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 1-10 of 12 results. Next

A118756 a(n) = smallest prime p such that p is the home prime (cf. A037274) of exactly n natural numbers.

Original entry on oeis.org

2, 23, 211, 379, 773, 3389, 23251, 3761, 178069, 77773, 379811, 378997, 747521, 2337691, 3789293, 3574657
Offset: 1

Views

Author

William Lindgren (william.lindgren(AT)sru.edu), Sep 04 2007, corrected Sep 15 2007

Keywords

Comments

a(17) > 10^7, a(18) = 3784463. - Robert G. Wilson v, Oct 01 2007

Examples

			23 is the home prime of both 6 and 23, thus a(2) = 23; 211 is the home prime of 4, 22 and 211, thus a(3) = 211.
More compactly: (2) -> 2, (6,23) -> 23, (4,22,211) -> 211,
(42,74,237,379) -> 379, (10,25,55,511,773) -> 773,
(118, 259, 737, 801, 1167, 3389) -> 3389,
{250, 1506, 2555, 3865, 5773, 6502, 23251} -> 23251,
{140, 332, 514, 566, 1281, 2257, 2283, 3761} -> 3761,
{4206, 7402, 10786, 16123, 23701, 25393, 67379, 137173, 178069} -> 178069,
{3786, 7262, 10078, 14513, 18417, 23631, 25039, 32449, 37877, 77773} -> 77773,
{41933, 50161, 56598, 103487, 192207, 206031, 216959, 239433, 307369, 363007, 379811} -> 379811,
{1798, 5982, 22931, 23997, 41315, 53033, 58263, 181293, 184102, 292051, 319421, 378997} -> 378997,
{722, 4938, 5718, 7646, 18929, 21919, 23823, 23953, 91277, 97941, 171409, 332647, 747521} -> 747521,
{87066, 128055, 138438, 153402, 175611, 226146, 358537, 465734, 588041, 675382, 866893, 1792003, 1564051, 2337691} -> 2337691,
{8691, 9725, 23585, 31437, 32897, 55389, 67491, 144995, 168163, 337499, 547617, 964849, 1875153, 3303841, 3789293} -> 3789293,
{5978, 10654, 27761, 40307, 47985, 111855, 156657, 172371, 202881, 250519, 357457, 379661, 407507, 488985, 1723971, 3574657} -> 3574657, etc. - _Robert G. Wilson v_, Oct 01 2007

Crossrefs

Cf. A037274, A133956, A133958, A133960, A133962, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978, A215408.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 10^7}];
lst = Sort@ lst; Table[d = n - 2; lsu = {}; Do[If[lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]] ]], {n, 188004 - d - 1}]; First@ Union@ lsu, {n, 3, 16}] (* Robert G. Wilson v, Oct 01 2007 *)

Formula

a(n) = A037274(A215408(n)). - Jonathan Sondow, Aug 09 2012

Extensions

a(7)-a(16) from Robert G. Wilson v, Sep 27 2007, Oct 01 2007

A133956 Complement of A133957.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 239, 251, 263, 269, 281, 293, 307, 349, 401, 409, 419, 421, 431, 433
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Home primes whose homeliness is 1.
Number of terms < 10^n: Pi(10^n)- {0, 2, 37, 274, 2087, 15472, 76940, ...}.

Examples

			Only {2} -> 2, {3} -> 3, etc. Whereas {6 & 23} -> 23 thus 23 has a homeliness of 2 and therefore is not a member of this sequence.

Crossrefs

Cf. A037274, A118756, A133957, A133958, A133960, A133962, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
Complement[ Prime@ Range@ 100, {23, 37, 211, 223, 227, 229, 233, 241, 257, 271, 277, 283, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 523, 541}]

A133958 Home primes whose homeliness is 2.

Original entry on oeis.org

37, 227, 229, 233, 241, 257, 271, 277, 283, 313, 317, 331, 347, 353, 359, 367, 383, 397, 523, 541, 557, 577, 719, 743, 761, 797, 1117, 1171, 1361, 1367, 1373, 1723, 1741, 1747, 1753, 1759, 1783, 1789, 1973, 1979, 1997, 2113, 2131, 2137, 2179, 2213, 2239
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 1, 26, 182, 1428, 10395, 84164, ... .

Examples

			Only {21 & 37} -> 37, etc.

Crossrefs

Cf. A037274, A118756, A133957, A133956, A133960, A133962, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 2 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n-1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[ Union@ lsu, 50]

A133960 Home primes whose homeliness is 3.

