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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

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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));

A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).

Original entry on oeis.org

14, 45, 74, 198, 5114, 10295, 65174, 1086194, 20485574, 40354813, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253
Offset: 1

Views

Author

Antti Karttunen, Oct 10 2019

Keywords

Comments

From David A. Corneth, Oct 12 2019: (Start)
Let k' be the arithmetic derivative of k. Then to find terms of the form k = p * q where p, q are prime, we could see that k' = p + q. Then as one of them needs to be two, say p, needs to be 2, we have q = A143293(m) - 2 a prime. This would give terms 2 * q.
If terms are of the form k = p * q * r where p, q, r are distinct primes then k' = p*q + p*r + q*r. For m we like, we could solve p*q + p*r + q*r = A143293(m). checking p * q below some bound, we can solve for r and get r = (A143293(m) - p*q) / (p + q). With some extra constraints and searching different prime signatures, one might confirm terms found are all below some chosen upper bound. (End)
See sequences A369239 and A369240 for more observations and insights about the terms of this sequence. - Antti Karttunen, Jan 22 2024

Links

Crossrefs

Sequence A369240 sorted into ascending order.
Cf. A003415, A143293, A327969, A327978, A328313, A369239.

Programs

  • PARI
    A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
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; };
isA328243(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(A276086(u)))); }; \\ This is very slow program!
k=0; for(n=1,A002620(A143293(6)),if(isA328243(n), k++; print1(n,", ")));

Formula

A327969(a(n)) <= 5 for all n.

Extensions

a(12)-a(25) from David A. Corneth and Giovanni Resta, Oct 12 2019

A328099 a(n) = min(A003415(n), A276086(n)).

Original entry on oeis.org

0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 7, 1, 21, 14, 19, 12, 35, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 49, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71, 1, 176, 108, 43, 1, 124, 22, 45, 32, 140, 1, 123, 20, 96, 34, 49, 24
Offset: 1

Views

Author

Antti Karttunen, Oct 07 2019

Keywords

Comments

Always choosing the lesser of A003415(n) and A276086(n) is often a good heuristic when trying to find the shortest path to zero. However, this doesn't always guarantee the optimal result. E.g., if we define b(0) = 0; and for n > 0, b(n) = 1+(a(n)), then we have b(8) = 8 > A327969(8) = 6, b(12) = 7 > A327969(12) = 5, and b(15) = 9 > A327969(15) = 6.

Links

Crossrefs

Cf. A003415, A048103, A129251, A276085, A276086, A327858, A327859, A327928, A327929, A327963, A327965, A327969, A328097, A328098, A328112.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 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; };
A328099(n) = min(A003415(n),A276086(n));

A328307 a(n) tells how many numbers m there are in range 0..n such that the k-th arithmetic derivative of A276086(m) is zero for some k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 25, 25, 26, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 28, 29, 29, 29, 29, 30, 30, 31, 32, 32, 32, 33, 33, 34, 35, 35, 36, 36, 36, 37, 37, 37, 37, 37, 38
Offset: 0

Views

Author

Antti Karttunen, Oct 12 2019

Keywords

Comments

a(n) = the number of such terms m in A328116 that m <= n.
Although in principle A276086 moves any n out of the "all hope lost" zone A100716 (where A328308 is always zero), back to its complement A048103, by comparing the ratio of this and A328309 it can be seen (see the Plot2-link in the Links-section) that such a transfer actually lessens the chances that by just iterating A003415 one could reach zero from there. Note also how the effect of the primorial base representation can be clearly seen in the folds and warps of that plot.

Links

Crossrefs

Partial sums of A328306, a left inverse of A328116.
Cf. A003415, A048103, A100716, A276086, A327969, A328306, A328308, A328309.

