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.

User: Saish S. Kambali

Saish S. Kambali's wiki page.

Saish S. Kambali has authored 7 sequences.

A371123 Numbers whose decimal representation contains the digital root of the product of their digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 29, 30, 31, 34, 37, 39, 40, 41, 43, 46, 49, 50, 51, 59, 60, 61, 64, 67, 69, 70, 71, 73, 76, 79, 80, 81, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
Offset: 1

Views

Author

Saish S. Kambali, Mar 11 2024

Keywords

Comments

All numbers with a 0 digit (A011540) are terms, since their product of digits is 0.
All numbers with a 9 digit (A011539) are terms, since their product of digits is a multiple of 9 and so has digital root 9 if no 0 digits, or 0 if any 0 digit.

Examples

			29 is a term because 2*9=18 and 1+8=9 and 29 contains digit 9.

Crossrefs

Cf. A007954, A010888, A011539, A011540, A119246.

Programs

  • Mathematica
    digRoot[n_] := If[n == 0, 0, Mod[n - 1, 9] + 1]; q[n_] := Module[{d = IntegerDigits[n]}, MemberQ[d, digRoot[Times @@ d]]]; Select[Range[0, 112], q] (* Amiram Eldar, Mar 11 2024 *)

A365591 Numbers k such that Sum_{i=1..k} prime(i) + i is prime.

Original entry on oeis.org

1, 5, 8, 17, 28, 33, 40, 41, 49, 52, 64, 65, 69, 77, 92, 93, 108, 109, 120, 121, 136, 137, 140, 144, 165, 200, 201, 204, 225, 229, 265, 269, 272, 280, 292, 312, 325, 332, 337, 344, 356, 361, 369, 376, 388, 457, 464, 473, 480, 529, 541, 548, 553, 556, 573, 577
Offset: 1

Views

Author

Saish S. Kambali, Sep 10 2023

Keywords

Comments

Numbers k such that A000217(k) + A007504(k) is prime. - Robert Israel, Sep 10 2023

Examples

			2+1 = 3, which is prime, so 1 is a term.
2+1 + 3+2 + 5+3 + 7+4 + 11+5 = 43, which is prime, so 5 is a term.

Crossrefs

Cf. A000217, A007504, A014688.

Programs

  • Maple
    P:= [seq(ithprime(i),i=1..1000)]:
S:= ListTools:-PartialSums(P):
select(i -> isprime(S[i]+i*(i+1)/2),[$1..1000]); # Robert Israel, Sep 10 2023
  • Mathematica
    With[{m = 600}, Position[Accumulate[Range[m] + Prime[Range[m]]], ?PrimeQ] // Flatten] (* _Amiram Eldar, Sep 10 2023 *)
  • PARI
    isok(k) = isprime(sum(i=1, k, i+prime(i))); \\ Michel Marcus, Sep 14 2023

A364877 Numbers k such that 2*pi(k) + k is a prime number.

Original entry on oeis.org

3, 5, 9, 17, 21, 23, 25, 31, 37, 41, 43, 45, 49, 57, 61, 65, 69, 85, 89, 91, 99, 103, 107, 109, 113, 119, 121, 129, 133, 135, 143, 151, 155, 159, 163, 165, 177, 185, 187, 191, 193, 195, 201, 213, 217, 219, 231, 235, 241, 243, 247, 251, 257, 267, 269, 273, 279
Offset: 1

Views

Author

Saish S. Kambali, Aug 11 2023

Keywords

Comments

All terms of this sequence are odd.
A231232 lists the prime terms of this sequence.

Examples

			k = 17 is a term: 2*pi(17) + 17 = 14 + 17 = 31, a prime number.

Links

Crossrefs

Cf. A000720, A231232.

Programs

  • Maple
    R:= NULL: count:= 0:
m:= 0:
for k from 1 while count < 100 do
  if isprime(k) then m:= m+1 fi;
  if isprime(2*m+k) then R:= R,k; count:= count+1 fi
od:
R; # Robert Israel, Oct 16 2023
  • Mathematica
    Select[Range[280], PrimeQ[2*PrimePi[#] + #] &] (* Amiram Eldar, Aug 11 2023 *)
  • PARI
    isok(k) = isprime(2*primepi(k) + k); \\ Michel Marcus, Aug 12 2023

Extensions

More terms from Jon E. Schoenfield, Aug 11 2023

A364807 Numbers k such that abs(k - Sum_{m=2..k} pi(prime(k)/m)) is a prime number, where pi(k) is number of primes <= k.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 18, 19, 21, 26, 29, 34, 48, 50, 56, 63, 69, 79, 84, 87, 95, 97, 99, 101, 110, 111, 132, 134, 139, 149, 151, 157, 160, 163, 164, 168, 171, 187, 201, 204, 209, 220, 222, 226, 227, 231, 244, 250, 256, 258, 268, 276, 282, 290, 292, 294, 296, 306
Offset: 0

Views

Author

Saish S. Kambali, Aug 08 2023

Keywords

Comments

Inspired by Ramanujan primes A104272.
Primes in common with A104272 are 2, 29, 97, 101, 149, 151, 227, ...; of those, the first twin prime pair is (149, 151).
pi(a(n)) ~ a(n)/log_2(n), where pi(a(n)) is number of primes <= a(n).

