A049345 -id:A049345 - cp's OEIS frontend

A324351 Square array read by antidiagonals: A(x,y) is the result from writing x and y in primorial base (A049345) and starting from their least significant ends, always choosing a minimal digit from each digit position, and converting back to decimal.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 1, 0, 3, 4, 3, 0, 1, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 2, 1, 0, 5, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 0, 0, 0, 2, 2, 2, 2, 6, 6, 2, 2, 2, 2, 0, 0, 0, 1, 0, 3, 4, 3, 6, 7, 6, 3, 4, 3, 0, 1, 0

Offset: 0

Antti Karttunen, Feb 25 2019

			The array A begins:
       0   1   2   3   4   5   6   7   8   9  10  11  12
  x/y  ------------------------------------------------------
   0:  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1:  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0, ...
   2:  0,  0,  2,  2,  2,  2,  0,  0,  2,  2,  2,  2,  0, ...
   3:  0,  1,  2,  3,  2,  3,  0,  1,  2,  3,  2,  3,  0, ...
   4:  0,  0,  2,  2,  4,  4,  0,  0,  2,  2,  4,  4,  0, ...
   5:  0,  1,  2,  3,  4,  5,  0,  1,  2,  3,  4,  5,  0, ...
   6:  0,  0,  0,  0,  0,  0,  6,  6,  6,  6,  6,  6,  6, ...
   7:  0,  1,  0,  1,  0,  1,  6,  7,  6,  7,  6,  7,  6, ...
   8:  0,  0,  2,  2,  2,  2,  6,  6,  8,  8,  8,  8,  6, ...
   9:  0,  1,  2,  3,  2,  3,  6,  7,  8,  9,  8,  9,  6, ...
  10:  0,  0,  2,  2,  4,  4,  6,  6,  8,  8, 10, 10,  6, ...
  11:  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  6, ...
  12:  0,  0,  0,  0,  0,  0,  6,  6,  6,  6,  6,  6, 12, ...
etc.
In primorial base, 5 is written as "21" (as 5 = 2*2 + 1*1) and 10 is written as "120" (as 10 = 1*6 + 2*2 + 0*1). Aligning them digit by digit (from the least significant end), and then always choosing a lesser digit leaves us with digits "020", which is 4 written in primorial base as 2*2 + 0*1 = 4, thus A(5,10) = A(10,5) = 4.

Cf. A001477 (central diagonal), A002110, A049345, A276085, A276086, A324350.

up_to = 65703; \\ = binomial(362+1,2)
A002110(n) = prod(i=1,n,prime(i));
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; };
A324351sq(row,col) = A276085(gcd(A276086(row),A276086(col)));
A324351list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A324351sq(a-col,col))); (v); };
v324351 = A324351list(up_to);
A324351(n) = v324351[1+n]; \\ Antti Karttunen, Feb 25 2019

A(x,y) = A276085(A324350(x,y)) = A276085(gcd(A276086(x), A276086(y))).

A328840 Numbers with no digit 1 in their primorial base expansion (A049345).

Original entry on oeis.org

0, 4, 12, 16, 18, 22, 24, 28, 60, 64, 72, 76, 78, 82, 84, 88, 90, 94, 102, 106, 108, 112, 114, 118, 120, 124, 132, 136, 138, 142, 144, 148, 150, 154, 162, 166, 168, 172, 174, 178, 180, 184, 192, 196, 198, 202, 204, 208, 420, 424, 432, 436, 438, 442, 444, 448, 480, 484, 492, 496, 498, 502, 504, 508, 510, 514, 522, 526

Offset: 1

Antti Karttunen, Nov 07 2019

Numbers for which the least missing nonzero digit (A329028) in their primorial base expansion is 1.

Antti Karttunen, Table of n, a(n) for n = 1..92160
Index entries for sequences related to primorial base.

Cf. A049345.
Positions of ones in A329028.
Cf. also A328574 and A329027.
Cf. A255411 for an analogous sequence.

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]]; FreeQ[s, 1]]; Select[Range[0, 600], q] (* Amiram Eldar, Mar 06 2024 *)

A329028(n) = { my(m=Map(), p=2); while(n, mapput(m,(n%p),1); n = n\p; p = nextprime(1+p)); for(k=1,oo,if(!mapisdefined(m,k),return(k))); };
isA328840(n) = (1 == A329028(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);

A380525 Squarefree numbers k such that for all factorizations of k as xy, the sum (x y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 47, 53, 59, 61, 62, 67, 70, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 134, 137, 139, 146, 149, 151, 154, 157, 158, 163, 167, 173, 179, 181, 186, 190, 191, 193, 194, 195, 197, 199, 206, 211, 218, 223, 227, 229, 233

Offset: 1

Antti Karttunen, Feb 04 2025

nonn

A380468 is a subsequence. This differs from it by containing also the terms 70, 154, 190, 195, 455, 574, 645, 1054, 1085, ...