Programs

  • PARI
    up_to = 65537;
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
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; };
A328308(n) = if(!n,1, while(n>1, n = A003415checked(n)); (n));
A328306(n) = A328308(A276086(n));
A328307list(up_to) = { my(v=vector(up_to), s=A328306(0)); for(i=1,up_to,s += A328306(i); v[i] = s); (v); };
v328307 = A328307list(up_to);
A328307(n) = if(!n,1,v328307[n]);

Formula

a(0) = 1; for n > 0, a(n) = a(n-1) + A328306(n).
For all n >= 0, a(A328116(n)) = n.

A351088 Numbers k such that A327860(k) is reachable from k by iterating the arithmetic derivative (A003415) and there are no terms with p^p-factors on the path there.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 30, 2310, 2556, 30030, 223092870
Offset: 1

Views

Author

Antti Karttunen, Feb 05 2022

Keywords

Comments

Sequence includes also the terms for which no iterations are needed (when k is already equal to A327860(k)), thus A328110 is a subsequence. The other terms (and also 1) seem to be the intersection of primorials (A002110) with sequence A099308. This includes terms A002110(A109628(n)), whose arithmetic derivatives are in A244622.
The numbers k for which A276086(k) is reachable from k by iterating A003415 form a subsequence of this sequence, but so far only one term is known: 6, for which A276086(6) = A003415(6) = 5. (See A351228). It would be interesting to know whether there are more such terms, especially terms that require more than one iteration of A003415.
Question: The eleven known terms are all sums of distinct primorials (in A276156), i.e., contain only digits 0's and 1's in primorial base. Is this a necessary property for the terms of this sequence (and also for A328110)? - Antti Karttunen, Feb 04 2024, corrected May 11 2024.

Crossrefs

Cf. A002110, A003415, A099308, A109628, A244622, A276086, A276156, A328110 (subsequence), A327860.
Cf. also A327969, A351089, A351228, A358215.

Programs

  • PARI
    A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s)); \\ Like A003415, but return zero also for n that have p^p-factor(s).
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
\\ This simple program doesn't check for any hypothetical p^p-free A003415-loops (they are so rare that they are conjectured not to exist at all):
isA351088(n) = if(!n, 1, my(g=A327860(n)); while(n>0, if(n==g, return(1)); n = A003415checked(n)); (n));

A328240 Numbers k such that the second arithmetic derivative of A276086(k) is prime.

Original entry on oeis.org

4, 12, 32, 35, 40, 46, 47, 65, 67, 68, 71, 73, 74, 76, 220, 221, 225, 226, 227, 250, 256, 257, 276, 283, 284, 420, 421, 425, 426, 436, 486, 489, 494, 2324, 2325, 2352, 2370, 2387, 2525, 2530, 2531, 2555, 2560, 2565, 2566, 2583, 2596, 2734, 2739, 2760, 2765, 2769, 2771, 2773, 2795, 2797, 2798, 2803, 4623, 4627, 4628
Offset: 1

Views

Author

Antti Karttunen, Oct 10 2019

Keywords

Comments

Numbers k for which A003415(A327860(k)) = A003415(A003415(A276086(k))) is a prime.
Numbers k such that A276086(k) is in A192192, or equally, k such that A327860(k) is in A157037.

Links

Crossrefs

Cf. A003415, A157037, A192192, A276086, A327860, A327969, A328233.
Subsequence of A328116 and of A328242.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
isA328240(n) = isprime(A003415(A327860(n)));

Formula

For all n, a(A327969(n)) <= 5.

A328313 For all such terms k in A143293 (partial sums of primorials) for which A129251(k) = 0, the term A276085(k) is included here.

Original entry on oeis.org

2, 4, 2312, 3217644767340672907899084554132
Offset: 1

Views

Author

Antti Karttunen, Oct 12 2019

Keywords

Comments

Construction: apply A276085 to the terms in the intersection of A143293 and A048103.
The next terms are quite big and can be found in the b-file. Note the nonmonotonic order, a(8) < a(5), a(6) and a(7).
A276086(a(n)) is in A143293, A276086(A276086(a(n))) is one of the primorials, A002110, and A276086(A276086(A276086(a(n)))) is a prime.
A327969(a(n)) <= 5 for all n.

