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 13 results. Next

A045983 Numbers k such that n or more consecutive integers starting at k have the same number of distinct prime divisors.

Original entry on oeis.org

1, 2, 2, 2, 54, 91, 141, 141, 44360, 48919, 218972, 526095, 526095, 526095, 17233173, 127890362, 29138958036, 118968284928, 118968284928, 585927201062, 585927201062, 585927201062, 585927201062, 313978488186061, 453918847597184, 453918847597184, 455626105596320
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

a(n) = smallest number k such that the n numbers from k through n+k-1 have the same number of prime divisors.
a(24) > 10^12. - Donovan Johnson, Mar 29 2013
a(28) > 2 * 10^15. - Toshitaka Suzuki, Jun 22 2025

Examples

			a(5) = 54 as 54, 55, 56, 57, 58 all have 2 prime divisors.

Crossrefs

Cf. A006049, A075028, A075031.

Programs

  • PARI
    v=vector(16); n=0; c1=0; for(k=1, 127890377, c2=omega(k); if(c1==c2, n++; if(v[n]==0, v[n]=k-n+1; print(n " " v[n])), n=1; c1=c2)) /* Donovan Johnson, Mar 29 2013 */

Extensions

a(18)-a(23) from Donovan Johnson, Mar 29 2013
a(24)-a(27) from Toshitaka Suzuki, Jun 22 2025

A075029 a(n) is the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are decreasing.

Original entry on oeis.org

1, 4, 45, 80, 28974, 28974, 8489103, 80314575, 5196065775, 77506573550, 166622964149389, 489289301397948
Offset: 1

Views

Author

Amarnath Murthy, Sep 02 2002

Keywords

Comments

a(11) > 10^12. - Donovan Johnson, Oct 13 2009
a(11) > 10^13. - Giovanni Resta, Jul 25 2013
a(13) > 2.64*10^15. - Jud McCranie, Mar 27 2019

Examples

			a(4) = 80, as tau(80) = 10 > tau(81) = 5 > tau(82) = 4 > tau(83) = 2, and no smaller sequence of 4 consecutive integers satisfies such a chain of inequalities.

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 45, p. 17, Ellipses, Paris 2008.

Crossrefs

Cf. A075028, A075031.
Cf. A000005 (numdiv).

Programs

  • PARI
    n=2; for(k=1, 100, while(1, m=n; s=1; while(numdiv(m)=k, print1(n-1, ", "); break); n=n+1))
  • PARI
    apply( {A075029(n)=for(k=4^(n-1),oo, my(d=numdiv(k)); d(d=numdiv(k+j)) || [k+=j-1, next(2)]); return(k))}, [1..6]) \\ For illustration - becomes slow for n >= 7. - M. F. Hasler, May 26 2025

Extensions

Edited by Ralf Stephan, Mar 21 2003
Definition corrected by Leroy Quet, Feb 23 2008
2 more terms from Lekraj Beedassy, Jul 13 2008
a(9)-a(10) from Donovan Johnson, Oct 13 2009
a(11)-a(12) from Jud McCranie, Mar 27 2019
Definition changed by Robert Israel, May 26 2025

A294528 a(n) is the smallest prime that begins a run of exactly n consecutive numbers having 2, 4, ..., 2*n divisors.

Original entry on oeis.org

2, 5, 61, 421, 1524085621
Offset: 1

Views

Author

Jon E. Schoenfield, Nov 01 2017

Keywords

Comments

No such run exists for any n > 5; for a proof, see Links.

Examples

			a(3) = 61 because 61 (prime), 62 = 2*31, and 63 = 3^2*7 have 2, 4, and 6 divisors, respectively (and 64 does not have exactly 8 divisors, so 63 is the last number in the run), and there is no smaller number having this property.
a(5) = 1524085621 because the 5 consecutive integers 1524085621..1524085625 have 2, 4, 6, 8, and 10 divisors, respectively (and 1524085626 does not have exactly 12 divisors), and there is no smaller number having this property.

