A276154 -id:A276154 - 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).

A276945 Square array A(row,col): A(row,1) = A276155(row), and for col > 1, A(row,col) = A276154(A(row,col-1)); Dispersion of primorial base left shift A276154.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 30, 36, 12, 5, 210, 240, 60, 14, 7, 2310, 2520, 420, 66, 32, 9, 30030, 32340, 4620, 450, 216, 38, 10, 510510, 540540, 60060, 4830, 2340, 246, 42, 11, 9699690, 10210200, 1021020, 62370, 30240, 2550, 270, 44, 13, 223092870, 232792560, 19399380, 1051050, 512820, 32550, 2730, 276, 62, 15

Offset: 1

Antti Karttunen, Sep 24 2016

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Entries in column k are all multiples of A002110(k-1). Dividing that factor out gives array A286625. - Antti Karttunen, Jun 30 2017

			The top left corner of the array:
   1,  2,   6,   30,   210,    2310,    30030,    510510
   3,  8,  36,  240,  2520,   32340,   540540,  10210200
   4, 12,  60,  420,  4620,   60060,  1021020,  19399380
   5, 14,  66,  450,  4830,   62370,  1051050,  19909890
   7, 32, 216, 2340, 30240,  512820,  9729720, 223603380
   9, 38, 246, 2550, 32550,  542850, 10240230, 233303070
  10, 42, 270, 2730, 34650,  570570, 10720710, 242492250
  11, 44, 276, 2760, 34860,  572880, 10750740, 243002760
  13, 62, 426, 4650, 60270, 1023330, 19429410, 446696250
  15, 68, 456, 4860, 62580, 1053360, 19939920, 456395940
  16, 72, 480, 5040, 64680, 1081080, 20420400, 465585120
  17, 74, 486, 5070, 64890, 1083390, 20450430, 466095630
  18, 90, 630, 6930, 90090, 1531530, 29099070, 669278610

Inverse permutation: A276946.
Transpose: A276943. One more than A286615.
Column 1: A276155.
Row 1: A002110.
Row 2: A276939.
Row 3: A088860 (2*A002110).
Row 11: 2*A276939 (row 2) from 16, 72, 480, 5040, 64680, ... onward.
Row 13: 3*A002110, from 18, 90, 630, 6930, 90090, ... onward.
Cf. A276154.
Cf. also arrays A286625, A276955.

(define (A276945 n) (A276945bi (A002260 n) (A004736 n)))
(define (A276945bi row col) (if (= 1 col) (A276155 row) (A276154 (A276945bi row (- col 1)))))

A(row,1) = A276155(row); for row > 1, A(row,col) = A276154(A(row,col-1)).

A276943 Square array A(row,col) read by antidiagonals: A(1,col) = A276155(col), and for row > 1, A(row,col) = A276154(A(row-1,col)); Dispersion of primorial base left shift A276154 (array transposed).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 12, 36, 30, 7, 14, 60, 240, 210, 9, 32, 66, 420, 2520, 2310, 10, 38, 216, 450, 4620, 32340, 30030, 11, 42, 246, 2340, 4830, 60060, 540540, 510510, 13, 44, 270, 2550, 30240, 62370, 1021020, 10210200, 9699690, 15, 62, 276, 2730, 32550, 512820, 1051050, 19399380, 232792560, 223092870

Offset: 1

Antti Karttunen, Sep 24 2016

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Entries on row n are all multiples of A002110(n-1).

			The top left corner of the array:
    1,    3,    4,    5,     7,     9,    10,    11,    13,    15,    16
    2,    8,   12,   14,    32,    38,    42,    44,    62,    68,    72
    6,   36,   60,   66,   216,   246,   270,   276,   426,   456,   480
   30,  240,  420,  450,  2340,  2550,  2730,  2760,  4650,  4860,  5040
  210, 2520, 4620, 4830, 30240, 32550, 34650, 34860, 60270, 62580, 64680

Inverse permutation: A276944.
Transpose: A276945.
Column 1: A002110, Row 1: A276155.
Cf. A276154.
Cf. also array A276953.

(define (A276943 n) (A276943bi (A002260 n) (A004736 n)))
(define (A276943bi row col) (if (= 1 row) (A276155 col) (A276154 (A276943bi (- row 1) col))))

A(1,col) = A276155(col); for row > 1, A(row,col) = A276154(A(row-1,col)).

