cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.

Original entry on oeis.org

3, 7, 9, 33, 37, 38, 211, 213, 218, 241, 242, 246, 247, 249, 2313, 2317, 2319, 2341, 2342, 2346, 2521, 2523, 2526, 2529, 2550, 2553, 2559, 30031, 30038, 30039, 30061, 30062, 30063, 30066, 30069, 30242, 30243, 30249, 30270, 30278, 30279, 32341, 32342, 32347, 32370, 32373, 32377, 32379, 32551, 32553, 510513, 510518, 510519
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2019

Keywords

Comments

Numbers n for which A327860(n) = A003415(A276086(n)) is a prime.
Numbers n such that A276086(n) is in A157037.
Terms come in distinct "batches", where in each batch they are "slightly more" than the nearest primorial (A002110) below. This is explained by the fact that for A276086(n) to be a squarefree (which is the necessary condition for A157037), n's primorial base expansion (A049345) must not contain digits larger than 1. Thus this is a subsequence of A276156.
Numbers n such that A327860(A276086(n)) = A003415(A276087(n)) is a prime [A276087(n) is in A157037] are much rarer: 2, 4, 30, 212, 421, 30045, 510511, 512820, 9729723, ...
For all terms k in this sequence, A327969(k) <= 4, and particularly A327969(k) = 2 when k is a prime. Otherwise, when k is not a prime, but A003415(k) is, A327969(k) = 3, while for other cases (when k is neither prime nor in A157037), we have A327969(k) = 4.

Links

Crossrefs

Cf. A002110, A003415, A051674, A157037, A276086, A327860, A327969, A327978, A328232, A328240.
Subsequence of A276156, of A328116, and of A328242.

Programs

  • PARI
    A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); };
isA328233(n) = isprime(A327860(n));

A328570 Index of the least significant zero digit in the primorial base expansion of n, when the rightmost digit is in the position 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5
Offset: 0

Views

Author

Antti Karttunen, Oct 20 2019

Keywords

Comments

Index of the least prime not dividing A276086(n), where A276086 converts the primorial base expansion of n into its prime product form.
Starting from x = n, repeatedly divide x by prime(1) (discarding the remainder), and set x to the integer quotient floor(x/prime(1)), then divide x with prime(2) (again discarding the remainder, and setting x to the integer quotient), etc., stopping as soon one of the primes is a divisor of the previous integer quotient (leaving zero remainder). a(n) is then the index of that prime, equal to 1 + the number of iterations done.

Examples

			For n = 2, we divide it with A000040(1) = 2, and it leaves zero remainder, so we have finished on the first round (needing no actual iterations), and thus a(2) = 1. Note that 2 in primorial base (A049345) is written as "10", and indeed the first zero from the right occurs at the position 1.
For n = 5, we first divide 5 with prime(1) = 2, and discarding the remainder, we are left with floor(5/2) = 2. Then we divide that 2 with prime(2) = 3, leaving floor(2/3) = 0 and remainder 2. And finally we divide 0 with prime(3) = 5, and that doesn't leave any remainder, thus we are finished on the third round, and a(5) = 3. Note that 5 in primorial base is written as "21", and allowing here a leading zero, written as "021", we see that it is in this case the least significant zero occurring at position 3 from the right.
For n = 43, we first divide it with prime(1) = 2, leaving a remainder 1 and integer quotient 21. Then we divide 21 with prime(2) = 3, which doesn't leave any remainder, thus we are finished on the second round, and a(43) = 2. Note that 43 is written as "1201" in primorial base, with the least significant zero occurring in the position 2.

Links

Crossrefs

Cf. A000720, A002110, A049345, A053669, A055396, A143293, A257993, A276086, A276087, A276088, A326810, A328475, A328476, A328569, A328578, A328580.

Programs

  • PARI
    A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };

Formula

a(n) = A000720(A326810(n)) = A257993(A276086(n)) = A055396(A276087(n)).
For all n >= 0, A002110(a(n)) = A328580(n), a(A276086(n)) = A328578(n).
For all odd n, A000040(a(n)) = A326810(n) > A276088(n).
For all n >= 0, A276086(n)/A002110(a(n)-1) = A328475(n) and A276086(n)-A002110(a(n)-1) = A328476(n).