Examples

			k=6 is a term: abs(6 - Sum_{m=2..6} pi(prime(k)/m)) = abs(6 - 3 - 2 - 2 - 1 - 1) = abs(-3) = 3, which is prime.

Crossrefs

Cf. A000720, A104272.

Programs

  • Mathematica
    Select[Range[320], PrimeQ[Abs[# - Sum[PrimePi[Prime[#]/m], {m, 2, #}]]] &] (* Amiram Eldar, Aug 08 2023 *)

A364698 Numbers k such that k! + k^2 + k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 11, 13, 1045
Offset: 1

Views

Author

Saish S. Kambali, Aug 03 2023

Keywords

Comments

a(11) > 10^4, if it exists. - Amiram Eldar, Aug 05 2023
a(11) > 2*10^4, if it exists. - Michael S. Branicky, Jul 05 2024

Examples

			For k = 2, 2! + 2^2 + 2 - 1 = 7 is a prime.

Crossrefs

Cf. A066143, A079649.

Programs

  • Mathematica
    Select[Range[1100], PrimeQ[#! + #^2 + # - 1] &] (* Amiram Eldar, Aug 03 2023 *)
  • PARI
    for (k=1,10^3,if(ispseudoprime(k! + k^2 + k - 1),print1(k,", ")))

Extensions

a(10) from Amiram Eldar, Aug 03 2023

A358028 Primes p = prime(9*t+1) such that the 9 consecutive primes prime(9*t+1) .. prime(9*t+9) arranged in a 3 X 3 array have at least 2 equal sums along the rows, columns or main diagonals.

Original entry on oeis.org

2, 29, 67, 107, 157, 257, 311, 367, 541, 599, 709, 769, 829, 967, 1021, 1549, 1741, 1811, 1879, 1973, 2609, 2677, 3019, 3541, 3677, 4051, 4217, 4271, 4517, 4597, 4663, 4931, 5227, 5303, 5399, 5449, 5623, 5683, 5839, 6079, 6229, 6301, 6361, 6451, 6949, 7253, 7351, 7477, 7537, 7589, 7673
Offset: 1

Views

Author

Saish S. Kambali, Nov 12 2022

Keywords

Comments

Primes are taken in successive blocks of 9 and arranged, for t>=0,
| prime(9*t+1) | prime(9*t+2) | prime(9*t+3) |
| prime(9*t+4) | prime(9*t+5) | prime(9*t+6) |
| prime(9*t+7) | prime(9*t+8) | prime(9*t+9) |
There are 8 lines altogether: 3 rows, 3 columns, and 2 main diagonals.
The sum of the first row is never duplicated since any other line has a greater sum.
The sum of the last row is never duplicated since any other line has a smaller sum.

Examples

			2 is a term since its block of 9 primes is
  | 2  | 3  | 5  |
  | 7  | 11 | 13 |
  | 17 | 19 | 23 |
which has among its lines (3 + 11 + 19) = (17 + 11 + 5).
67 is a term since its block of 9 primes (the 3rd block) is 67..103,
  | 67 | 71 | 73 |
  | 79 | 83 | 89 |
  | 97 | 101| 103|
which has 67+83+103 = 97+83+73.

Crossrefs

Cf. A105093.
Subsequence of A031918 (by definition).

Programs

  • Mathematica
    a = {}
row = {{1, 4, 7}, {2, 5, 8}, {3, 6, 9}};
col = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
dia = {{1, 3}, {5, 5}, {9, 7}};
Duplicates[l_] :=
Block[{i}, i[n_] := (i[n] = n; Unevaluated@Sequence[]); i /@ l]
Do[If[Duplicates[
    Flatten[{Total[Prime[row + 9 n]], Total[Prime[col + 9 n]],
      Total[Prime[dia + 9 n]]}]] != {},
  AppendTo[a, Prime[9 n + 1]]], {n, 0, 110}]
a (* Gerry Martens, Nov 12 2022 *)

Extensions

More terms from Gerry Martens, Nov 12 2022

A357679 a(n) = prime(n)*(prime(n-1) + prime(n+1)).

Original entry on oeis.org

21, 50, 112, 220, 364, 544, 760, 1104, 1566, 2046, 2664, 3280, 3784, 4512, 5618, 6726, 7686, 8844, 9940, 10950, 12324, 13944, 16020, 18430, 20200, 21424, 22684, 23980, 26668, 30988, 34584, 36990, 39754, 43210, 46206, 49298, 52812, 56112, 59858, 63366, 66970, 71434, 74884, 77224, 81192
Offset: 2

Views

Author

Saish S. Kambali, Oct 09 2022

Keywords

Comments

21 is the only semiprime term.
All terms after 21 are even.

Crossrefs

Cf. A000040, A048448, A006094.
Cf. A338529 (first differences).

Programs

  • Mathematica
    Array[Prime[# + 1] (Prime[#] + Prime[# + 2]) &, 45] (* Michael De Vlieger, Oct 09 2022 *)
  • PARI
    a(n) = prime(n)*(prime(n-1) + prime(n+1)) \\ Michel Marcus, Oct 09 2022

Formula

a(n) = A000040(n)*A048448(n).
a(n) = A006094(n) + A006094(n-1).

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