A328317 -id:A328317 - cp's OEIS frontend

A328316 Iterates of A276086 starting from 0.

0, 1, 2, 3, 6, 5, 18, 125, 43218, 258413198822535882125

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

The unique infinite sequence such that a(0) = 0, a(n) = A276085(a(n+1)) for n >= 0, and A129251(a(n)) = 0 for n >= 1, i.e., all nonzero terms must be in A048103.

a(10) is 240 decimal digits long (can be found in b-file), and a(11) is too big to fit even into a b-file as it is 32700 decimal digits long, but it can be found in the given a-file.

Antti Karttunen, Table of n, a(n) for n = 0..10
Antti Karttunen, Terms a(0) .. a(11), without running index
Index entries for sequences related to primorial base

Cf. A002110, A048103, A129251, A276085, A276086, A328317 (the smallest prime not dividing a(n)), A328318, A328319 (digit sum in primorial base), A328322 (max. digit), A328323.
Cf. A153013, and also A109162, A179016, A219666, A259934 for more or less analogous sequences.
Cf. also A328313.

A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
NestList[A276086, 0, 10] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));

a(0) = 0; and for n > 0, a(n) = A276086(a(n-1)).

A328585 Numbers n for which A257993(n) is equal to A257993(A276086(A276086(n))), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 103, 105, 107, 108, 109, 111, 113, 115, 117, 119, 121, 123, 125, 126, 127, 129, 131, 133, 135, 137, 138, 139

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Numbers n for which A257993(n) is equal to A328578(n).

All odd numbers are included, as A257993(2n+1) = A328578(2n+1) = 1 for all n >= 0.

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

Union of A005408 (odd numbers) and A328586 (even terms).
Positions of zeros in A328590.
Cf. A257993, A276086, A328578, A328587, A328588.
Cf. also A328316, A328317.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328585(n) = (A328578(n) == A257993(n));

A328632 Numbers k such that A276086(k) == 1 (mod 6), where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 12, 24, 30, 42, 54, 60, 72, 84, 90, 102, 114, 120, 132, 144, 150, 162, 174, 180, 192, 204, 216, 228, 246, 258, 276, 288, 306, 318, 336, 348, 366, 378, 396, 408, 420, 432, 444, 450, 462, 474, 480, 492, 504, 510, 522, 534, 540, 552, 564, 570, 582, 594, 600, 612, 624, 636, 648, 666, 678, 696, 708, 726, 738, 756, 768, 786, 798, 816

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers k >= 0 for which A328578(k) = A257993(A276086(A276086(k))) = 2, where A276086 converts the primorial base expansion of k into its prime product form, and A257993 returns the index of the least prime not present in its argument. - The original, equivalent definition.
Numbers k for which A276087(k) is an even number, but not a multiple of three.
All terms are multiples of 6, and thus apart from the initial zero, this is a subsequence of A328587, numbers k for which A257993(A276086(A276086(k))) is less than A257993(k).

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

Row 2 of A328631.
After the initial zero, setwise difference A328587 \ A328762. Also setwise difference A008588 \ A358843.
Positions of 1's in A358840 and A358841 (characteristic function), positions of 2's in A328578.
Cf. A257993, A276086, A328578, A358845 (= a(n)/6).
Cf. also A328317.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328632(n) = (2==A328578(n));

isA326832(n) = A358841(n);
A358841(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (1==(m%6)); }; \\ Antti Karttunen, Dec 12 2022

{k | A358840(k) == 1}. - Antti Karttunen, Dec 02 2022

Definition replaced with a simpler one and the original definition moved to the comments section by Antti Karttunen, Dec 03 2022

A326810 The smallest prime that does not divide the prime product form (A276086) of the primorial base expansion of n.

Original entry on oeis.org

2, 3, 2, 5, 2, 5, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 5, 2, 5, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11, 2, 11, 2, 3, 2, 5, 2, 5, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11, 2, 11, 2, 3, 2, 5, 2, 5, 2, 3, 2, 11, 2, 11, 2, 3, 2, 11

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A053669, A143293, A276086, A276087, A276088, A328476, A328570, A328579, A328580.

Cf. also A328316, A328317, A328613.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Table[Block[{p = 2}, While[Mod[#, p] == 0, p = NextPrime@ p]; p] &@ Apply[Times, Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #}] &@ IntegerDigits[n, b], {n, 0, 105}]] (* Michael De Vlieger, Oct 22 2019 *)

A326810(n) = { my(i=1, p=2); while(n && (n%p), n = n\p; p = nextprime(1+p)); (p); };

a(n) = A053669(A276086(n)).
a(n) = A000040(A328570(n)).
a(n) = A020639(A276087(n)) = A020639(A328613(n)).
a(n) = A276087(n) / A276086(A328476(n)).
For all odd n, a(n) > A276088(n).
For all n >= 0, a(A276086(n)) = A328579(n).
For all n >= 1, A328317(n) = a(A328316(n-1)).

A328322 Maximal digit value used when A328316(n) is written in primorial base; maximal prime exponent in A328316(1+n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 7, 49, 430, 74814

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114, A328316, A328317, A328318, A328319, A328323.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328322(n) = A328114(A328316(n));
\\ Or alternatively as, more slowly:
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328322(n) = A051903(A328316(1+n));

a(n) = A328114(A328316(n)).

a(n) = A051903(A328316(1+n)).

