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 10 results.

A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 124, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157
Offset: 1

Views

Author

Amiram Eldar, May 14 2021

Keywords

Examples

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

Links

Crossrefs

Cf. A048105, A064115, A335328, A344315.
Similar sequences: A005237, A006049, A343819, A344312, A344313.

Programs

  • Mathematica
    nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[200], nd[#] == nd[# + 1] &]

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);

A374671 Positive numbers k such that k! and (k+1)! have an equal number of infinitary divisors.

Original entry on oeis.org

8, 19, 23, 44, 45, 57, 67, 76, 80, 83, 84, 85, 105, 107, 116, 120, 123, 140, 141, 146, 158, 161, 165, 174, 177, 187, 201, 208, 214, 225, 235, 239, 241, 243, 244, 246, 247, 263, 269, 272, 277, 284, 297, 309, 315, 321, 322, 325, 337, 339, 341, 342, 344, 360, 363
Offset: 1

Views

Author

Amiram Eldar, Jul 16 2024

Keywords

Comments

Positive numbers such that k! and (k+1)! have an equal number of Fermi-Dirac factors (A064547).
Positive numbers k such that A037445(k!) = A037445((k+1)!).
Positive numbers k such that A064547(k!) = A064547((k+1)!).
Positive numbers k such that A177329(k) = A177329(k+1).

Examples

			8 is a term since A037445(8!) = A037445(9!) = 64.

Links

Crossrefs

Cf. A037445, A064547, A177329, A343819, A374672, A374673, A374674.

Programs

  • Mathematica
    s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 400], s[#] == s[# + 1] &]
  • PARI
    s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}

A374672 Numbers k such that k! has more infinitary divisors than (k+1)!.

Original entry on oeis.org

5, 9, 17, 27, 33, 34, 35, 43, 48, 51, 53, 59, 65, 68, 69, 75, 77, 87, 91, 97, 98, 99, 103, 115, 119, 125, 129, 134, 135, 139, 147, 149, 151, 155, 163, 164, 171, 179, 183, 189, 194, 195, 197, 199, 203, 211, 215, 221, 227, 229, 230, 231, 237, 245, 249, 257, 259
Offset: 1

Views

Author

Amiram Eldar, Jul 16 2024

Keywords

Comments

Numbers k such that k! has more Fermi-Dirac factors (A064547) than (k+1)!.
Numbers k such that A037445(k!) > A037445((k+1)!).
Numbers k such that A064547(k!) > A064547((k+1)!).
Numbers k such that A177329(k) > A177329(k+1).

Examples

			5 is a term since A037445(5!) = 16 > A037445(6!) = 8.

Links

Crossrefs

Cf. A037445, A064547, A177329, A343819, A374671, A374673, A374674.

Programs

  • Mathematica
    s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 300], s[#] > s[# + 1] &]
  • PARI
    s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > s2, print1(k-1, ", ")); s1 = s2);}

A343818 a(n) is the least number k such that k and k+1 both have n Fermi-Dirac factors (A064547).

Original entry on oeis.org

2, 14, 104, 2079, 21735, 3341624, 103488384, 6110171144
Offset: 1

Views

Author

Amiram Eldar, Apr 30 2021

Keywords

Comments

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), a(n) is also the least number k such that k and k+1 both have 2^n infinitary divisors.
a(9) > 2*10^11, if it exists.

Examples

			a(1) = 2 since A064547(2) = A064547(3) = 1.
a(2) = 14 since A064547(14) = A064547(15) = 2.

Crossrefs

Cf. A064547, A343819.
Similar sequences: A045920, A052215, A075036, A093548, A115186.

Programs

  • Mathematica
    fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; seq[m_] := Module[{s = Table[0, {m}], c = 0, n = 1, fd1, fd2}, fd1=fd[n]; While[c < m, fd2 = fd[++n]; If[fd1 == fd2 && fd1 <= m && s[[fd1]] == 0, s[[fd1]] = n-1; c++]; fd1=fd2]; s]; seq[5]

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);

A372693 Numbers k such that A372692(k) = A372692(k+1) > 1.

Original entry on oeis.org

7380, 18755, 24804, 25631, 26299, 27467, 32799, 44891, 49196, 49725, 50940, 53603, 59652, 64386, 71027, 79739, 85788, 89300, 94275, 103212, 105056, 105875, 124992, 129348, 132011, 138060, 141899, 147100, 149435, 155484, 158147, 164196, 170324, 175571, 181620, 184283
Offset: 1

Views

Author

Amiram Eldar, May 10 2024

Keywords

Comments

The numbers k such that A372692(k) = A372692(k+1) = 1 are in A372690.

Links

Crossrefs

Cf. A036537, A372690, A372692.
Subsequence of A068781.
A372694 is a subsequence.
Similar sequences: Cf. A002961, A064125, A293183, A306985, A343819, A348346.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 0 &)])); s[1] = 1; s[n] := s[n] = Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1);
Select[Range[10^5], (s1 = s[#]) > 1 && s1 == s[# + 1] &]
  • PARI
    s(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = apply(x -> 1 - x, binary(f[i, 2])); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > 1 && s1 == s2, print1(k - 1, ", ")); s1 = s2);}
Showing 1-10 of 10 results.

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