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 1-5 of 5 results.

A329149 Integers that do not appear in A329147, in increasing order.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 24, 26, 28, 29, 31, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103
Offset: 1

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Author

Bernard Schott, Jan 06 2020

Keywords

Comments

As A329147 is not surjective, also numbers k such that equation A329147(x) = k has no solution.
No term in A329147 can have digits 4, 6 or 8; also no term in A329147 begins with 9, 10, 12, 15, 29, 39, 59 or 79.

Examples

			As A329147(x) = 28 has no solution, 28 is a term.
As A329147(21) = 32, 32 is not a term.

Crossrefs

Cf. A329147.

A329150 Increasingly ordered list without repetition of numbers k that are obtained, when, for nonnegative integers n, each nonzero digit d is replaced with prime(d).

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 13, 17, 19, 20, 22, 23, 25, 27, 30, 32, 33, 35, 37, 50, 52, 53, 55, 57, 70, 72, 73, 75, 77, 110, 112, 113, 115, 117, 130, 132, 133, 135, 137, 170, 172, 173, 175, 177, 190, 192, 193, 195, 197, 200, 202, 203, 205, 207, 211, 213, 217, 219, 220
Offset: 1

Views

Author

Bernard Schott, Jan 11 2020

Keywords

Comments

As A329147 is not increasing, not decreasing, not surjective, not injective, another possible name could be: Terms of A329147 in increasing order without repetition.
No term can have digits 4, 6 or 8; also, no term begins with 9, 10, 12, 15, 29, 39, 59 or 79.
First primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 53, ...
First squares are 0, 25, 225, 2025, 2500, 3025, 7225, ...
First palindromes are 0, 2, 3, 5, 7, 11, 22, 33, 55, 77, 202, 222, 232, ...

Examples

			Prime(1) = 2, prime(5) = 11, so A329147(15) = 211 and 211 is a term.

Links

Crossrefs

Cf. A329147, A329149 (complement).

Programs

  • PARI
    f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = my(list = List(), m); for (n=0, nn, m = f(n); if (m <= nn, listput(list, m));); Set(list); \\ Michel Marcus, Mar 26 2023

A361750 Terms of A329150 that have several preimages.

Original entry on oeis.org

23, 223, 230, 232, 233, 235, 237, 323, 523, 723, 1123, 1323, 1723, 1923, 2023, 2223, 2230, 2232, 2233, 2235, 2237, 2300, 2302, 2303, 2305, 2307, 2311, 2313, 2317, 2319, 2320, 2322, 2323, 2325, 2327, 2330, 2332, 2333, 2335, 2337, 2350, 2352, 2353, 2355, 2357, 2370, 2372, 2373, 2375, 2377
Offset: 1

Views

Author

Bernard Schott, Mar 23 2023

Keywords

Comments

From k = 1 to 9, A329147(k) has more than one preimage only for k = 9 since A329147(9) = A329147(12) = 23; so, all terms of this sequence have in their decimal expansion the two-digit integer '23'; it follows that when || denotes concatenation, the terms of this sequence are of the form:
X||23 as 223, 323, 523, 723, 1123, 1323, 1723, 1923, 2023, 2223, ... or,
23||Y as 230, 232, 233, 235, 237, 2300, 2302, 2303, 2305, 2307, 2311, ... or,
X||23||Y as 2230, 2232, 2233, 2235, 2237, 3230, 3232, 3233, ...
Every term of this sequence has 2^q preimages, and the smallest terms with exactly 2^q preimages for q >= 1 are 23, 2323, 232323, ... with q times the 2-digit number 23 in its decimal expansion; hence, the 8 preimages of 232323 are 999, 9912, 9129, 91212, 1299, 12912, 12129 and 121212.

Examples

			One example for each type:
23||2 = 232 = A329147(91) = A329147(121).
7||23 = 723 = A329147(49) = A329147(412).
2||23||5 = 2235 = A329147(193) = A329147(1123).
23||23 = 2323 = A329147(99) = A329147(129) = A329147(1212) = A329147(912).

Crossrefs

Cf. A329147, A329149, A329150.

