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 1-8 of 8 results.

A361935 Numbers k such that k and k+1 are both primitive unitary abundant numbers (definition 2, A302574).

Original entry on oeis.org

2457405145194, 2601523139214, 3320774552094, 3490250769005, 3733421997305, 3934651766045, 3954730124345, 4514767592334, 4553585751714, 4563327473705, 5226433847634
Offset: 1

Views

Author

Amiram Eldar, Mar 31 2023

Keywords

Comments

There are no more terms below 10^13.
There are no numbers k such that k and k+1 are both unitary abundant numbers with definition 1 (A302573) below 10^13.

Crossrefs

Subsequence of A034683, A302574 and A331412.
Cf. A302573.
Similar sequences: A283418, A330872.

Programs

  • Mathematica
    f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);
puabQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
Select[Import["https://oeis.org/A331412/b331412.txt", "Table"][[;; , 2]], puabQ[#] && puabQ[# + 1] &]

A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 498, 534, 582, 606, 618, 630, 642, 654, 660, 678, 726, 750, 762, 780, 786, 822, 834, 840, 894, 906, 942, 978, 990, 1002, 1014, 1020, 1038, 1074, 1086
Offset: 1

Views

Author

Amiram Eldar, Oct 12 2019

Keywords

Comments

Differs from A302574(n) at n >= 30.
Equivalently, numbers that equal to the sum of their proper unitary divisors, with one of them taken with a minus sign.
The unitary version of A111592.
The squarefree terms are also admirable numbers (A111592). The nonsquarefree terms are 150, 294, 420, 630, 660, 726, 750, 780, 840, 990, ...
The unitary abundant numbers (A034683) that are not unitary admirable numbers are: 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 924, 930, 966, ...

Examples

			150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper unitary divisors with one of them, 6, taken with a minus sign.

Links

Crossrefs

Cf. A034448, A077610, A111592, A302574.
Subsequence of A034683 and A290466.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); aQ[n_] := (ab = usigma[n] - 2n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && CoprimeQ[2*n/ab, ab/2]; Select[Range[1086], aQ]

A302573 Primitive unitary abundant numbers (definition 1): unitary abundant numbers (A034683) all of whose proper unitary divisors are unitary deficient.

Original entry on oeis.org

70, 840, 924, 1092, 1386, 1428, 1430, 1596, 1638, 1870, 2002, 2090, 2142, 2210, 2394, 2470, 2530, 2970, 2990, 3190, 3230, 3410, 3510, 3770, 4030, 4070, 4510, 4730, 5170, 5390, 5830, 13860, 15015, 16380, 17160, 18480, 19635, 20020, 21420, 21840, 21945, 22440
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2018

Keywords

Comments

The unitary analog of A071395.
Prasad & Reddy proved that n is a primitive unitary abundant number if and only if 0 < usigma(n) - 2n < 2n/p^e, where p^e is the largest prime power that divides n.

Examples

			70 is primitive unitary abundant since it is unitary abundant (usigma(70) = 144 > 2*70), and all of its unitary divisors are unitary deficient. The smaller unitary abundant numbers, 30, 42, 66, are not primitive, since in each 6 is a unitary divisor, and 6 is not unitary deficient.

References

  • J. Sandor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, p. 115.

Links

Crossrefs

Cf. A034448, A034683, A071395, A129487, A302574.

Programs

  • Mathematica
    maxPower[n_]:=Max[Power @@@ FactorInteger[n]];usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; d[n_]:=usigma[n]-2n; punQ[n_] := d[n]>0 && d[n]< 2n/maxPower[n]; Select[Range[1000], punQ]

A339940 Primitive coreful abundant numbers: coreful abundant numbers having no coreful abundant aliquot divisor.

Original entry on oeis.org

72, 108, 200, 784, 900, 1764, 1936, 2704, 2744, 4356, 4900, 6084, 9248, 10404, 11552, 12996, 16928, 19044, 26912, 30276, 34596, 47432, 49284, 60500, 60516, 61504, 66248, 66564, 79524, 84500, 87616, 99225, 101124, 107584, 113288, 118336, 125316, 133956, 141376
Offset: 1

Views

Author

Amiram Eldar, Dec 23 2020

Keywords

Comments

Analogous to A091191 as A057723 is analogous to A000203.
All the coreful abundant numbers (A308053) are multiples of terms of this sequence.

