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 1-7 of 7 results.

A084414 Duplicate of A069691.

Original entry on oeis.org

101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 1103, 1117, 1123, 2131, 2141, 1151, 1163, 1171, 1181, 1193, 1201, 1213, 1223, 1231, 1249, 1259, 2267, 1277, 1283, 1291, 1301, 1319, 1321, 2333, 2341, 2351, 1361, 1373, 1381, 1399, 1409
Offset: 0

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Author

Keywords

A084413 Smallest prime with "n" as central digit(s), or -1 if no such prime exists.

Original entry on oeis.org

101, 113, 2, 3, 149, 5, 163, 7, 181, 191, 1103, 11, 1123, 13, 2141, 1151, 1163, 17, 1181, 19, 1201, 1213, 1223, 23, 1249, 1259, 2267, 1277, 1283, 29, 1301, 31, 1321, 2333, 2341, 2351, 1361, 37, 1381, 1399, 1409, 41, 1423, 43, 1447, 1451, 2467, 47, 1481
Offset: 0

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Author

Zak Seidov, Jun 24 2003

Keywords

Comments

For n=5620 and n=7358, adding only one digit to both sides do not result in any prime. - Robert Price, Sep 16 2023

Examples

			a(1)=113 because 113 is the smallest prime with "1" as central digit.
a(10)=1103 because 1103 is the smallest prime with "10" as central digits.

Links

Crossrefs

Cf. A032734, A069691.

Programs

  • PARI
    findex(n) = {for (i = 1, 9, forstep (j=1, 9, 2, v = j + 10*n + i*10^(1+length(Str(n))); if (isprime(v), return (v));););}
a(n) = if (isprime(n), n, findex(n)); \\ code assumes only 1 digit on each side of 'n' will be necessary. See A032734 for numbers that need a better script. - Michel Marcus, Oct 05 2013

Formula

If n is prime, then a(n) = n, else a(n) = A069691(n). - Michel Marcus, Oct 05 2013

Extensions

Escape clause added by N. J. A. Sloane, Nov 11 2020

A038370 Smallest prime with "n^2" as central digit(s).

Original entry on oeis.org

101, 113, 149, 191, 1163, 1259, 1361, 1493, 2647, 1811, 11003, 11213, 11443, 11699, 11969, 12251, 12569, 12893, 13241, 13613, 14009, 14411, 14843, 15299, 15761, 16253, 16763, 17291, 27847, 18413, 19001, 29611, 210241, 110899, 311561, 112253, 112967
Offset: 0

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Author

Zak Seidov, Jun 25 2003

Keywords

Comments

1195 is the smallest n such that 2 digits are needed on both sides of 1195^2 = 1428025 to give prime 10142802503. - Michel Marcus, Jun 05 2013

Crossrefs

Cf. A084413, A069691.

Programs

  • PARI
    a(n) = {a = 1; ok = 0; while (!ok, lena = #Str(a); lenn = #Str(n^2); for (b=0, 10^lena - 1, p = b + 10^lena*n^2 + 10^(lenn+lena)*a; if (isprime(p), return (p)););a++;);} \\ Michel Marcus, Jun 05 2013

Extensions

Corrected by Robert G. Wilson v, Jun 27 2003
a(19) corrected by Sean A. Irvine, Jan 12 2021

A181739 Smallest semiprime with internal digits = n; or 0 if no such number exists.

Original entry on oeis.org

106, 111, 121, 133, 141, 155, 161, 177, 183, 194, 1101, 1111, 1121, 1133, 1141, 1154, 1165, 1174, 1186, 1191, 1202, 1211, 1226, 1234, 1241, 1253, 1261, 1271, 1282, 1293, 1306, 1313, 1322, 1333, 1343, 1351, 1363, 1371, 1382, 1391, 1401, 1411, 2426
Offset: 0

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Author

Jonathan Vos Post, Jan 31 2011

Keywords

Comments

This is to A069691 as semiprimes A001358 are to primes A000040.
By placing one digit on both sides of n (1..9 on the left and on the right) one gets 81 different numbers that might be semiprimes. If none of these numbers is a semiprime then a(n) = 0.
The smallest n such that a(n) = 0 is 20056492. - Donovan Johnson, Feb 01 2011
If one or more digits are allowed on both sides of n, the smallest semiprime containing 20056492 is 10200564926 = 2*5100282463.