Original entry on oeis.org

211, 223, 311, 337, 373, 389, 547, 571, 1123, 1153, 1319, 1931, 2237, 2311, 2341, 2371, 2557, 2719, 2797, 2953, 2971, 3167, 3181, 3191, 3257, 3313, 3329, 3347, 3449, 3457, 3461, 3463, 3517, 3541, 3547, 3557, 3571, 3643, 3701, 3709, 3727, 3733, 3739
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 8, 65, 437, 3233, 24579, ... .

Examples

			Only {4, 22, 211} -> 211, etc.

Crossrefs

Cf. A037274, A118756, A133959, A133956, A133957, A133958, A133962, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 3 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 45]

A133962 Home primes whose homeliness is 4.

Original entry on oeis.org

379, 2347, 2383, 2389, 3331, 3359, 3373, 3719, 3767, 4397, 4759, 7331, 7457, 7523, 7547, 7823, 11251, 13883, 17137, 17317, 19157, 19181, 22367, 22397, 23131, 23173, 23197, 23311, 23593, 23677, 23767, 23911, 29101, 31063, 31123, 31181, 31189
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 1, 16, 146, 1085, 8522, ... .

Examples

			Only (42,74,237,379) -> 379, etc.

Crossrefs

Cf. A037274, A118756, A133961, A133956, A133957, A133958, A133960, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 4 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 45]

A133964 Home primes whose homeliness is 5.

Original entry on oeis.org

773, 1129, 3137, 3251, 3593, 3797, 7541, 7853, 23167, 23557, 23719, 23743, 23773, 23971, 31973, 33191, 33331, 33769, 33863, 37307, 37573, 37691, 37997, 39733, 39929, 47797, 53101, 61613, 67349, 67709, 71129, 71171, 71443, 71719, 73181
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 1, 8, 44, 417, 3299, ... .

Examples

			Only (10, 25, 55, 511, 773) -> 773, etc.

Crossrefs

Cf. A037274, A118756, A133963, A133956, A133957, A133958, A133960, A133962, A133966, A133968, A133970, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 5 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 45]

A133968 Home primes whose homeliness is 7.

Original entry on oeis.org

23251, 23761, 33797, 36389, 37643, 37951, 37967, 77711, 113779, 131777, 132749, 134129, 233347, 233617, 233881, 237547, 293863, 311677, 311821, 313619, 314627, 317743, 319547, 331319, 331907, 333139, 333271, 333457, 333791, 333911, 337517
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 0, 0, 8, 80, 667, ... .

Crossrefs

Cf. A037274, A118756, A133967, A133956, A133957, A133958, A133960, A133962, A133964, A133966, A133970, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 7 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 35]

A133970 Home primes whose homeliness is 8.

Original entry on oeis.org

3761, 13367, 31397, 132953, 233743, 233911, 313829, 333857, 333923, 337397, 337487, 337661, 337907, 357727, 370451, 373753, 374159, 375407, 375979, 379433, 379859, 383557, 397673, 397751, 397829, 399241, 409823, 473659, 616789, 733331
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 0, 1, 3, 33, 318, ... .

Crossrefs

Cf. A037274, A118756, A133969, A133956, A133957, A133958, A133960, A133962, A133964, A133966, A133968, A133972, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 8 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 35]

A133972 Home primes whose homeliness is 9.

Original entry on oeis.org

178069, 229751, 313613, 313739, 317741, 317903, 337457, 337853, 352489, 374587, 375743, 375997, 379103, 379187, 379397, 379997, 389971, 719239, 733391, 742283, 1128901, 1143113, 1148593, 1392143, 1911319, 2231051, 2331997, 2333329
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 0, 0, 0, 20, 146, ... .

Crossrefs

Cf. A037274, A118756, A133971, A133956, A133957, A133958, A133960, A133962, A133964, A133966, A133968, A133970, A133974, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 9 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 35]

A133974 Home primes whose homeliness is 10.

Original entry on oeis.org

77773, 312619, 359929, 364627, 374531, 749711, 771941, 1355021, 1389281, 1929311, 1991153, 2336263, 2337547, 2339929, 2373823, 2389853, 2397127, 3137993, 3139547, 3139733, 3139907, 3235471, 3312539, 3313631, 3316661
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2007

Keywords

Comments

Number of terms < 10^n: 0, 0, 0, 0, 1, 7, 82, ... .

Crossrefs

Cf. A037274, A118756, A133973, A133956, A133957, A133958, A133960, A133962, A133964, A133966, A133968, A133970, A133972, A133976, A133978.

Programs

  • Mathematica
    lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
d = 10 - 2; lsu = {}; Do[ If[ lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]]]], {n, 188004 - d - 1}]; Take[Union@ lsu, 35]
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