Links

Crossrefs

Cf. A075028, A284596, A341213.

A075031 a(n) is the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are nonincreasing.

Original entry on oeis.org

1, 2, 20, 20, 714, 714, 25550, 90180, 142803, 809300, 27195648, 27195648, 6973441007, 37962822225, 37962822225, 114296059262, 265228019405583, 394047434860662, 2493689139940250
Offset: 1

Views

Author

Amarnath Murthy, Sep 02 2002

Keywords

Comments

tau(k) >= tau(k+1) >= ... >= tau(k+n-1).
Next term is > 2000000. - David Wasserman, May 06 2005
a(17) > 10^12. [Donovan Johnson, Oct 13 2009]
a(20) > 2.64x10^15. - Jud McCranie, Mar 27 2019

Examples

			a(3)=a(4) = 20 as tau(20) > tau(21) = tau(22) > tau(23).

Crossrefs

Cf. A075028, A075029, A075046.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
a(11)-a(16) from Donovan Johnson, Oct 13 2009
a(17)-a(19) from Jud McCranie, Mar 27 2019

A075046 a(n) = the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are in nondescending order.

Original entry on oeis.org

1, 1, 1, 1, 241, 241, 12853, 12853, 234613, 376741, 78312721, 125938261, 4019167441, 16586155153, 35237422882, 1296230533473, 42301168491121, 61118966262061
Offset: 1

Views

Author

Amarnath Murthy, Sep 03 2002

Keywords

Comments

tau(k) <= tau(k+1) <= ... <= tau(k+n-1).
a(16) > 10^12. - Donovan Johnson, Oct 13 2009
a(17) > 10^13. - Giovanni Resta, Apr 12 2017
a(19) > 2.64*10^15. - Jud McCranie, Mar 27 2019
If a(n) > 1, then A013632(a(n)) >= n. Might be useful to help speed up brute force search. - Chai Wah Wu, May 04 2017

Examples

			a(5) = 241 = a(6) as tau(241) = 2 < tau(242) = tau(243) = tau(244) = tau(245) = 6 < tau(246).

Crossrefs

Cf. A075028, A075029, A075031, A045983, A006073, A075044, A055927, A075047.

Programs

  • Mathematica
    k = 1; Do[ While[t = Table[ DivisorSigma[0, i], {i, k, k + n - 1}]; t != Sort[t], k++ ]; Print[k], {n, 1, 11}]

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
a(11) from Robert G. Wilson v, Sep 07 2003
a(12)-a(15) from Donovan Johnson, Oct 13 2009
a(16) from Fred Schneider, Mar 29 2017
a(17)-a(18) from Jud McCranie, Mar 27 2019

A188631 Numbers k such that d(k-1) < d(k) < d(k+1) where d(k) is the number of divisors of k.

Original entry on oeis.org

62, 63, 74, 164, 188, 194, 195, 207, 255, 275, 278, 279, 284, 314, 356, 362, 363, 398, 399, 404, 422, 423, 428, 455, 458, 459, 483, 494, 524, 539, 584, 614, 615, 662, 674, 692, 734, 747, 758, 759, 764, 824, 854, 867, 890, 927, 935, 944, 956, 998, 999
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 05 2011

Keywords

Links

Crossrefs

Cf. A000005, A188540, A075028.

Programs

  • Magma
    d:=func; [ n: n in [2..1000] | d(n-1) lt d(n) and d(n) lt d(n+1)]; // Bruno Berselli, Apr 05 2011
  • Mathematica
    Select[Range[2,1000], DivisorSigma[0, # - 1] < DivisorSigma[0, #] < DivisorSigma[0, # + 1] &] (* T. D. Noe, Apr 05 2011 *)

Formula

A000005(a(n)-1) < A000005(a(n)) < A000005(a(n)+1).

