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.

Showing 1-6 of 6 results.

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.

A369642 Composite numbers k, not squarefree semiprimes, such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

9, 16, 28, 30, 45, 108, 112, 136, 189, 198, 210, 212, 225, 236, 244, 246, 282, 290, 361, 374, 399, 435, 507, 1480, 1940, 2132, 2212, 2308, 2356, 2524, 2655, 2766, 2802, 3018, 3054, 3501, 3590, 3771, 3938, 4225, 4454, 4755, 4809, 5005, 5763, 6123, 6771, 9024, 9936, 10295, 11881, 12221, 16296, 22491, 24389, 26865
Offset: 1

Views

Author

Antti Karttunen, Jan 31 2024

Keywords

Comments

Composite numbers k, not squarefree semiprimes, such that A327859(k) = A276086(A003415(k)) is squarefree number, or equally, k' is in A276156.
Squares that appear in this sequence: 9, 16, 225, 361, 4225, 11881, 1371241, 1635841, 225930961, 228644641, 229189321, 262083721, ...

Links

Crossrefs

Sequence A369641 without any terms of A006881.
Cf. A003415, A276086, A276156, A327859, A369647 (subsequence after its two initial terms).
Nonsquarefree terms all occur in A369639.

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); };
A369640(n) = if(n<2 || isprime(n), 0, ismaxprimobasedigit_at_most(A003415(n),1));
isA369642(n) = (((bigomega(n)>2)||(bigomega(n)>omega(n))) && A369640(n));

A369649 Numbers k in A276156 (sums of distinct primorial numbers) where the maximal exponent in the prime factorization of k attains a novel value.

Original entry on oeis.org

1, 2, 8, 9, 32, 240, 30272, 510720, 223635968, 6469693440, 6470203776, 200560520192, 200793823232, 304250487160832, 13082767811575808, 13090182069805056, 32602248665739755520, 1955964710091685625856, 117289009331951114780672, 557940862715864858896105472, 558058119122955571275235328, 40729680631838190048559235072
Offset: 1

Views

Author

Antti Karttunen, Feb 03 2024

Keywords

Examples

			                  k   factorization               max.exp         A049345(k)
                  1                                  0                 1
                  2 = 2^1,                           1,               10
                  8 = 2^3,                           3,              110
                  9 = 3^2,                           2,              111
                 32 = 2^5,                           5,             1010
                240 = 2^4 * 3 * 5,                   4,            11000
              30272 = 2^6 * 11 * 43,                 6,          1011010
             510720 = 2^8 * 3 * 5 * 7 * 19,          8,         10010000
          223635968 = 2^9 * 577 * 757,               9,       1011111110
         6469693440 = 2^12 * 3 * 5 * 7^3 * 307,     12,      10000010000
         6470203776 = 2^7 * 3 * 1151 * 14639,        7,      10010001100
       200560520192 = 2^10 * 43 * 4554881,          10,     100001001010
       200793823232 = 2^11 * 98043859,              11,     101111000010
    304250487160832 = 2^14 * 113 * 164336071,       14,   10001011010010
  13082767811575808 = 2^15 * 167 * 2390744843,      15,  100010110101110
  13090182069805056 = 2^13 * 3^4 * 5939 * 3321677,  13,  101000000010100.
Max. exp. column, which is equal to A051903(k) is most probably a permutation of nonnegative integers.
Note that the last column is equal to A007088(A369648(n)).

Links

Crossrefs

Cf. A007088, A049345, A051903, A276156, A351073, A369648.
Cf. also A369647.

Formula

a(n) = A276156(A369648(n)).

A369645 Numbers k for which the difference A051903(k) - A328114(k) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, and A328114 is the maximal digit in the primorial base expansion of n.

