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 1-4 of 4 results.

A225669 Slowest-growing sequence of odd primes whose reciprocals sum to 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 967, 101419, 2000490719, 106298338760698351, 586903266015193517540253132922939, 3494365451928289992209032562272585187947069047023572601254975717
Offset: 1

Views

Author

Jonathan Sondow, May 11 2013

Keywords

Comments

See comments, references, and links in A075442 = slowest-growing sequence of primes whose reciprocals sum to 1.
a(n) = 3, 5, 7, 11, 13, 17, 19, 23, 967, ..., so A225671(2) = 23.

Examples

			Since 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 < 1, the first eight odd primes are members. The ninth is not, because adding 1/29 pushes the sum over 1.

References

  • Popular Computing (Calabasas, CA), Problem 175: A Sum of a Different Kind, Vol. 5 (No. 50, May 1977), p. PC50-8.

Links

Crossrefs

Cf. A000058, A075442, A046689, A136616, A181503, A225670, A225671.

Programs

  • Mathematica
    a[n_] := a[n] = Block[{sm = Sum[1/(a[i]), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 2; Array[a, 14]

A225670 Slowest-growing sequence of odd primes p where 1/(p+1) sums to 1 without actually reaching it.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 2539, 936599, 127852322431, 510819260848900502567, 1553192364608434843485965159509450536731, 52119893982548112392303882371161186032080710958633917215400463948724068502699
Offset: 1

Views

Author

Jonathan Sondow, May 11 2013

Keywords

Comments

Is there a finite set of odd primes p where 1/(p+1) sums exactly to 1? (This would be an analog of 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(11+1) + 1/(23+1) = 1 -- see A000058.)

Links

Crossrefs

Similar to A075442, A181503, A225669.
Cf. A000058.
See also A046689.

Programs

  • Mathematica
    a[n_] := a[n] = Block[ {sm = Sum[ 1/(a[i] + 1), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 2; Array[ a, 20]

A226527 Slowest-growing sequence of 3-almost primes (trientprimes) where 1/(tp+1) sums to 1 without actually reaching it.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244, 245, 246, 255, 258, 261, 266, 268, 273, 275, 279, 282, 284, 285, 286, 290, 292, 310, 316, 318, 322, 325, 332, 333, 338, 343, 345, 354, 356, 357, 363, 366, 369, 370, 374, 385, 387, 388, 399, 402, 404, 406, 410, 412, 418, 423, 425, 426, 428, 429, 430, 434, 435, 436, 8662, 44335708, 1251938572491943, 1505273212784203338150808798466, 680617602541158152398258079780439819108542271775727566330763
Offset: 1

Views

Author

Aaron Meyerowitz, Jonathan Vos Post, and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

See comments in A226526.
Deviates from A014612 after the 110th term.

Crossrefs

Cf. A181503, A226526.

Programs

  • Mathematica
    kAlmostPrimeQ[n_, k_: 2] := Plus @@ Last /@ FactorInteger@ n == k (* For those who have Mmca v or later, you could use PrimeOmega@ n == k *) NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextkAlmostPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]

A226526 Slowest-growing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 69, 1497, 259465, 4852747709, 3429487924785490781, 305153651313989042415043589313598477, 21932475414742921908206321699222250910796483151080020353252738457741771
Offset: 1

Views

Author

Aaron Meyerowitz, Jonathan Vos Post, and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

The semiprime analogous to A181503.
Because the semiprimes are sparser than the primes in the beginning, the sequence contains more of the lesser semiprimes than the analogous sequence of primes. In fact, one has to get to the seventeenth semiprime before it, 49,is not present, whereas in A181503, one only has to get to the sixth prime before it, 13, is not present.
If you change 1/(a(n)+1) to simply 1/a(n) the sequence becomes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 355, 16627, 76723511, 17218740226618333, 374886275842473712491638217368219, 9036922116709843444667289331349853231276337589593114741410804131,....

Examples

			1/(4+1) + 1/(6+1) + 1/(9+1) + … 1/(46+1) + 1/(69+1) is still less than 1. Instead of 1/69, if one were to use any semiprime between 46 and 69, {} the sum would then exceed 1.

Crossrefs

Cf. A181503, A226527.

Programs

  • Mathematica
    semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2 (* For those who have Mmca v or later, you could use PrimeOmega@ n == 2 *) NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextSemiPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]
Showing 1-4 of 4 results.

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