A328586 Even numbers n for which A257993(n) is equal to A257993(A276086(A276086(n))), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

6, 18, 36, 48, 66, 78, 96, 108, 126, 138, 156, 168, 186, 198, 222, 234, 252, 264, 282, 294, 312, 324, 342, 354, 372, 384, 402, 414, 426, 438, 456, 468, 486, 498, 516, 528, 546, 558, 576, 588, 606, 618, 642, 654, 672, 684, 702, 714, 732, 744, 762, 774, 792, 804, 822, 834, 846, 858, 876, 888, 906, 918, 936, 948, 966, 978, 996, 1008

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

All terms are multiples of 6, but very few multiples of 5 (and thus of 10) are present: the first ones are at a(169) = 2520 and a(254) = 3780. Among the first 10000 terms, there are only 28 ending with decimal digit 0, while those that end with either 2 or 4 are 2450 both, and with either 6 or 8, both have 2536 each.

Other multiples of six are in A328587 and A328589.

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

Cf. A257993, A276086, A328578, A328316, A328317, A328633.

Even terms in A328585. Cf. also A328587, A328588, A328589.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328586(n) = ((!(n%2))&&(A328578(n) == A257993(n)));

A328318 Number of nonzero digits in representation of A328316(n) in primorial base; Number of distinct prime factors in A328316(1+n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 5, 16, 104, 7447

Offset: 0

Antti Karttunen, Oct 14 2019

Index entries for sequences related to primorial base

Cf. A001221, A267263, A276086, A328316, A328317, A328319.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A267263(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += !!d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A328318(n) = A267263(A328316(n));
\\ Or alternatively, more slowly as:
A328318(n) = omega(A328316(1+n));

a(n) = A267263(A328316(n)).

a(n) = A001221(A328316(1+n)).

A328319 Sum of digits when A328316(n) is written in primorial base; number of prime factors in A328316(1+n), counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 7, 21, 159, 12927, 132571335

Offset: 0

Antti Karttunen, Oct 14 2019

Index entries for sequences related to primorial base

Cf. A001221, A276086, A276150, A328316, A328317, A328318, A328322.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A328319(n) = A276150(A328316(n));
\\ Or alternatively, more slowly as:
A328319(n) = bigomega(A328316(1+n));

a(n) = A276150(A328316(n)).

a(n) = A001222(A328316(1+n)).

A328323 a(n) = gcd(A328316(n),A328316(1+n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 2401, 861018571670257

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

All terms are odd, because the parity of A328316 alternates.

a(10) is 208 decimal digits long, and can be found in the b-file.

Antti Karttunen, Table of n, a(n) for n = 0..10

Cf. A276086, A324198, A328316, A328317.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A328323(n) = gcd(A328316(n),A328316(1+n));
\\ Or alternatively, as:
A324198(n) = gcd(n, A276086(n));
A328323(n) = A324198(A328316(n));

a(n) = A324198(A328316(n)) = gcd(A328316(n),A328316(1+n)).

A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

2, 6, 18, 34, 36, 48, 66, 78, 96, 108, 122, 126, 138, 154, 156, 168, 186, 198, 212, 222, 234, 244, 252, 264, 282, 294, 312, 324, 332, 342, 354, 364, 372, 384, 402, 414, 422, 426, 438, 454, 456, 468, 486, 498, 516, 528, 542, 546, 558, 574, 576, 588, 606, 618, 632, 642, 654, 664, 672, 684, 702, 714, 732, 744, 752, 762, 774, 784, 792, 804

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 6, but not of 5.
Question: Is the even bisection of A328316, starting from A328316(4) as: 6, 18, 43218, ..., a subsequence of this sequence? See also A328317.
Subsequence such that both k and A276087(k) are in this sequence starts as: 2, 6, 18, 34, 36, 48, 66, 154, 156, 186, 234, 244, 294, 312, 324, 354, 364, 384, 426, 438, 454, 456, 542, 546, 558, 588, 606, ...
When A276086 is applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, implying that the number's primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), the original number will be converted to a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401", with the least significant zero in position A328578(n), which is seen to be always either 3 or 2.

			294 = 7^2 * 3 * 2 has primorial base expansion (A049345) "12400", which, when converted to a prime product form (A276086) yields 11^1 * 7^2 * 5^4 * 3^0 * 2^0 = 336875. This in turn has primorial base representation [11,2,9,1,0,2,1], which when converted to prime product form gives 17^11 * 13^2 * 11^9 * 7^1 * 5^0 * 3^2 * 2^1 = 1720796647657111567992931482, which has the required property of being a multiple of 6 but not of 5, thus 294 is included in this sequence.

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

Row 3 of A328631.
Cf. A049345, A257993, A276086, A276087, A328578.
Cf. also A328316, A328317, A328589, A328762.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328633(n) = (3==A328578(n));

isA328633(n) = { my(u=A276086(A276086(n))); ((u%5)&&!(u%6)); };

cp's OEIS Frontend

A328316 Iterates of A276086 starting from 0.

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A328585 Numbers n for which A257993(n) is equal to A257993(A276086(A276086(n))), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

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A328632 Numbers k such that A276086(k) == 1 (mod 6), where A276086 is the primorial base exp-function.

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A326810 The smallest prime that does not divide the prime product form (A276086) of the primorial base expansion of n.

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A328322 Maximal digit value used when A328316(n) is written in primorial base; maximal prime exponent in A328316(1+n).

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A328586 Even numbers n for which A257993(n) is equal to A257993(A276086(A276086(n))), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

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A328318 Number of nonzero digits in representation of A328316(n) in primorial base; Number of distinct prime factors in A328316(1+n).

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A328319 Sum of digits when A328316(n) is written in primorial base; number of prime factors in A328316(1+n), counted with multiplicity.

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A328323 a(n) = gcd(A328316(n),A328316(1+n)).