A276155 Complement of A276154; numbers that cannot be obtained by shifting left the primorial base representation (A049345) of some number.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 108, 109

Offset: 1

Antti Karttunen, Aug 24 2016

The first 25 terms, when viewed in primorial base (A049345) look as: 1, 11, 20, 21, 101, 111, 120, 121, 201, 211, 220, 221, 300, 301, 310, 311, 320, 321, 400, 401, 410, 411, 420, 421, 1001.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Complement: A276154.
Row 1 of A276943 and A286623. Column 1 of A276945 and A286625.
Cf. A002110, A049345.
Cf. A005408, A057588, A061720, A143293, A286630 (subsequences).
For the first 17 terms coincides with A273670.

nn = 109; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Complement[Range@ nn, Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}]] (* Version 10.2, or *)
nn = 109; 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]]]; Complement[Range@ nn, Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, nn}]] (* Michael De Vlieger, Aug 26 2016 *)

A323879 Number of divisors d of n such that A276154(d) divides n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A002110, A276154.

Cf. also A323878.

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))));
A323879(n) = sumdiv(n,d,!(n%A276154(d)));

a(n) = Sum_{d|n} [A276154(d)|n], where [ ] is the Iverson bracket.

A353574 a(n) = A342002(n) / gcd(A342002(n), A342002(A276154(n))).

Original entry on oeis.org

1, 1, 5, 1, 7, 1, 7, 2, 31, 13, 41, 1, 9, 11, 37, 2, 47, 1, 11, 7, 43, 19, 53, 1, 13, 17, 49, 11, 59, 1, 9, 5, 41, 17, 55, 2, 59, 71, 247, 53, 317, 19, 73, 46, 289, 127, 359, 13, 87, 113, 331, 74, 401, 11, 101, 67, 373, 169, 443, 1, 11, 13, 47, 5, 61, 17, 69, 43, 277, 121, 347, 2, 83, 107, 319, 71, 389, 31, 97, 16

Offset: 1

Antti Karttunen, Apr 29 2022

Compare the scatter plot to that of A342002.

Cf. A003415, A003557, A003961, A342001, A342002, A353572, A353573, A353575.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342001(n) = (A003415(n) / A003557(n));
A353571(n) = A342001(A003961(n));
A353574(n) = { my(u=A342001(A276086(n))); u/gcd(u, A353571(A276086(n))); };

a(n) = A342002(n) / A353573(n) = A342002(n) / gcd(A342002(n), A353572(n)).

A324384 a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 11, 12, 1, 2, 1, 8, 1, 18, 1, 4, 7, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 8, 1, 2, 7, 12, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 52, 1, 6, 1, 56, 1, 2, 1, 60, 1, 2, 1, 16, 1, 6, 1, 4, 1, 14, 1, 24, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 8, 1, 90, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 8, 1

Offset: 0

Antti Karttunen, Feb 26 2019

nonn

For a very few primes, a(p) > 1 (then by necessity a(p) = p). In range 2 .. 2^25 there are three: 2, 11, 119039.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A002110, A049345, A276085, A276086, A276151, A276152, A276154, A323879, A324198, A324350, A324351.

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))));
\\ Alternatively, A276154 can be defined with A276085, A276086 and A003961:
A002110(n) = prod(i=1,n,prime(i));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A276154(n) = A276085(A003961(A276086(n)));
A324384(n) = gcd(n,A276154(n));

a(n) = gcd(n, A276154(n)).

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset: 1

Antti Karttunen, Aug 21 2016

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)-1).
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

A left inverse of A276086.
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
Other identities.
For all n >= 0:
a(A276086(n)) = n.
a(A000040(1+n)) = A002110(n).
a(A002110(1+n)) = A143293(n).
From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)
a(A283477(n)) = A283985(n).
a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:
a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:
a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
The sum or difference of the rhs-sequences is A108951:
a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
Here the two sequences are inverse permutations of each other:
a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
a(A346101(n)) = A289234(n). [Self-inverse]
Other correspondences:
a(A324350(x,y)) = A324351(x,y).
a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
(End)

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

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]

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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A276945 Square array A(row,col): A(row,1) = A276155(row), and for col > 1, A(row,col) = A276154(A(row,col-1)); Dispersion of primorial base left shift A276154.

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A276943 Square array A(row,col) read by antidiagonals: A(1,col) = A276155(col), and for row > 1, A(row,col) = A276154(A(row-1,col)); Dispersion of primorial base left shift A276154 (array transposed).

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A276155 Complement of A276154; numbers that cannot be obtained by shifting left the primorial base representation (A049345) of some number.

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A323879 Number of divisors d of n such that A276154(d) divides n.

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A353574 a(n) = A342002(n) / gcd(A342002(n), A342002(A276154(n))).

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A324384 a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

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A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

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