A328464 -id:A328464 - cp's OEIS frontend

A328466 Row 3 of A328464: a(n) = A276156(8n - 4) / 6.

1, 6, 36, 41, 386, 391, 421, 426, 5006, 5011, 5041, 5046, 5391, 5396, 5426, 5431, 85086, 85091, 85121, 85126, 85471, 85476, 85506, 85511, 90091, 90096, 90126, 90131, 90476, 90481, 90511, 90516, 1616616, 1616621, 1616651, 1616656, 1617001, 1617006, 1617036, 1617041, 1621621, 1621626, 1621656, 1621661, 1622006, 1622011

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 3 of A328464.

Cf. A002110, A276156, A328461.

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); };
A328466(n) = (A276156((8*n)-4) / 6);

a(n) = (1/6) * A276156(8*n - 4).

A328467 Row 4 of A328464: a(n) = A276156(16n - 8) / 30.

Original entry on oeis.org

1, 8, 78, 85, 1002, 1009, 1079, 1086, 17018, 17025, 17095, 17102, 18019, 18026, 18096, 18103, 323324, 323331, 323401, 323408, 324325, 324332, 324402, 324409, 340341, 340348, 340418, 340425, 341342, 341349, 341419, 341426, 7436430, 7436437, 7436507, 7436514, 7437431, 7437438, 7437508, 7437515, 7453447, 7453454, 7453524

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 4 of A328464.

Cf. A002110, A276156, A328461.

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); };
A328467(n) = (A276156((16*n)-8) / 30);

a(n) = (1/30) * A276156(16*n - 8).

A328468 Row 5 of A328464: a(n) = A276156(32n - 16) / 210.

Original entry on oeis.org

1, 12, 144, 155, 2432, 2443, 2575, 2586, 46190, 46201, 46333, 46344, 48621, 48632, 48764, 48775, 1062348, 1062359, 1062491, 1062502, 1064779, 1064790, 1064922, 1064933, 1108537, 1108548, 1108680, 1108691, 1110968, 1110979, 1111111, 1111122, 30808064, 30808075, 30808207, 30808218, 30810495, 30810506, 30810638, 30810649

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 5 of A328464.

Cf. A002110, A276156, A328461.

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); };
A328468(n) = (A276156((32*n)-16) / 210);

a(n) = (1/210) * A276156(32*n - 16).

A328463 Transpose of square array A328464.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 16 2019

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 antidiagonals of array

Cf. A328464.

\\ Needs also code from A328464:
A328463sq(n,k) = A328464sq(k,n);
A328463list(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] = A328463sq(col,(a-(col-1))))); (v); };
v328463 = A328463list(up_to);
A328463(n) = v328463[n];

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).

A008864 a(n) = prime(n) + 1.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

N. J. A. Sloane, R. K. Guy

Sum of divisors of prime(n). - Labos Elemer, May 24 2001
For n > 1, there are a(n) more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions, which are counted in A239396 and A239394, respectively. - T. D. Noe, Mar 31 2014
These are the numbers which are in A239708 or in A187813, but excluding the first 3 terms of A187813, i.e., a number m is a term if and only if m is a term > 2 of A187813, or m is the sum of two distinct powers of 2 such that m - 1 is prime. This means that a number m is a term if and only if m is a term > 2 such that there is no base b with a base-b digital sum of b, or b = 2 is the only base for which the base-b digital sum of m is b. a(6) is the only term such that a(n) = A187813(n); for n < 6, we have a(n) > A187813(n), and for n > 6, we have a(n) < A187813(n). - Hieronymus Fischer, Apr 10 2014
Does not contain any number of the format 1 + q + ... + q^e, q prime, e >= 2, i.e., no terms of A060800,  A131991, A131992, A131993 etc. [Proof: that requires 1 + p = 1 + q + ... + q^e, or p = q*(1 + ... + q^(e-1)). This is not solvable because the left hand side is prime, the right hand side composite.] - R. J. Mathar, Mar 15 2018
1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is odd. - Amiram Eldar, Jan 23 2021

C. W. Trigg, Problem #1210, Series Formation, J. Rec. Math., 15 (1982), 221-222.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021).