Links

Crossrefs

Cf. A002110, A048103, A129251, A143293, A276085, A276086, A327969, A328243.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
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; };
k=0; for(n=1,12,t = A143293(n); u = A276085(t); if(A276086(u) == t, k++; print1(u,", ")));

A341518 Numbers k such that the primorial base representation of their arithmetic derivative does not contain digits larger than 1.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 23, 28, 29, 30, 31, 37, 41, 43, 45, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 83, 87, 89, 97, 101, 103, 107, 108, 109, 112, 113, 127, 131, 136, 137, 139, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 189, 191, 193, 197, 198, 199, 203, 209, 210, 211, 212, 217
Offset: 1

Views

Author

Antti Karttunen, Feb 28 2021

Keywords

Comments

Numbers k for which A328390(k) <= 1, numbers k such that A003415(k) is in A276156.
Numbers k such that A327859(k) = A276086(A003415(k)) is squarefree.

Links

Crossrefs

Cf. A003415, A005117, A049345, A276086, A276156, A327859, A328114, A328390.
Positions of nonzero terms in A341517.
Subsequences: A000040, A327978, A328232, A369647 (terms k where A051903(k) obtains novel values).
Cf. also A327969.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
isA341518(n) = ismaxprimobasedigit_at_most(A003415(n),1); \\ Antti Karttunen, Feb 03 2024

Formula

For all n > 2, A328390(a(n)) = A328114(A003415(a(n))) = 1.

A351067 Number of integers between the n-th and the (n+1)-th primorial such that the maximal exponent in their prime factorization is larger than the maximal digit in their primorial base expansion.

Original entry on oeis.org

0, 3, 11, 52, 291, 1681, 11506, 89347
Offset: 1

Views

Author

Antti Karttunen, Feb 02 2022

Keywords

Comments

a(n) is the number of terms of A350075 in range A002110(n) .. A002110(1+n)-1.
The ratio a(n) / A061720(n) develops as:
n = 1: 0 / 4 = 0
2: 3 / 24 = 0.125
3: 11 / 180 = 0.061111...
4: 52 / 2100 = 0.247619...
5: 291 / 27720 = 0.010498...
6: 1681 / 480480 = 0.003499...
7: 11506 / 9189180 = 0.001252...
8: 89347 / 213393180 = 0.000419...

Examples

			Between A002110(2) = 6 and A002110(3) = 30, there are exactly three numbers that satisfy the condition: 8, 9, 16, therefore a(2) = 3.

Links

Crossrefs

Cf. A002110, A051903, A276086, A328114, A350075, A351068 (partial sums), A351069.
Cf. also A327969.

Programs

Formula

a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} [A328114(k) < A051903(k)], where [ ] is the Iverson bracket.
For all n, a(n) < A351069(n).

A351069 Number of integers in range A002110(n) .. A002110(1+n)-1 such that the maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization, where A002110(n) gives the n-th primorial.

Original entry on oeis.org

3, 6, 29, 122, 633, 3587, 24091, 184924
Offset: 1

Views

Author

Antti Karttunen, Feb 02 2022

Keywords

Comments

a(n) is the number of terms of A351038 (numbers k satisfying A328114(k) <= A051903(k)) in range A002110(n) .. A002110(1+n)-1.

Examples

			There are six numbers in range A002110(2) .. A002110(3)-1 [in 6 .. (30-1)] that satisfy the condition: 6, 7, 8, 9, 12, 16, therefore a(2) = 6.

Crossrefs

Cf. A002110, A051903, A276086, A328114, A351038, A351039, A351067, A351070 (partial sums).
Cf. also A327969.

Programs

Formula

a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} A351039(k).
For all n, a(n) > A351067(n).
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