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 1-6 of 6 results.

A344312 Number k such that k and k+1 have the same number of exponential divisors (A049419).

Original entry on oeis.org

1, 2, 5, 6, 8, 10, 13, 14, 21, 22, 24, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 120, 122, 124, 125, 129, 130, 133, 135, 137, 138, 141
Offset: 1

Views

Author

Amiram Eldar, May 14 2021

Keywords

Examples

			1 is a term since A049419(1) = A049419(2) = 1.
8 is a term since A049419(8) = A049419(9) = 2.

Links

Crossrefs

Cf. A049419.
Similar sequences: A005237, A006049, A343819, A344313, A344314.

Programs

  • Mathematica
    f[p_, e_] := DivisorSigma[0, e]; ed[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], ed[#] == ed[# + 1] &]

A344313 Number k such that k and k+1 have the same number of bi-unitary divisors (A286324).

Original entry on oeis.org

2, 3, 4, 14, 15, 20, 21, 26, 27, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 68, 74, 75, 76, 81, 85, 86, 91, 92, 93, 94, 98, 99, 104, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 145, 146, 147, 158, 171, 177, 187, 189, 201, 202, 205, 206, 212, 213, 214
Offset: 1

Views

Author

Amiram Eldar, May 14 2021

Keywords

Examples

			2 is a term since A286324(2) = A286324(3) = 2.
14 is a term since A286324(14) = A286324(15) = 4.

Links

Crossrefs

Cf. A286324, A293183, A304463.
Similar sequences: A005237, A006049, A343819, A344312, A344314.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e + 1, e]; bd[1] = 1; bd[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], bd[#] == bd[# + 1] &]

A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 242, 243, 244, 332, 387, 507, 548, 603, 604, 724, 735, 819, 844, 908, 931, 963, 1035, 1075, 1083, 1196, 1251, 1274, 1275, 1324, 1412, 1449, 1467, 1556, 1587, 1665, 1675, 1772, 1924, 1925, 1952, 1988, 2324, 2331, 2511, 2523, 2524, 2540
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

First differs from A049103 at n=17.
Numbers k such that A348341(k) = A348341(k+1) > 0.
The terms are restricted to have a positive number of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

Examples

			44 is a term since A348341(44) = A348341(45) = 2 > 0.

Links

Crossrefs

Cf. A036537, A049103, A348341, A348346.
Subsequence of A162643.
Similar sequences: A005237, A006049, A343819, A344312, A344313, A344314.

Programs

  • Mathematica
    nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Select[Range[2500],(nid1 = nid[#]) > 0 && nid1 == nid[# + 1] &]
  • PARI
    A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])));
isA348345(n) = { my(u=A348341(n)); (u>0&&(A348341(1+n)==u)); }; \\ Antti Karttunen, Oct 13 2021

A355710 Numbers k such that k and k+1 have the same number of 5-smooth divisors.

Original entry on oeis.org

2, 21, 33, 34, 38, 57, 85, 86, 93, 94, 104, 116, 122, 141, 145, 146, 154, 158, 171, 177, 182, 189, 201, 205, 213, 214, 218, 237, 265, 266, 273, 296, 302, 321, 326, 332, 334, 338, 344, 357, 362, 381, 385, 387, 393, 394, 398, 417, 445, 446, 453, 454, 475, 476, 482
Offset: 1

Views

Author

Amiram Eldar, Jul 15 2022

Keywords

Comments

Numbers k such that A355583(k) = A355583(k+1).

Examples

			2 is a term since A355583(2) = A355583(3) = 2.

Links

Crossrefs

Cf. A355583, A355709 (3-smooth analog).
Subsequences: A355711, A355712.
Similar sequences: A005237, A006049, A332386, A343819, A344312, A344313, A344314, A348345.

Programs

  • Mathematica
    s[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Select[Range[500], s[#] == s[#+1] &]
  • PARI
    s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); for(k = 2, 500, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

A344315 a(n) is the least number k such that A048105(k) = A048105(k+1) = 2*n, and 0 if it does not exist.

Original entry on oeis.org

1, 27, 135, 2511, 2295, 6975, 5264, 12393728, 12375, 2200933376, 108224, 257499, 170624, 3684603215871, 4402431, 2035980763136, 126224, 41680575, 701443071, 46977524, 1245375, 2707370000, 4388175, 3129761024, 1890944
Offset: 0

Views

Author

Amiram Eldar, May 14 2021

Keywords

Comments

There are no two consecutive numbers with an odd number of non-unitary divisors, since A048105(k) is odd only if k is a perfect square.
a(25) <= 1965640805422351777791, a(26) <= 3127059999. In general, a(n) <= A215199(n+1). - Daniel Suteu, May 20 2021

Examples

			a(0) = 1 since A048105(1) = A048105(2) = 0.
a(1) = 27 since A048105(27) = A048105(28) = 2.

Crossrefs

Cf. A048105, A075036, A093548, A309181, A344314.

Programs

  • Mathematica
    nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; seq[max_] := Module[{s = Table[0, {max}], k = 2, c = 0, nd1 = 0}, While[c < max, If[(nd2 = nd[k]) == nd1 && nd2 < 2*max && s[[nd2/2 + 1]] == 0, c++; s[[nd2/2 + 1]] = k - 1]; nd1 = nd2; k++]; s]; seq[7]
  • PARI
    A048105(n) = numdiv(n) - 2^omega(n);
isok(n,k) = A048105(k) == 2*n && A048105(k+1) == 2*n;
a(n) = for(k=1, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, May 16 2021

Extensions

a(13)-a(24) confirmed by Martin Ehrenstein, May 20 2021

A355709 Numbers k such that k and k+1 have the same number of 3-smooth divisors.

Original entry on oeis.org

2, 14, 21, 33, 38, 44, 50, 57, 69, 74, 80, 86, 93, 99, 105, 110, 116, 122, 129, 135, 141, 146, 158, 165, 171, 177, 182, 194, 201, 213, 218, 230, 237, 249, 254, 260, 266, 273, 285, 290, 296, 302, 309, 315, 321, 326, 332, 338, 345, 357, 362, 374, 381, 387, 393, 398
Offset: 1

Views

Author

Amiram Eldar, Jul 15 2022

Keywords

Comments

Numbers k such that A072078(k) = A072078(k+1).
This sequence is infinite since it includes all the numbers of the form 3*(2^(2*k+1)-1), with k>=1.

Examples

			2 is a term since A072078(2) = A072078(3) = 2.

Links

Crossrefs

Cf. A072078, A355710 (5-smooth analog).
Similar sequences: A005237, A006049, A332386, A343819, A344312, A344313, A344314, A348345.

Programs

  • Mathematica
    s[n_] := Times @@ (1 + IntegerExponent[n, {2, 3}]); Select[Range[400], s[#] == s[#+1] &]
  • PARI
    s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1);
s1 = s(1); for(k = 2, 400, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);
Showing 1-6 of 6 results.

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