Extensions

Corrected and extended by T. D. Noe, Apr 05 2011

A284596 a(n) is the minimum number that is the first of n consecutive integers with an increasing number of divisors.

Original entry on oeis.org

2, 1, 73, 61, 35521, 11371, 7392171, 168776043, 1584614377, 38045133481
Offset: 1

Views

Author

Fred Schneider, Mar 29 2017

Keywords

Comments

Comment from Giovanni Resta, Apr 02 2017: (Start)
In A075028 the chain has to be at least of length k, whereas here it has to be of length exactly k.
Here a(2) = 1, because d(1)=1, d(2)=2, d(3)=2, so the first chain of 2 starts at 1.
(End)
Calculated with a brute force C++ program.
a(11) > 10^13. - Giovanni Resta, Apr 14 2017

Examples

			61 => 61^1 => 2 divisors
62 => 2^1 * 31^1 => 4 divisors
63 => 3^2 * 7 => 6 divisors
64 => 2^6 => 7 divisors
65 => 5^1 * 13^1 => 4 divisors.
So 61 is the first of four consecutive numbers with an increasing number of divisors. 65 breaks that chain. 61 is the minimum such number so it is the 4th number in the sequence.

Crossrefs

See A075028 for another version.

Extensions

Name clarified by Jon E. Schoenfield, Apr 28 2019

A364715 Numbers k such that d(k) < d(k+1) < d(k+2), where d(n) is the number of divisors of n.

Original entry on oeis.org

61, 62, 73, 163, 187, 193, 194, 206, 254, 274, 277, 278, 283, 313, 355, 361, 362, 397, 398, 403, 421, 422, 427, 454, 457, 458, 482, 493, 523, 538, 583, 613, 614, 661, 673, 691, 733, 746, 757, 758, 763, 823, 853, 866, 889, 926, 934, 943, 955, 997, 998, 1003, 1027
Offset: 1

Views

Author

Seiichi Manyama, Aug 04 2023

Keywords

Crossrefs

Cf. A000005, A075028, A364659, A364718.
Cf. A074775, A364716, A364717.

Programs

  • PARI
    isok(n) = numdiv(n)

A364716 Numbers k such that d(k) < d(k+1) < d(k+2) < d(k+3), where d(n) is the number of divisors of n.

Original entry on oeis.org

61, 193, 277, 361, 397, 421, 457, 613, 757, 997, 1213, 1237, 1453, 1657, 1867, 1873, 1933, 2137, 2347, 2593, 2797, 2917, 3013, 3183, 3217, 3361, 3427, 3481, 3517, 3697, 3721, 3805, 4057, 4083, 4177, 4261, 4603, 4621, 4717, 4771, 4813, 4957, 5029, 5041, 5101, 5107, 5223
Offset: 1

Views

Author

Seiichi Manyama, Aug 04 2023

Keywords

Crossrefs

Cf. A000005, A075028, A364662, A364719.
Cf. A074775, A364715, A364717.

Programs

  • PARI
    isok(n) = numdiv(n)

A364717 Numbers k such that d(k) < d(k+1) < d(k+2) < d(k+3) < d(k+4), where d(n) is the number of divisors of n.

Original entry on oeis.org

11371, 11372, 35521, 38281, 45613, 48121, 50821, 50822, 52321, 52322, 54421, 54422, 59341, 59342, 71821, 79621, 86873, 87181, 117841, 125737, 127852, 130021, 130022, 132051, 132206, 133396, 151082, 153221, 173221, 180001, 184973, 186481, 195541, 195542, 196171, 196172
Offset: 1

Views

Author

Seiichi Manyama, Aug 04 2023

Keywords

Crossrefs

Cf. A000005, A075028, A364720.
Cf. A074775, A364715, A364716.

Programs

  • PARI
    isok(n) = numdiv(n)
Showing 1-10 of 13 results. Next

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