Original entry on oeis.org

1, 2, 8, 32, 256, 2560, 30720, 32768, 4194304, 20971520, 58720256, 234881024, 536870912, 1342177280
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2024

Keywords

Examples

			           k   factorization   max.exp.  in primorial  max digit  diff
                                             base
           1                       0,            1,       1,      -1
           2 = 2^1,                1,           10,       1,       0
           8 = 2^3,                3,          110,       1,       2
          32 = 2^5,                5,         1010,       1,       4
         256 = 2^8,                8,        11220,       2,       6
        2560 = 2^9 * 5^1,          9,       111120,       2,       7
       30720 = 2^11 * 3^1 * 5^1,  11,      1032000,       3,       8
       32768 = 2^15,              15,      1120110,       2,      13
     4194304 = 2^22,              22,     83876020,       8,      14
    20971520 = 2^22 * 5^1,        22,    231462310,       6,      16
    58720256 = 2^23 * 7^1,        23,    610501410,       6,      17
   234881024 = 2^25 * 7^1,        25,   1141710210,       7,      18
   536870912 = 2^29,              29,   296AA71010,      10,      19
  1342177280 = 2^28 * 5^1,        28,   6071712310,       7,      21.
On the penultimate row, letter "A" in the primorial base expansion stands for ten (10 in decimal), as 2^29 = 0*prime(0)# + 1*prime(1)# + 0*prime(2)# + 1*prime(3)# + 7*prime(4)# + 10*prime(5)# + 10*prime(6)# + 6*prime(7)# + 9*prime(8)# + 2*prime(9)#, where prime(n)# = A002110(n).

Links

Crossrefs

Positions of records for -A350074(n).
Cf. A002110, A049345, A051903, A328114.
Cf. also A369646, A369647.
After the initial 1, subsequence of A351038, after the two initial terms, subsequence of A350075.

Programs

  • PARI
    A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A350074(n) = (A328114(n) - A051903(n));
m=A350074(1); print1(1,", "); for(n=2,oo,x=A350074(n); if(x

A369646 Numbers k such that the difference A051903(k) - A328114(A003415(k)) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, A328114 is the maximal digit in the primorial base expansion of n, and A003415 is the arithmetic derivative.

Original entry on oeis.org

1, 8, 16, 832, 1024, 95232, 131072, 2097152, 1006632960, 1090519040
Offset: 1

Views

Author

Antti Karttunen, Feb 02 2024

Keywords

Examples

			           k   factorization    max.exp.  k' in primorial  max digit  diff
                                                  base
           1                        0,              0,        0,       0
           8 = 2^3,                 3,            200,        2,       1
          16 = 2^4,                 4,           1010,        1,       3
         832 = 2^6 * 13^1,          6,         111120,        2,       4
        1024 = 2^10,               10,         222310,        3,       7
       95232 = 2^10 * 3^1 * 31^1,  10,       10021220,        2,       8
      131072 = 2^17,               17,       23132010,        3,      14
     2097152 = 2^21,               21,      252354100,        5,      16
  1006632960 = 2^26 * 3^1 * 5^1,   26,    23194866010,        9,      17
  1090519040 = 2^24 * 5^1 * 13^1,  24,    22053155300,        5,      19.
Here k' stands for the arithmetic derivative of k, A003415(k). Primorial base expansion is obtained with A049345.

Links

Crossrefs

Positions of records for -A351097(n).
After the initial 1, a subsequence of A351098.
Cf. A002110, A003415, A049345, A051903, A328114.
Cf. also A369645, A369647.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A351097(n) = (A328114(A003415(n))-A051903(n));
m=A351097(1); print1(1,", "); for(n=2,oo,x=A351097(n); if(x

A369648 Indices of novel terms in A351073, where A351073 is the maximal exponent in the prime factorization of the numbers that are sums of distinct primorial numbers.

Original entry on oeis.org

1, 2, 6, 7, 10, 24, 90, 144, 766, 1040, 1164, 2122, 3010, 8914, 17838, 20500, 87472, 243252, 312058, 1118346, 1347998, 2167146, 3569780, 6038946
Offset: 1

Views

Author

Antti Karttunen, Feb 03 2024

Keywords

Comments

Indices k at which A351073(k) for the first time attains a new distinct value.

Examples

			See examples in A369649.

Links

Crossrefs

Cf. A002110, A051903, A276156, A351073, A369649 (corresponding values of A276156).
Cf. also A369647.

Programs

  • PARI
    A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276156(n) = { my(s=0, p=1, r=1); while(n, if(n%2, s += r); n>>=1; p = nextprime(1+p); r *= p); (s); };
A351073(n) = A051903(A276156(n));
m=Map(); for(n=1,2^25,e=A351073(n);if(!mapisdefined(m,e),mapput(m,e,n);print1(n,", ")));
Showing 1-6 of 6 results.

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