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.

A328770 Numbers in whose primorial base expansion any digit is at most half of the maximal allowed digit for that position.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 30, 32, 36, 38, 42, 44, 60, 62, 66, 68, 72, 74, 90, 92, 96, 98, 102, 104, 210, 212, 216, 218, 222, 224, 240, 242, 246, 248, 252, 254, 270, 272, 276, 278, 282, 284, 300, 302, 306, 308, 312, 314, 420, 422, 426, 428, 432, 434, 450, 452, 456, 458, 462, 464, 480, 482, 486, 488, 492, 494, 510, 512, 516, 518, 522
Offset: 1

Views

Author

Antti Karttunen, Oct 31 2019

Keywords

Comments

Equally, numbers in whose primorial base expansion there are no digits more than ((prime(k)-1)/2), where prime(k) is the modulus for the digit position k = 1 + maximal allowed digit for that position.
Differs from A276154, for example, this sequence does not contain term 120.

Examples

			2 is included, as in the primorial base (A049345) it is written as "10", thus 2 is included in the sequence as the maximal value that can occur in the second rightmost digit (in the primorial base representation) is 2 (as in "20" = 4 or "21" = 5 for example).

Links

Crossrefs

Subsequence of A276154 (because of Bertrand's postulate).
Cf. A002110, A049345, A276086, A328834, A328849.

Programs

  • Mathematica
    q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; AllTrue[s/(Prime[Range[1, Length[s]]] - 1), # <= 1/2 &]]; Select[Range[0, 600], q] (* Amiram Eldar, Mar 13 2024 *)
  • PARI
    isA328770(n) = { my(p=2); while(n, if((n%p)>((p-1)/2), return(0)); n = n\p; p = nextprime(1+p)); (1); };

Formula

a(n) = A328849(n)/2.
Because doubling these numbers in primorial base does not generate any carries, it follows that:
A276086(a(n)+a(n)) = A276086(a(n)) * A276086(a(n)) = A328834(n)^2.

A379963 Numbers k such that A276086(k)+1 is a perfect square (A000290), where A276086 is the primorial base exp-function.

Original entry on oeis.org

2, 8, 34, 36, 214, 248, 254, 2318, 2350, 2520, 2564, 2776, 5076, 30038, 30092, 30480, 32374, 510542, 510728, 510746, 512886, 515134, 540540, 540818, 542862, 542888, 1021442, 9699702, 9699722, 9699772, 9699788, 9702010, 9702256, 9729938, 9734358, 10210414, 10217558, 10240472, 10240724, 19401924, 19429870, 19912238
Offset: 1

Views

Author

Antti Karttunen, Jan 24 2025

Keywords

Examples

			A276086(34) = 63, +1 = 64 = 8^2, therefore 34 is included.
A276086(36) = 35, +1 = 36 = 6^2, therefore 36 is included.

Links

Crossrefs

Subsequence of A379962.
Cf. A000290, A002110, A276086.
Cf. also A379965 and A328849 (numbers k such that A276086(k) is a square).

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A379963(n) = issquare(1+A276086(n));

A328834 Square root of the prime factor form (A276086) of the primorial base expansion, computed for such numbers for which it is a square.

Original entry on oeis.org

1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 49, 147, 245, 735, 1225, 3675, 343, 1029, 1715, 5145, 8575, 25725, 11, 33, 55, 165, 275, 825, 77, 231, 385, 1155, 1925, 5775, 539, 1617, 2695, 8085, 13475, 40425, 3773, 11319, 18865, 56595, 94325, 282975, 121, 363, 605, 1815, 3025, 9075, 847, 2541, 4235, 12705, 21175, 63525, 5929, 17787
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2019

Keywords

Comments

All terms are odd.

Links

Crossrefs

Cf. A000196, A276086, A328849, A328770.

Programs

  • Mathematica
    With[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Select[Array[Sqrt@ Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[#, b] &, 10^3, 0], IntegerQ]] (* Michael De Vlieger, Oct 30 2019 *)

Formula

a(n) = A276086(A328770(n)) = A000196(A276086(A328849(n))).

A328850 Squares in whose primorial base expansion only even digits appear.