A328403 a(n) = A276086(A276086(A276086(n))), where A276086(n) converts primorial base expansion of n into its prime product form.

Original entry on oeis.org

3, 6, 5, 18, 7, 43218, 125, 1050, 16807, 10, 161051, 1320550, 7105308412125, 7357350, 265837, 835182761270, 2292646180979, 146410, 258413198822535882125, 107718961350, 1045726245479028578048875581177070937, 32310040537956402036523898190, 528950766088569054716797662440479
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Links

Crossrefs

Cf. A276086, A276087, A328316, A328394, A328398, A328405, A328406.

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[Nest[f, #, 3] &, 23, 0]] (* Michael De Vlieger, Oct 17 2019 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328403(n) = A276086(A276086(A276086(n)));

Formula

a(n) = A276086(A276087(n)) = A276086(A276086(A276086(n))).
A051903(a(n)) = A328394(n).
A061395(a(n)) = A328405(n).
A328114(a(n)) = A328398(n).
A235224(a(n)) = A328406(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

Views

Author

Antti Karttunen, Oct 27 2019

Keywords

Comments

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).

Links

Crossrefs

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.

Programs

Formula

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

Extensions

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

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Links

Crossrefs

Cf. A053669, A143293, A276086, A276087, A276088, A328476, A328570, A328579, A328580.
Cf. also A328316, A328317, A328613.

Programs

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

Formula

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)).

A328406 The length of primorial base expansion (number of significant digits) of A276086(A276086(A276086(n))), where A276086(n) converts primorial base expansion of n into its prime product form.

Original entry on oeis.org

2, 3, 2, 3, 3, 7, 4, 5, 6, 3, 7, 8, 12, 8, 7, 12, 12, 7, 17, 11, 25, 21, 24, 84, 49, 63, 94, 67, 49, 97, 4, 6, 8, 9, 7, 10, 6, 14, 13, 4, 14, 11, 22, 22, 19, 20, 66, 16, 23, 40, 20, 19, 50, 105, 81, 87, 104, 71, 49, 81, 12, 10, 34, 21, 9, 16, 11, 23, 16, 17, 85, 49, 71, 27, 44, 21, 93, 87, 39, 58, 171, 50, 205, 112, 54, 78, 78
Offset: 0

Views

Author

Antti Karttunen, Oct 16 2019

Keywords

Links

Crossrefs

Cf. A235224, A276086, A276087, A328398, A328403, A328404, A328405.

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[IntegerLength[Nest[f, #, 3], b] &, 87, 0]] (* Michael De Vlieger, Oct 17 2019 *)
  • PARI
    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); };
A328403(n) = A276086(A276086(A276086(n)));
A328406(n) = A235224(A328403(n));

Formula

a(n) = A235224(A328403(n)) = A328404(A276087(n)) = A328405(A276086(n)).
For all n, A000040(a(n)) > A328398(n).

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

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Links

Crossrefs

Cf. A002110, A053589, A276086, A276087, A276151, A326810, A328475.
Cf. A143293 (indices of other zeros after a(0)=0).

Programs

  • PARI
    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));

Formula

a(n) = A276151(A276086(n)).
A276086(a(n)) = A276087(n) / A326810(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

Views

Author

Antti Karttunen, Nov 10 2024

Keywords

Comments

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)

Examples

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

Links

Crossrefs

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.

Programs

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

Views

Author

Antti Karttunen, Oct 16 2019

Keywords

Comments

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)?

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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))));

Formula

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

Views

Author

Antti Karttunen, Oct 16 2019

Keywords

Links

Crossrefs

Cf. A061395, A143293, A235224, A276086, A276087, A328389, A328405, A328406.
Cf. A328402 (number of times each n occurs in this sequence).

Programs

  • Mathematica
    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 *)
  • PARI
    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));

Formula

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).
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