A143293 -id:A143293 - cp's OEIS frontend

A328475 Convert the primorial base expansion of n into its prime product form, then divide by the largest primorial which divides that product: a(n) = A111701(A276086(n)).

1, 1, 3, 1, 9, 3, 5, 5, 15, 1, 45, 3, 25, 25, 75, 5, 225, 15, 125, 125, 375, 25, 1125, 75, 625, 625, 1875, 125, 5625, 375, 7, 7, 21, 7, 63, 21, 35, 35, 105, 1, 315, 3, 175, 175, 525, 5, 1575, 15, 875, 875, 2625, 25, 7875, 75, 4375, 4375, 13125, 125, 39375, 375, 49, 49, 147, 49, 441, 147, 245, 245, 735, 7, 2205, 21, 1225, 1225, 3675, 35, 11025, 105

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A002110, A053589, A111701, A276086, A328476, A328399 (rgs-transform).

Cf. A143293 (indices of 1's after a(0)=1).

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328475(n) = A111701(A276086(n));

a(n) = A111701(A276086(n)).

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 *)

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)).

A328476 Convert the primorial base expansion of n into its prime product form, then subtract the largest primorial which divides that product: a(n) = A276151(A276086(n)).

Original entry on oeis.org

0, 0, 2, 0, 8, 12, 4, 8, 14, 0, 44, 60, 24, 48, 74, 120, 224, 420, 124, 248, 374, 720, 1124, 2220, 624, 1248, 1874, 3720, 5624, 11220, 6, 12, 20, 36, 62, 120, 34, 68, 104, 0, 314, 420, 174, 348, 524, 840, 1574, 2940, 874, 1748, 2624, 5040, 7874, 15540, 4374, 8748, 13124, 26040, 39374, 78540, 48, 96, 146, 288, 440, 876, 244, 488

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Index entries for sequences related to primorial base

Cf. A002110, A053589, A276086, A276087, A276151, A326810, A328475.

Cf. A143293 (indices of other zeros after a(0)=0).

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328476(n) = A276151(A276086(n));

a(n) = A276151(A276086(n)).

A276086(a(n)) = A276087(n) / A326810(n).

A328613 Starting with m = 1, p = 2 and x = n, iterate m -> m * p^valuation(x,p), x -> floor(x/p), p -> {the next prime after p}, until x is zero. Then a(n) = product of the values of m and p that were reached at the end of the iteration.

Original entry on oeis.org

2, 3, 10, 5, 20, 5, 42, 21, 56, 7, 14, 7, 84, 21, 14, 7, 112, 7, 126, 63, 28, 7, 14, 7, 168, 21, 14, 7, 28, 7, 330, 165, 1760, 55, 110, 55, 396, 99, 22, 11, 88, 11, 66, 33, 44, 11, 22, 11, 528, 33, 22, 11, 44, 11, 594, 297, 88, 11, 22, 11, 660, 165, 110, 55, 3520, 55, 66, 33, 44, 11, 22, 11, 792, 99, 22, 11, 44, 11, 66, 33, 176, 11

Offset: 0

Antti Karttunen, Oct 22 2019

nonn

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

Cf. A000040, A007814, A020639, A055396, A049345, A143293, A235224, A276086, A326810, A328570, A328612.

A328613(n) = { my(m=1, p=2); while(n, m *= p^valuation(n,p); n = n\p; p = nextprime(1+p)); (m*p); };

a(n) = A000040(1+A235224(n)) * A328612(n).
For all n >= 0, A055396(a(n)) = A328570(n) and A020639(a(n)) = A326810(n).
For all n >= 1, A007814(a(n)) = A007814(n).
For all n >= 1, a(A143293(n-1)) = A000040(1+n).