Examples

			a(23) = 1234 = 2 * 617 has the embedded substring 1"23"4.

Links

Crossrefs

Cf. A001358, A032734, A069691.

Programs

  • Maple
    a:= proc(n) local i, j, k;
      for i to 9 do
        for j to 9 do
          k:= parse(cat(i, n, j));
          if not isprime(k) and add(t[2], t=ifactors(k)[2])=2
             then return k fi
        od
      od; return 0;
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2011

Extensions

More terms from Alois P. Heinz, Feb 01 2011

A069692 Smallest triangular number with value of the internal digits = n; or 0 if no such number exists.

Original entry on oeis.org

105, 210, 120, 136, 741, 153, 465, 171, 780, 190, 6105, 0, 1128, 30135, 2145, 5151, 3160, 1176, 4186, 0, 80200, 2211, 1225, 0, 3240, 5253, 7260, 1275, 0, 10296, 20301, 9316, 1326, 700336, 2346, 5356, 0, 1378, 7381, 0, 3403, 2415, 0, 1431, 6441, 9453
Offset: 0

Views

Author

Amarnath Murthy, Apr 06 2002

Keywords

Comments

Subset of A000217. - R. J. Mathar, Apr 07 2006

Examples

			a(13) = 30135 as the internal digits are 013 with value 13.

Crossrefs

Cf. A000217, A069691.

Programs

  • Maple
    midl := proc(n) local nshf,resul,dig ; resul := 0 ; dig := 0 ; nshf := iquo(n,10) ; while nshf >= 10 do resul := 10^dig* irem(nshf,10)+resul ; nshf := iquo(nshf,10) ; dig := dig+1 ; od ; RETURN(resul) ; end: for n from 1 to 40 do for t from 1 to 1000000 do tri := t*(t+1)/2 ; if midl(tri) = n then printf("%a,",tri) ; break ; elif t = 1000000 then printf("0,") ; fi ; od: od : # R. J. Mathar, Apr 07 2006

Extensions

Corrected and extended by R. J. Mathar, Apr 07 2006

A072348 Number of primes whose decimal representation has the form rns where r and s are single digits.

Original entry on oeis.org

15, 13, 13, 15, 14, 16, 15, 15, 13, 14, 11, 10, 12, 9, 5, 12, 10, 10, 8, 8, 12, 13, 13, 13, 9, 11, 8, 12, 12, 13, 11, 10, 11, 10, 11, 9, 9, 11, 10, 12, 5, 11, 11, 12, 12, 11, 12, 10, 11, 9, 7, 10, 14, 11, 14, 11, 9, 10, 10, 11, 11, 8, 12, 10, 12, 12, 11, 13, 11, 15, 11, 11, 10, 12
Offset: 0

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Author

Reinhard Zumkeller, Jul 17 2002

Keywords

Comments

a(2437) = 0 and a(n)>0 for n<2437, see A069691(2437) = 0;
a(1660) = 1 and a(n)>1 for n<1660, A069691(1660) = 216607;
a(n) <= 16 for all n, a(5) = 16, conjecture: a(n)<16 for n>5.

Examples

			The 16 prime sandwiches for n=5: 151, 157, 251, 257, 353, 359, 457, 557, 653, 659, 751, 757, 853, 857, 859 and 953.

Extensions

Offset corrected by Sean A. Irvine, Sep 24 2024

A084430 Smallest prime with "n^3" as central digit(s).

Original entry on oeis.org

101, 113, 181, 1277, 2647, 11251, 12161, 23431, 15121, 17291, 410009, 213319, 117281, 421973, 127447, 233759, 240967, 249131, 158329, 168599, 180001, 192611, 1106489, 1121671, 2138249, 2156251, 1175767, 1196837, 2219527, 2243893, 1270001, 3297911, 2327681
Offset: 0

Views

Author

Zak Seidov, Jun 27 2003

Keywords

Crossrefs

Cf. A038370, A084413, A069691.

Programs

  • PARI
    a(n) = {n = n^3; for (i = 1, 9, forstep (j=1, 9, 2, v = j + 10*n + i*10^(1+length(Str(n))); if (isprime(v), return (v)); ); ); } \\ Michel Marcus, Oct 05 2013

Extensions

More terms from Michel Marcus, Oct 05 2013
Showing 1-7 of 7 results.

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