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.

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A230042 Palindromic primes with strictly increasing product of digits.

Original entry on oeis.org

2, 3, 5, 7, 181, 191, 353, 373, 383, 727, 757, 787, 797, 19891, 19991, 34843, 35753, 36563, 37573, 38783, 74747, 75557, 76667, 77977, 78787, 78887, 79997, 1987891, 1988891, 1998991, 3479743, 3487843, 3569653, 3586853, 3589853, 3689863, 3698963, 3799973
Offset: 1

Views

Author

Shyam Sunder Gupta, Oct 06 2013

Keywords

Comments

a(1)=2; a(n+1) is the smallest palindromic prime with product of digits > product of digits of a(n).

Examples

			a(6) = 191, product of digits is 9; a(7) = 353, product of digits is 45 and 45 > 9.

Links

Crossrefs

Cf. A002385, A230041, A007954, A053666.

Programs

  • Mathematica
    a = {}; t = 0; Do[z = n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[z], s = Apply[Times, IntegerDigits[z]]; If[s > t, t = s; AppendTo[a, z]]], {n, 10^4}]; a
nxt[{p_,d_}]:=Module[{n=NextPrime[p]},While[!PalindromeQ[n]||Times@@ IntegerDigits[ n]<=d,n=NextPrime[n]];{n,Times@@IntegerDigits[n]}]; NestList[nxt,{2,2},40][[All,1]] (* Harvey P. Dale, Sep 30 2018 *)

A255005 a(n) = the digit sum of prime(n) + the digit product of prime(n).

Original entry on oeis.org

4, 6, 10, 14, 3, 7, 15, 19, 11, 29, 7, 31, 9, 19, 39, 23, 59, 13, 55, 15, 31, 79, 35, 89, 79, 2, 4, 8, 10, 8, 24, 8, 32, 40, 50, 12, 48, 28, 56, 32, 80, 18, 20, 40, 80, 100, 6, 19, 39, 49, 26, 68, 15, 18, 84, 47, 125, 24, 114, 27, 61, 68, 10, 8, 16, 32, 16
Offset: 1

Views

Author

Vincenzo Librandi, Feb 12 2015

Keywords

Examples

			Prime(5)=11 and (1*1) + (1+1) = 3 so a(5) = 3.
Prime(10)=29 and (2*9) + (2+9) = 29 so a(10) = 29.

Links

Crossrefs

Subsequence of A061762.
Cf. A007605, A053666.

Programs

  • Magma
    [&*Intseq(NthPrime(n))+&+Intseq(NthPrime(n)): n in [1..80]];
  • Mathematica
    Table[Plus @@ IntegerDigits[Prime[n]] + Times @@ IntegerDigits[Prime[n]], {n, 80}] (* Bruno Berselli, Feb 12 2015 *)
dsdp[n_]:=Module[{idpn=IntegerDigits[Prime[n]]},Total[idpn]+Times@@idpn]; dsdp/@Range[70] (* Harvey P. Dale, Mar 05 2017 *)

Formula

a(n) = A007605(n) + A053666(n).

A273402 Squarefree numbers n such that n is divisible by the product of digits of prime(n).

Original entry on oeis.org

5, 6, 7, 21, 30, 105, 190, 318, 462, 1974, 1995, 3390, 10546, 26886, 86618, 733533, 3624222, 6363183, 13767810, 17334030, 37290610, 56196114, 56196174, 56198055, 56225085, 56388130, 56676030, 60985974, 84686126
Offset: 1

Views

Author

K. D. Bajpai, May 21 2016

Keywords

Examples

			30 is in the sequence because 30 = 2*3*5 that is squarefree; prime(30) = 113; 1*1*3 = 3 that divides 30.
105 is in the sequence because 105 = 3*5*7 that is squarefree; prime(105) = 571; 5*7*1 = 35 that divides 105.

Crossrefs

Cf. A005117, A005349, A007602, A038186, A053666, A013929.

Programs

  • Mathematica
    A273402 = {}; Do[p = Prime[n]; k = Times @@ IntegerDigits[p]; If[k > 0 && Divisible[n, k] && SquareFreeQ[n], AppendTo[A273402, n]], {n, 1, 10^8}]; A273402

A341634 Smallest prime whose product of digits (A007954) is the n-th 7-smooth number = A002473(n), with a(0) = 101.

Original entry on oeis.org

101, 11, 2, 3, 41, 5, 23, 7, 181, 19, 251, 43, 127, 53, 281, 29, 541, 37, 83, 11551, 139, 47, 523, 1481, 157, 149, 12451, 67, 59, 283, 11177, 2551, 239, 1187, 1453, 79, 881, 257, 89, 1553, 2851, 199, 347, 563, 1483, 277, 14551, 1753, 269, 827, 853, 15551, 367
Offset: 0

Views

Author

Bernard Schott, Feb 16 2021

Keywords

Comments

For n>=1, equals A107698 without the zeros.
101 is the smallest prime with the digit 0, so A007954(101) = 0 but as 0 is not a 7-smooth number, it is chosen a(0) = 101.

