121
Index entries for sequences related to primorial base.

Subsequence of A381034, thus subsequence of the intersection A177711 and A381035.
Cf. A003557, A007947, A049345, A276086, A328572, A381036 (characteristic function).
Cf. also A380527.

PARI
```
is_A381037 = A381036;
```

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)).

cp's OEIS Frontend

A276156 Numbers obtained by reinterpreting base-2 representation of n in primorial base: a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1 + A276154(a(n)).

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A381034 Numbers that have a digit > 1 in their primorial base expansion, and that are multiples of the corresponding radix prime of the least significant such digit.

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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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A381037 Numbers with a digit > 1 in their primorial base expansion that are multiples of the corresponding radix primes of all such digits.

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

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