A377871 Numbers k such that neither k nor A276085(k) has divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 41, 42, 43, 45, 46, 47, 50, 53, 58, 59, 61, 62, 63, 66, 67, 70, 71, 73, 74, 75, 78, 79, 82, 83, 86, 89, 90, 94, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 113, 114, 117, 118, 122, 125, 126, 127, 130, 131, 134, 137, 138, 139, 142

Offset: 1

Antti Karttunen, Nov 10 2024

nonn

Range of A276087, where A276087(n) = A276086(A276086(n)) [the twofold application of the primorial base exp-function].
A276087(0) = 2, and for n >= 0, A276087(A143293(n)) = A000040(n+2), therefore all primes are included.
From Antti Karttunen, Nov 17 2024: (Start)
Even semiprimes > 4 form a subsequence, because A006862 (Euclid numbers) is a subsequence of A048103. Note that A276087(A376416(n)) = A276086(A006862(n)) = A100484(1+n). On the other hand, none of the odd semiprimes, A046315, occur here, because they are all included in A369002, and thus in A377873. Similarly, A276092 after its initial 1 is a subsequence, because A057588 (Kummer numbers) is also a subsequence of A048103.
For k=1..6, there are 6, 52, 486, 4775, 46982, 467372 terms <= 10^k. Question: Does this sequence have an asymptotic density?
(End)

			A276087(A002110(10)) = A276086(A276086(A002110(10))) = A276086(A000040(10+1)) = A276086(31) = 14, therefore 14 is included in this sequence.

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

Intersection of A048103 and A377869.
Sequence A276087 sorted into ascending order.
Cf. A377870 (characteristic function).
Subsequences: A000040, A100484 (after its initial 4), A276092 (after its initial 1).
Cf. A006862, A143293, A276086, A376416.

PARI
```
\\ See A377870.
```

A328398 Maximal digit value in primorial base expansion of A276086(A276086(A276086(n))), where A276086(n) converts primorial base expansion of n into its prime product form.

Original entry on oeis.org

1, 1, 2, 3, 1, 7, 4, 5, 7, 2, 7, 12, 35, 14, 11, 15, 15, 11, 49, 19, 88, 64, 81, 403, 198, 248, 405, 271, 166, 449, 2, 3, 6, 7, 11, 25, 5, 30, 32, 3, 37, 8, 66, 53, 49, 49, 302, 40, 73, 116, 48, 47, 177, 495, 351, 391, 518, 338, 188, 331, 15, 16, 109, 65, 13, 39, 11, 37, 25, 44, 371, 181, 300, 87, 154, 44, 440, 396, 131

Offset: 0

Antti Karttunen, Oct 16 2019

nonn

2's occur at 2, 9, 30, 2312, 2559, 32589, ... (cf. A143293).
In range n = 0 .. 32768, a(n) attains the maximum possible value A000040(A328406(n))-1 only at n=2 and n=2804, when it must be the value of the most significant digit in the primorial base expansion of A328403(n).
When comparing the scatter plots of this sequence and those of A328389 and A328394, although the overall shape gets more blurred on each iteration of A276086, it is easy to see by informal inductive reasoning that the low values of the sequences should occur at about same positions.
Question: Are there any 1's after a(0), a(1) and a(4)?

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

Cf. A002110, A143293, A276086, A276087, A328114, A328389, A328394, A328403, A328406.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[Max@ IntegerDigits[Nest[f, #, 3], b] &, 79, 0]] (* Michael De Vlieger, Oct 17 2019 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328114(n) = { my(s=0, p=2); while(n, s = max(s,n%p); n = n\p; p = nextprime(1+p)); (s); };
A328398(n) = A328114(A276086(A276086(A276086(n))));

a(n) = A328114(A328403(n)) = A328389(A276087(n)) = A328394(A276086(n)).

For all n, a(n) < A000040(A328406(n)).

A328404 The length of primorial base expansion (number of significant digits) of A276086(n), where A276086(n) converts primorial base expansion of n into its prime product form.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 5, 5, 5, 5, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 5, 5, 5, 5, 6, 6, 5, 6, 6, 6, 6, 7, 6, 6, 6, 7

Offset: 0

Antti Karttunen, Oct 16 2019

nonn

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

Cf. A061395, A143293, A235224, A276086, A276087, A328389, A328405, A328406.