Examples

			83 is prime, A007954(83) = 8*3 = 24 that is the 18th 7-smooth number, and as no prime < 83 has a product of digits = 24, a(18) = 83.

Links

Crossrefs

Cf. A002473, A004954, A007954, A053666, A107698, A335331.

Programs

  • Mathematica
    pod[n_] := Times @@ IntegerDigits[n]; seq[max_] := Module[{sm7 = Join[{0}, Select[Range[max], Max[FactorInteger[#][[;; , 1]]] <= 7 &]], m, s, n, c, i, ind}, m = Length[sm7]; s = Table[0, {m}]; n = 1; c = 0; While[c < m, n = NextPrime[n]; i = pod[n]; If[MemberQ[sm7, i], ind = Position[sm7, i][[1, 1]]]; If[s[[ind]] == 0, c++; s[[ind]] = n]]; s]; seq[150] (* Amiram Eldar, Feb 16 2021 *)

Formula

a(n) = A107698(A002473(n)) for n>=1. - Amiram Eldar, Feb 17 2021

Extensions

More terms from Amiram Eldar, Feb 16 2021

A344127 Primes p such that (p mod s) and (p mod t) are consecutive primes, where s is the sum of the digits of p and t is the product of the digits of p.

Original entry on oeis.org

23, 29, 313, 397, 431, 661, 941, 1129, 1193, 1223, 1277, 1613, 2621, 2791, 3461, 4111, 4159, 12641, 12911, 14419, 15271, 19211, 21611, 21773, 22613, 26731, 29819, 31181, 31511, 41381, 61211, 74611, 111191, 115811, 121181, 121727, 141161, 141221, 141269, 145513, 157523, 171713, 173141, 173891
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, May 09 2021

Keywords

Comments

Since p mod 0 is not defined, the digit 0 is not allowed.

Examples

			a(3) = 313 is a term because with s = 3+1+3 = 7 and t = 3*1*3 = 9, 313 mod 7 = 5 and 313 mod 9 = 7 are consecutive primes.

Links

Crossrefs

Cf. A007605, A053666, A136251.

Programs

  • Maple
    filter:= proc(p) local L,s,t,q;
  L:= convert(p,base,10);
  s:= convert(L,`+`);
  t:= convert(L,`*`);
  if t = 0 then return false fi;
  q:= p mod s;
  isprime(q) and (p mod t) = nextprime(q)
end proc:
select(filter, [seq(ithprime(i),i=1..20000)]);
  • PARI
    isok(p) = if (isprime(p), my(d=digits(p)); vecmin(d) && isprime(q=(p%vecsum(d))) && isprime(r=(p%vecprod(d))) && (nextprime(q+1)==r)); \\ Michel Marcus, May 10 2021

A165957 Product of the digits of the n-th nonprime.

Original entry on oeis.org

1, 4, 6, 8, 9, 0, 2, 4, 5, 6, 8, 0, 2, 4, 8, 10, 12, 14, 16, 0, 6, 9, 12, 15, 18, 24, 27, 0, 8, 16, 20, 24, 32, 36, 0, 5, 10, 20, 25, 30, 35, 40, 0, 12, 18, 24, 30, 36, 48, 54, 0, 14, 28, 35, 42, 49, 56, 0, 8, 16, 32, 40, 48, 56, 64, 0, 9, 18, 27, 36, 45, 54, 72, 81, 0, 0, 0, 0, 0, 0, 0, 1, 2
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 01 2009

Keywords

Crossrefs

Cf. A018252, A053666.

Formula

a(n) = A007954(A018252(n)).

Extensions

Entries checked, keyword:base added by R. J. Mathar, Oct 05 2009

A230085 Smallest of four consecutive primes whose product of digits is equal and nonzero.

Original entry on oeis.org

336737123, 812444239, 1731191219, 2187575239, 2549315123, 2672459219, 2721498343, 2778476123, 2781452239, 2924114819, 2925926819, 3232115219, 3441686219, 3579455219, 3617846123, 3755345219, 3943951637
Offset: 1

Views

Author

Shyam Sunder Gupta, Oct 08 2013

Keywords

Examples

			336737123 is in the sequence because 336737123, 336737161, 336737213 and 336737231 are consecutive primes and the product of the digits of each = 47628.

Crossrefs

Cf. A230082, A230083, A230084, A007954, A053666.

Programs

  • Mathematica
    a = {}; m = 1; s = 1; Do[If[(y = Apply[Times, IntegerDigits[x = Prime[n]]]) == s  && s != 0,  m = m + 1; If[m > 3, AppendTo[a, Prime[n - 3]]], m = 1]; s = y, {n, 1, 200000000}]; a
pdeQ[{a_,b_,c_,d_}]:=Module[{u=Union[Times@@@(IntegerDigits/@{a,b,c,d})]}, Length[ u] ==1&&u[[1]]>0]; Transpose[Select[Partition[Prime[Range[ 19*10^7]],4,1],pdeQ]][[1]] (* Harvey P. Dale, Jan 27 2015 *)
Previous Showing 21-27 of 27 results.

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