Cf. A000040, A060800, A131991, A131992, A131993, A141468.
Cf. A007953, A079696, A187813, A239703, A239708.
Column 1 of A341605, column 2 of A286623 and of A328464.
Partial sums of A125266.

a008864 = (+ 1) . a000040
-- Reinhard Zumkeller, Sep 04 2012, Oct 08 2012

[NthPrime(n)+1: n in [1..70]]; // Vincenzo Librandi, Jul 30 2016

A008864:=n->ithprime(n)+1; seq(A008864(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2014

Prime[Range[70]]+1 (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

forprime(p=2,1e3,print1(p+1", ")) \\ Charles R Greathouse IV, Jun 16 2011

A008864(n) = (1+prime(n)); \\ Antti Karttunen, Mar 14 2021

[nth_prime(n) +1 for n in (1..70)] # G. C. Greubel, May 20 2019

a(n) = prime(n) + 1 = A000040(n) + 1.
a(n) = A000005(A034785(n)) = A000203(A000040(n)). - Labos Elemer, May 24 2001
a(n) = A084920(n) / A006093(n). - Reinhard Zumkeller, Aug 06 2007
A239703(a(n)) <= 1. - Hieronymus Fischer, Apr 10 2014
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) ~ n*log(n).
Product_{n>=1} (1 + 2/(a(n)*(a(n) - 2))) = 5/2. (End)

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)).

Original entry on oeis.org

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).

A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 30, 32, 36, 38, 42, 44, 60, 62, 66, 68, 72, 74, 90, 92, 96, 98, 102, 104, 120, 122, 126, 128, 132, 134, 210, 212, 216, 218, 222, 224, 240, 242, 246, 248, 252, 254, 270, 272, 276, 278, 282, 284, 300, 302, 306, 308, 312, 314, 330, 332, 336, 338, 342, 344, 420, 422, 426, 428, 432, 434, 450, 452, 456, 458, 462, 464, 480, 482, 486, 488

Offset: 0

Antti Karttunen, Aug 24 2016

			   n   A049345  with one zero           converted back
                appended to the right   to decimal = a(n)
---------------------------------------------------------
   0       0            00                     0
   1       1            10                     2
   2      10           100                     6
   3      11           110                     8
   4      20           200                    12
   5      21           210                    14
   6     100          1000                    30
   7     101          1010                    32
   8     110          1100                    36
   9     111          1110                    38
  10     120          1200                    42
  11     121          1210                    44
  12     200          2000                    60
  13     201          2010                    62
  14     210          2100                    66
  15     211          2110                    68
  16     220          2200                    72

Antti Karttunen, Table of n, a(n) for n = 0..30029
Index entries for sequences related to primorial base

Complement: A276155.
Cf. A002110, A003961, A049345, A276085, A276086, A276151, A276152, A286629 [= a(A061720(n-1))], A324384 [= gcd(n, a(n))], A323879, A328770 (a subsequence).
Cf. also A276156, A328461, A328464.
Dispersion array and its transpose: A276943, A276945, with primorials divided out: A286623, A286625.
Analogous to A153880.

nn = 75; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[#] <= nn &]]; Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, 75}] (* Michael De Vlieger, Aug 26 2016 *)

A276154(n) = A276085(A003961(A276086(n))); \\ Antti Karttunen, Mar 15 2021

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n)))); \\ Antti Karttunen, Mar 15 2021

(definec (A276154 n) (if (zero? n) n (+ (A276152 n) (A276154 (A276151 n)))))

a(0) = 0; for n >= 1, a(n) = A276152(n) + a(A276151(n)).

a(n) = A276085(A003961(A276086(n))). - Antti Karttunen, Mar 15 2021

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)).

A023523 a(n) = prime(n)*prime(n-1) + 1.

Original entry on oeis.org

3, 7, 16, 36, 78, 144, 222, 324, 438, 668, 900, 1148, 1518, 1764, 2022, 2492, 3128, 3600, 4088, 4758, 5184, 5768, 6558, 7388, 8634, 9798, 10404, 11022, 11664, 12318, 14352, 16638, 17948, 19044, 20712, 22500, 23708, 25592, 27222, 28892

Offset: 1

Clark Kimberling

nonn

This sequence assumes prime(0) = 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A008578.

From a(2) = 7 onward, column 5 of A286623, column 3 of A328464.

a023523 n = a023523_list !! (n-1)
a023523_list =  map (+ 1) $ zipWith (*) a000040_list a008578_list
-- Reinhard Zumkeller, Oct 09 2012

[&+[(NthPrime(n-1)*NthPrime(n)+1)]: n in [1..1000]]; // Vincenzo Librandi, Dec 23 2010

f[n_]:=Prime[n]*Prime[n+1]+1; Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)
Join[{3},Times@@@Partition[Prime[Range[40]],2,1]+1] (* Harvey P. Dale, Oct 02 2012 *)

cp's OEIS Frontend

A328466 Row 3 of A328464: a(n) = A276156(8n - 4) / 6.

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A328467 Row 4 of A328464: a(n) = A276156(16n - 8) / 30.

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A328468 Row 5 of A328464: a(n) = A276156(32n - 16) / 210.

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A328463 Transpose of square array A328464.

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

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A008864 a(n) = prime(n) + 1.

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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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A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

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