Cf. A328402 (number of times each n occurs in this sequence).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120]}, Array[IntegerLength[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b], b] &, 105, 0]] (* Michael De Vlieger, Oct 17 2019 *)

A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328404(n) = A235224(A276086(n));

a(n) = A235224(A276086(n)) = A061395(A276087(n)).
For all n, a(A143293(n-1)) = n+1.
For all n, A000040(a(n)) > A328389(n).

A328405 The length of primorial base expansion (number of significant digits) of A276086(A276086(n)), where A276086(n) converts primorial base expansion of n into its prime product form.

Original entry on oeis.org

2, 2, 3, 2, 4, 4, 3, 4, 4, 3, 5, 5, 5, 6, 6, 6, 5, 5, 7, 6, 9, 8, 10, 14, 11, 12, 14, 12, 12, 15, 3, 4, 5, 4, 5, 6, 4, 5, 7, 3, 8, 5, 9, 9, 8, 7, 12, 7, 8, 12, 8, 7, 12, 14, 16, 15, 15, 15, 11, 12, 5, 6, 8, 7, 7, 8, 5, 7, 9, 9, 14, 12, 12, 9, 12, 7, 15, 15, 12, 12, 18, 13, 20, 17, 11, 13, 15, 14, 17, 13, 8, 9, 11, 14, 11, 13, 11, 10, 10, 10

Offset: 0

Antti Karttunen, Oct 16 2019

nonn

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

Cf. A061395, A143293, A235224, A276086, A276087, A328394, A328403, A328404, A328406.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[IntegerLength[Nest[f, #, 2], b] &, 100, 0]] (* Michael De Vlieger, Oct 17 2019 *)

A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
A328405(n) = A235224(A276087(n));

a(n) = A235224(A276087(n)) = A061395(A328403(n)).

For all n, A000040(a(n)) > A328394(n).

A328614 Number of 1-digits in primorial base expansion of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 22 2019

			In primorial base (A049345), 87 is written as "2411" because 87 = 2*A002110(3) + 4*A002110(2) + 1*A002110(1) + 1*A002110(0) = 2*30 + 4*6 + 1*2 + 1*1. Only two of these digits are "1"'s, thus a(87) = 2.

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

Cf. A049345, A056169, A267263, A276086, A328615, A328616.
Cf. A143293 (positions of records after initial zero).
Cf. also A257511.

a[n_] := Module[{k = n, p = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, If[r == 1, s++]; p = NextPrime[p]]; s]; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)

A328614(n) = { my(s=0, p=2); while(n, s += (1==(n%p)); n = n\p; p = nextprime(1+p)); (s); };

a(n) = A056169(A276086(n)).
a(n) = A267263(n) - A328615(n).
For n >= 1, a(A143293(n-1)) = n. [This is the first occurrence of each n]

cp's OEIS Frontend

A328475 Convert the primorial base expansion of n into its prime product form, then divide by the largest primorial which divides that product: a(n) = A111701(A276086(n)).

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

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A328476 Convert the primorial base expansion of n into its prime product form, then subtract the largest primorial which divides that product: a(n) = A276151(A276086(n)).

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A328613 Starting with m = 1, p = 2 and x = n, iterate m -> m * p^valuation(x,p), x -> floor(x/p), p -> {the next prime after p}, until x is zero. Then a(n) = product of the values of m and p that were reached at the end of the iteration.

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A377871 Numbers k such that neither k nor A276085(k) has divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

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A328398 Maximal digit value in primorial base expansion of A276086(A276086(A276086(n))), where A276086(n) converts primorial base expansion of n into its prime product form.

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A328404 The length of primorial base expansion (number of significant digits) of A276086(n), where A276086(n) converts primorial base expansion of n into its prime product form.

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A328405 The length of primorial base expansion (number of significant digits) of A276086(A276086(n)), where A276086(n) converts primorial base expansion of n into its prime product form.

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