Original entry on oeis.org

0, 4, 16, 64, 144, 196, 484, 900, 1024, 1444, 1764, 2116, 2304, 4624, 5184, 5476, 6084, 6724, 13924, 14400, 14884, 18496, 19044, 20164, 23104, 23716, 24964, 28224, 29584, 61504, 65536, 66564, 70756, 73984, 79524, 80656, 85264, 88804, 90000, 121104, 131044, 135424, 139876, 186624, 195364, 204304, 209764, 242064, 260100, 264196
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2019

Keywords

Comments

Squares in A328849, squares such that also the prime factor form (A276086) of their primorial base expansion is a square,

Examples

			12^2 = 144 is written as "4400" in primorial base (A049345), as 4*A002110(3) + 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*30 + 4*6 = 144, thus its prime code encoding, A276086(144) = prime(4)^4 * prime(3)^4 = 7^4 * 5^4 = 1500625 is also a square, and 144 is included in this sequence.

Links

Crossrefs

Cf. A328838 (gives the square roots).
Intersection of A000290 and A328849.
Cf. A002110, A010052, A049345, A276086.

Programs

  • Mathematica
    q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; AllTrue[s, EvenQ]]; Select[Range[0, 520]^2, q] (* Amiram Eldar, Mar 06 2024 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328850(n) = (issquare(n) && issquare(A276086(n)));

Formula

a(n) = A000290(A328838(n)).

A328838 Numbers such that in the primorial base expansion of their squares only even digits appear.

Original entry on oeis.org

0, 2, 4, 8, 12, 14, 22, 30, 32, 38, 42, 46, 48, 68, 72, 74, 78, 82, 118, 120, 122, 136, 138, 142, 152, 154, 158, 168, 172, 248, 256, 258, 266, 272, 282, 284, 292, 298, 300, 348, 362, 368, 374, 432, 442, 452, 458, 492, 510, 514, 548, 558, 562, 574, 608, 616, 652, 660, 698, 704, 708, 1018, 1020, 1042, 1054, 1080, 1082, 1096, 1124
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2019

Keywords

Examples

			For n = 4, its square 16 is written as "220" in primorial base (A049345), as 2*A002110(2) + 2*A002110(1) + 0*A002110(0) = 2*6 + 2*2 = 16, thus 4 is included in this sequence.

Links

Crossrefs

Cf. A000196, A002110, A010052, A049345, A276086, A328849, A328850.

Programs

  • Mathematica
    q[n_] := Module[{k = n^2, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; AllTrue[s, EvenQ]]; Select[Range[0, 1200], q] (* Amiram Eldar, Mar 06 2024 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328838(n) = (issquare(A276086(n*n)));

Formula

a(n) = A000196(A328850(n)).

A363242 Numbers whose primorial-base representation contains only odd digits.

Original entry on oeis.org

1, 3, 9, 21, 39, 51, 99, 111, 159, 171, 249, 261, 309, 321, 369, 381, 669, 681, 729, 741, 789, 801, 1089, 1101, 1149, 1161, 1209, 1221, 1509, 1521, 1569, 1581, 1629, 1641, 1929, 1941, 1989, 2001, 2049, 2061, 2559, 2571, 2619, 2631, 2679, 2691, 2979, 2991, 3039
Offset: 1

Views

Author

Amiram Eldar, May 23 2023

Keywords

Comments

All the terms above 1 are odd multiples of 3.
The partial sums of the primorials (A143293) are terms, since the primorial-base representation of A143293(n) is n+1 1's.

Examples

			3 is a term since its primorial-base presentation, 11, has only odd digits.
21 is a term since its primorial-base presentation, 311, has only odd digits.

Links

Crossrefs

Cf. A002110, A049345, A328849.
Subsequence: A143293.
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary), A351894 (factorial base).

Programs

  • Mathematica
    With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[1, nmax, 2], AllTrue[prmBaseDigits[#], OddQ] &]]
  • PARI
    is(n) = {my(p = 2); if(n < 1, return(0)); while(n > 0, if((n%p)%2 == 0, return(0)); n \= p; p = nextprime(p+1)); return(1);}
Showing 1-6 of 6 results.

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