Programs

  • Mathematica
    p[n_] := If[n > 0, Prime[n], 0]; seq[ndigmax_] := Module[{t = Table[FromDigits[ Flatten @ IntegerDigits @ (p /@ IntegerDigits[n]) ], {n, 1, 10^ndigmax - 1}]}, Sort @ Select[Tally[t], First[#] < 10^ndigmax && Last[#] > 1 &][[;; , 1]]]; seq[4] (* Amiram Eldar, Mar 23 2023 *)
  • PARI
    f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = {my(map = Map()); for (n=1, nn, my(m = f(n)); if (mapisdefined(map, m), mapput(map, m, 1+mapget(map, m)), mapput(map, m, 0));); my(list = List()); for (n=1, nn, if (mapisdefined(map, n), my(v = mapget(map, n)); if (#v > 0, listput(list, n)););); Vec(list);} \\ Michel Marcus, Mar 24 2023

A361821 Perfect powers in A329150.

Original entry on oeis.org

25, 27, 32, 225, 2025, 2197, 2500, 3025, 3375, 7225, 11025, 13225, 21952, 22500, 27000, 27225, 55225, 70225, 112225, 133225, 172225, 195112, 202500, 207025, 235225, 250000, 255025, 302500, 319225, 511225, 555025, 570025, 722500, 1102500, 1113025, 1177225, 1311025
Offset: 1

Views

Author

Bernard Schott, Mar 25 2023

Keywords

Comments

No term has a digit 4, 6 or 8.
Subsequences of squares are listed in Crossrefs.

Examples

			32 is a term since A329147(21) = 32 = 2^5.
2197 is a term since A329147(194) = 2197 = 13^3.
235225 is a term since A329147(123113) = 235225 = 485^2.

Crossrefs

Intersection of A001597 and A329150.
Cf. A329147.
Subsequences of squares with specified digits: A058426 (0,2,5), A053919 (2,3,5), A030485 (2,5,7), A191486 (2,3,5,7).

Programs

  • Mathematica
    p[n_] := If[n > 0, Prime[n], 0]; ppQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; seq[ndigmax_] := Module[{t = Table[FromDigits[Flatten@ IntegerDigits@ (p /@ IntegerDigits[n])], {n, 0, 10^ndigmax - 1}]}, Union@ Select[t, 0 < # < 10^ndigmax && ppQ[#] &]]; seq[6] (* Amiram Eldar, Mar 26 2023 *)
  • PARI
    f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = my(list = List(), m); for (n=0, nn, m = f(n); if ((m <= nn) && ispower(m), listput(list, m));); vecsort(Set(list)); \\ Michel Marcus, Mar 26 2023

A361820 Palindromes in A329150.

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 22, 33, 55, 77, 202, 222, 232, 252, 272, 303, 313, 323, 333, 353, 373, 505, 525, 535, 555, 575, 707, 717, 727, 737, 757, 777, 1111, 2002, 2112, 2222, 2332, 2552, 2772, 3003, 3113, 3223, 3333, 3553, 3773, 5005, 5115, 5225, 5335, 5555, 5775, 7007, 7117
Offset: 1

Views

Author

Bernard Schott, Mar 25 2023

Keywords

Comments

If m is a palindrome with no digit greater than 5 in A118597, then A329147(m) is a term, but there exist terms that are not of this form as 313, 717, ...

Examples

			232 is a term which has two preimages since A329147(91) = A329147(121) = 232.
313 = A329147(26) is a term whose preimage is not in A118597.
2002 is a term since A329147(1001) = 2002.
2112 is a term since A329147(151) = 2112.
27172 = A329147(1471) is a term whose preimage is not in A118597.

Crossrefs

Intersection of A002113 and A329150.
Cf. A118597, A329147, A361750, A361821.
Subsequences: A002276, A002277, A002279, A002281, A099814.

Programs

  • Mathematica
    p[n_] := If[n > 0, Prime[n], 0]; seq[ndigmax_] := Module[{t = Table[FromDigits[ Flatten@ IntegerDigits@ (p /@ IntegerDigits[n])], {n, 0, 10^ndigmax - 1}]}, Union@ Select[t, # < 10^ndigmax && PalindromeQ[#] &]]; seq[4] (* Amiram Eldar, Mar 26 2023 *)
  • PARI
    ispal(n) = my(d=digits(n)); d==Vecrev(d);
f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = my(list = List(), m); for (n=0, nn, m = f(n); if ((m <= nn) && ispal(m), listput(list, m));); vecsort(Set(list)); \\ Michel Marcus, Mar 26 2023
Showing 1-5 of 5 results.

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