			For n=70, there are four factorizations into two factors: 1*70, 2*35, 5*14, 7*10, and thus, applying the formula (x' * y) + (x * y') we obtain
  0*70 + 1*70' = A003415(70) = 59, and A049345(59) = 1421.
  1*35 + 2*35' = 35 + 2*12, i.e., 1021 + 400 in primorial base, (giving 1421)
  1*14 + 5*14' = 14 + 5*9, i.e., 210 + 1211 in primorial base,
  1*10 + 7*10' = 10 + 7*7, i.e., 120 + 1301 in primorial base,
and as all these sums are carryless, 70 is included in this sequence.
For n = 1518 = 2*3*11*23, we obtain eight factorizations into two factors:
x*y:    | 1*1518   2*759   3*506   6*253  11*138   22*69   23*66   33*46
--------+----------------------------------------------------------------
x' * y  |      0   34111   22410   60021    4300   41411    2100   30210 (in primorial base)
x  * y' |  66421   32310   44011    6400   62121   25010   64321   36211
--------+----------------------------------------------------------------
Sum     |  66421   66421   66421   66421   66421   66421   66421   66421 = A049345(A003415(1518)), and as all these sums are carryless, 1581 is included in this sequence.

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

Intersection of A005117 and A358673.
Cf. A003415, A049345, A380524 (characteristic function).
Subsequences: A380468, A380526 (terms with at least 5 prime factors).

PARI
```
is_A380525 = A380524;
```

A380535 Numbers such that the least significant nonzero digit in their primorial base representation (A049345) is greater than 1.

Original entry on oeis.org

4, 10, 12, 16, 18, 22, 24, 28, 34, 40, 42, 46, 48, 52, 54, 58, 60, 64, 70, 72, 76, 78, 82, 84, 88, 90, 94, 100, 102, 106, 108, 112, 114, 118, 120, 124, 130, 132, 136, 138, 142, 144, 148, 150, 154, 160, 162, 166, 168, 172, 174, 178, 180, 184, 190, 192, 196, 198, 202, 204, 208, 214, 220, 222, 226, 228, 232, 234, 238, 244, 250

Offset: 1

Antti Karttunen, Feb 11 2025

Equally, numbers k such that A327860(k) [equally, A329029(k)] is a multiple of A053669(k), where A327860 is the arithmetic derivative of the primorial base exp-function, and A053669(k) gives the least prime not dividing k. See comments in A329029.

The asymptotic density of this sequence is 1 - A064648 = 0.294769828... . - Amiram Eldar, Feb 17 2025

			   n, A049345(n), A276088(n)
  ---------------------------------------------
   4       20       2, thus 4 is present,
  10      120       2, thus 10 is present,
  11      121       1, thus 11 is not present,
  12      200       2, thus 12 is present,
  14      210       1, thus 14 is not present,
  16      220       2, thus 16 is present,
  18      300       3, thus 18 is present.

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

Cf. A049345, A053669, A064648, A276088, A327860, A329029, A380527, A380534 (characteristic function).

Cf. also A342018.

q[n_] := Module[{k = n, p = 2, r}, While[{k, r} = QuotientRemainder[k, p]; k > 0 && r == 0, p = NextPrime[p]]; r > 1]; Select[Range[250], q] (* Amiram Eldar, Feb 17 2025 *)

PARI
```
is_A380535 = A380534;
```

{k such that A276088(k) > 1}.

A373606 Sum of the even-indexed digits in the primorial base representation (A049345) of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 18 2024

nonn

			A049345(85) = 2401, and the sum of digits at even positions (with the rightmost position having index 0) is 1+4 = 5, thus a(85) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..16170
Index entries for sequences related to primorial base

Cf. A049345, A276150, A373605, A373607.

A373606(n) = { my(p=2, i=1, s=0); while(n, if(i%2, s += (n%p)); n = n\p; p = nextprime(1+p); i = !i); (s); };

a(n) = A276150(n) - A373607(n).

a(n) = A373605(n) + A373607(n).

A373607 Sum of the odd-indexed digits in the primorial base representation (A049345) of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 18 2024

nonn

			A049345(85) = 2401, and the sum of digits at the odd positions  (with the rightmost digit having index 0) is 0+2 = 2, thus a(85) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..16170
Index entries for sequences related to primorial base

Cf. A049345, A276150, A373605, A373606.

A373607(n) = { my(p=2, i=0, s=0); while(n, if(i%2, s += (n%p)); n = n\p; p = nextprime(1+p); i = !i); (s); };

a(n) = A276150(n) - A373606(n).

a(n) = A373606(n) - A373605(n).