Links

Crossrefs

Cf. A057723, A091191, A302574, A307959, A308053.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); cabQ[n_] := s[n] > 2*n; pricabQ[n_] := cabQ[n] && AllTrue[Most @ Divisors[n], !cabQ[#] &]; Select[Range[10^5], pricabQ]

A372299 Primitive infinitary abundant numbers (definition 2): infinitary abundant numbers (A129656) having no proper infinitary divisors that are infinitary abundant numbers.

Original entry on oeis.org

24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 642, 654, 660, 678, 726, 756, 762, 780, 786, 822, 834, 894, 906, 924, 942, 945, 960, 978, 990
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2024

Keywords

Examples

			24 is a term since it is an infinitary abundant number and none of its proper infinitary divisors, {1, 2, 3, 4, 6, 8, 12}, are infinitary abundant numbers.
The least infinitary abundant number that is not primitive is 120. It has 3 infinitary divisors, 24, 30, and 40, that are also infinitary abundant numbers.

Links

Crossrefs

Subsequence of A129656.
A372298 is a subsequence.
Cf. A077609, A129657.
Similar sequences: A091191, A302574, A339940.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; iabQ[n_] := isigma[n] > 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], !iabQ[#] &]]; Select[Range[1000], q]
  • PARI
    isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) > 2*x, idivs(n)) == [];

A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.

Original entry on oeis.org

56, 104, 196, 304, 364, 368, 464, 532, 644, 812, 1036, 1184, 1204, 1316, 1376, 1484, 1504, 1696, 1708, 1952, 1988, 2044, 2212, 2492, 2716, 2828, 2884, 2996, 3164, 3496, 3668, 3836, 3892, 4172, 4228, 4408, 4544, 4564, 4672, 4676, 4844, 5056, 5068, 5336, 5404, 5516
Offset: 1

Views

Author

Amiram Eldar, Feb 04 2023

Keywords

Comments

If m is a term then k*m is a term of A360332 for all k in A320628.
Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.

Links

Crossrefs

Subsequence of A360332.
Cf. A320628.
Similar sequences: A006038, A091191, A249263, A302574, A360355.

Programs

  • Mathematica
    f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[6000], primQ]
  • PARI
    isab(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}
is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

A357606 Primitive terms of A357605: numbers in A357605 with no proper divisor in A357605.

Original entry on oeis.org

36, 48, 80, 120, 162, 168, 200, 224, 264, 270, 280, 300, 312, 352, 378, 392, 408, 416, 450, 456, 500, 552, 588, 594, 630, 696, 700, 702, 744, 750, 882, 888, 918, 968, 980, 984, 1026, 1032, 1050, 1088, 1100, 1128, 1216, 1232, 1242, 1272, 1300, 1372, 1416, 1452
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

Numbers k such that A162296(k) > 2*k but for all the aliquot divisors d of k (i.e., d | k, d < k), A162296(d) <= 2*d.
If k is a term then all the positive multiples of k are terms of A357605.
The least odd term is a(144) = 4725.

Examples

			36 is a term since A162296(36) = 79 > 2*36, but for all the divisors d of 36, 1, 2, 3, 4, 6, 9, 12 and 18, A162296(d) <= 2*d. E.g., A162296(18) = 28 < 2*18.

Links

Crossrefs

Cf. A162296.
Subsequence of A005101, A013929 and A357605.
Similar sequences: A091191, A302574.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; q[1] = False; primQ[n_] := q[n] && AllTrue[Most @ Divisors[n], ! q[#] &]; Select[Range[1500], primQ]

A360355 Primitive terms of A360328: terms of A360328 with no proper divisor in A360328.

Original entry on oeis.org

7425, 8415, 46035, 76725, 101475, 182655, 355725, 669735, 1411425, 1606275, 2352375, 2891295, 3592215, 3650625, 4079295, 4861575, 5053455, 5870205, 6093225, 6636465, 6920595, 7732395, 8750835, 9120375, 9783675, 9850005, 9958905, 10155375, 11298375, 11532375, 12120075
Offset: 1

Views

Author

Amiram Eldar, Feb 04 2023

Keywords

Comments

If m is a term then k*m is a term of A360328 for all k in A076610.
Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only prime-indexed prime factors.

Crossrefs

Subsequence of A360328.
Cf. A076610.
Similar sequences: A006038, A091191, A249263, A302574, A360356.

Programs

  • Mathematica
    f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[10^6], primQ]
  • PARI
    isab(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)) > 2*n;}
is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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