A328462 -id:A328462 - 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).

A328461 a(n) = A276156(n) / A002110(A007814(n)).

Original entry on oeis.org

1, 1, 3, 1, 7, 4, 9, 1, 31, 16, 33, 6, 37, 19, 39, 1, 211, 106, 213, 36, 217, 109, 219, 8, 241, 121, 243, 41, 247, 124, 249, 1, 2311, 1156, 2313, 386, 2317, 1159, 2319, 78, 2341, 1171, 2343, 391, 2347, 1174, 2349, 12, 2521, 1261, 2523, 421, 2527, 1264, 2529, 85, 2551, 1276, 2553, 426, 2557, 1279, 2559, 1, 30031, 15016, 30033, 5006

Offset: 1

Antti Karttunen, Oct 16 2019

nonn

A276156(n) converts the binary expansion of n to a number whose primorial base representation has the same digits of 0's and 1's, thus each one of its terms is a unique sum of distinct primorial numbers. In this sequence that sum is then divided by the largest primorial that divides it, which only depends on the position of the least significant 1-bit in the binary expansion of the original n, that is, the 2-adic valuation of n.

Cf. A000265, A002110, A007814, A111701, A276154, A276156.
Cf. A328462 (bisection, also row 1 of array A328464 which shows the same information in tabular form).
Cf. A328471, A328472, A328473, A328474.

A002110(n) = prod(i=1,n,prime(i));
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); };
A328461(n) = (A276156(n)/A002110(valuation(n,2)));

a(n) = A276156(n) / A002110(A007814(n)).

a(n) = A111701(A276156(n)).

A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 9, 16, 6, 1, 31, 19, 36, 8, 1, 33, 106, 41, 78, 12, 1, 37, 109, 386, 85, 144, 14, 1, 39, 121, 391, 1002, 155, 222, 18, 1, 211, 124, 421, 1009, 2432, 235, 324, 20, 1, 213, 1156, 426, 1079, 2443, 4200, 341, 438, 24, 1, 217, 1159, 5006, 1086, 2575, 4213, 7430, 457, 668, 30, 1, 219, 1171, 5011, 17018, 2586, 4421, 7447, 12674, 691, 900, 32, 1

Offset: 1

Antti Karttunen, Oct 16 2019

Array is read by falling antidiagonals with n (row) and k (column) ranging as: (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Row n contains all such sums of distinct primorials whose least significant summand is A002110(n-1), with each sum divided by that least significant primorial, which is also the largest primorial which divides that sum.

			Top left 9 X 11 corner of the array:
1: | 1,  3,   7,   9,    31,    33,    37,    39,    211,    213,    217
2: | 1,  4,  16,  19,   106,   109,   121,   124,   1156,   1159,   1171
3: | 1,  6,  36,  41,   386,   391,   421,   426,   5006,   5011,   5041
4: | 1,  8,  78,  85,  1002,  1009,  1079,  1086,  17018,  17025,  17095
5: | 1, 12, 144, 155,  2432,  2443,  2575,  2586,  46190,  46201,  46333
6: | 1, 14, 222, 235,  4200,  4213,  4421,  4434,  96578,  96591,  96799
7: | 1, 18, 324, 341,  7430,  7447,  7753,  7770, 215442, 215459, 215765
8: | 1, 20, 438, 457, 12674, 12693, 13111, 13130, 392864, 392883, 393301
9: | 1, 24, 668, 691, 20678, 20701, 21345, 21368, 765050, 765073, 765717

Cf. A328463 (transpose).
Cf. A000265, A002110, A007814, A135764, A276154, A276156,
Rows 1 - 5: A328462, A328465, A328466, A328467, A328468.
Column 2: A008864.
Column 3: A023523 (after its initial term).
Column 4: A286624.
Cf. also arrays A276945, A286625.

up_to = 105;
A002110(n) = prod(i=1,n,prime(i));
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); };
A328464sq(n,k) = (A276156((2^(n-1)) * (k+k-1)) / A002110(n-1));
A328464list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A328464sq(col,(a-(col-1))))); (v); };
v328464 = A328464list(up_to);
A328464(n) = v328464[n];

A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1).

a(n) = A328461(A135764(n)). [When all sequences are considered as one-dimensional]

A328465 Row 2 of A328464: a(n) = A276156(4n - 2) / 2.

Original entry on oeis.org

1, 4, 16, 19, 106, 109, 121, 124, 1156, 1159, 1171, 1174, 1261, 1264, 1276, 1279, 15016, 15019, 15031, 15034, 15121, 15124, 15136, 15139, 16171, 16174, 16186, 16189, 16276, 16279, 16291, 16294, 255256, 255259, 255271, 255274, 255361, 255364, 255376, 255379, 256411, 256414, 256426, 256429, 256516, 256519, 256531

Offset: 1

Antti Karttunen, Oct 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to primorial base

Row 2 of A328464.

Cf. A002110, A276156, A328461, A328462, A328466, A328467, A328468.

A002110(n) = prod(i=1,n,prime(i));
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); };
A328465(n) = (A276156((4*n)-2) / 2);

a(n) = (1/2) * A276156(4*n - 2).

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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A328461 a(n) = A276156(n) / A002110(A007814(n)).

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A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

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A328465 Row 2 of A328464: a(n) = A276156(4n - 2) / 2.

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