A381035 Numbers such that the least significant nonzero digit in their primorial base representation (A049345) is 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 96, 97, 98, 99, 101, 103, 104, 105, 107, 109, 110, 111

Offset: 1

Antti Karttunen, Feb 17 2025

Numbers k such that A327860(k) is not a multiple of A053669(k), where A327860 is the arithmetic derivative of the primorial base exp-function, and A053669(k) gives the least prime not dividing k.

The asymptotic density of this sequence is 0.70523017... (A064648). - Amiram Eldar, Feb 17 2025

			   n, A049345(n), A276088(n)
  ---------------------------------------------
   4       20       2, thus 4 is not present,
  11      121       1, thus 11 is present,
  14      210       1, thus 14 is present.

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

Complement of A380535 (apart from 0 which is in neither).
Cf. A049345, A064648, A276088, A380534.
Subsequences: A276156, A290249, A381034.

q[n_] := Module[{k = n, p = 2, r}, While[{k, r} = QuotientRemainder[k, p]; k > 0 && r == 0, p = NextPrime[p]]; r == 1]; Select[Range[120], q] (* Amiram Eldar, Feb 17 2025 *)

A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); };
is_A381035(n) = (1==A276088(n));

{k such that A276088(k) = 1}.

A324550 Primes written in primorial base (A049345).

Original entry on oeis.org

10, 11, 21, 101, 121, 201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301, 10001, 10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221

Offset: 1

Antti Karttunen, Mar 11 2019

When the primorial base representation is expressed with decimal digits as here, the sequence stays unambiguous only up to the 317th prime, 2099, written as 96421, because after that primorial base digits larger than 9 would be needed.
By writing down terms from a(6) to a(46) (primes 13 .. 199):
  201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301,
and then from a(48) to a(80) (primes 223 .. 409):
  10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221, 11321, 11421, 12001, 12101, 12121, 12201, 12321, 13101, 13121, 13201, 13221, 14001, 14101, 14221, 14301, 14321, 14421, 15101, 15201, 15301, 15321, 15421, 16101, 16121, 16301,
it is clearly seen that if n is a prime, then p+n is also likely to be prime, where p is the next higher primorial (A002110) > n. See also A324656.

Antti Karttunen, Table of n, a(n) for n = 1..317
Index entries for sequences related to primorial base.

Cf. A000040, A018239, A049345, A324642, A324656, A324657.

Cf. also A004676, A214617, A286941.

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

A324550(n) = A049345(prime(n)); \\ For A049345, see under that entry.

a(n) = A049345(A000040(n)).

A358671 Numbers k such that for all factorizations of k as x*y, the sum x+y is carryfree when the addition is done in the primorial base, A049345.

Original entry on oeis.org

2, 4, 6, 14, 18, 24, 26, 28, 38, 42, 52, 54, 62, 72, 74, 76, 78, 86, 96, 98, 114, 122, 124, 126, 134, 146, 148, 158, 172, 186, 194, 206, 218, 222, 244, 254, 258, 268, 278, 292, 302, 314, 316, 326, 362, 366, 386, 388, 398, 402, 412, 422, 434, 436, 438, 446, 458, 474, 482, 508, 518, 542, 554, 556, 558

Offset: 1

Antti Karttunen, Nov 26 2022

Numbers k such that there are no factorization of k into such a pair of natural numbers x and y that would generate any carries when added together in the primorial base.

			8 is not included, because while factorization 1*8 would yield a carry-free sum ("1" and "110" added together gives "111" = 9 in primorial base, A049345), factorization 2*4 would not, as 2+4 (= "10" + "20") and 2 is the max. allowed digit in the second rightmost place.

Index entries for sequences related to primorial base

Cf. A038548, A049345, A276086, A329041, A358233, A358670 (characteristic function).

Cf. also A358673.

PARI
```
isA358671(n) = A358670(n);
```

{k | A358233(k) = A038548(k)}.

cp's OEIS Frontend

A324351 Square array read by antidiagonals: A(x,y) is the result from writing x and y in primorial base (A049345) and starting from their least significant ends, always choosing a minimal digit from each digit position, and converting back to decimal.

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A328840 Numbers with no digit 1 in their primorial base expansion (A049345).

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A370132 Numbers with no digit larger than 2 in primorial base, A049345.

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A380525 Squarefree numbers k such that for all factorizations of k as x*y, the sum (x * y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.

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A380535 Numbers such that the least significant nonzero digit in their primorial base representation (A049345) is greater than 1.

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Mathematica

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A373606 Sum of the even-indexed digits in the primorial base representation (A049345) of n.

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A373607 Sum of the odd-indexed digits in the primorial base representation (A049345) of n.

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A381035 Numbers such that the least significant nonzero digit in their primorial base representation (A049345) is 1.

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A380525 Squarefree numbers k such that for all factorizations of k as xy, the sum (x y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.