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.

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A325819 a(n) = A324213(A228058(n)).

Original entry on oeis.org

26, 84, 78, 168, 118, 146, 242, 182, 208, 276, 200, 398, 396, 322, 438, 344, 390, 412, 536, 628, 432, 338, 582, 472, 558, 840, 512, 824, 640, 726, 1022, 852, 914, 628, 744, 616, 1178, 1018, 858, 1140, 856, 760, 990, 936, 1490, 1014, 1564, 1482, 1104, 1096, 1196, 1138, 1008, 1550, 1556, 1180, 1474, 1158, 1508, 858, 2020
Offset: 1

Views

Author

Antti Karttunen, May 29 2019

Keywords

Comments

If a(n) > 2 for all n, then there are no odd perfect numbers. See also the conjectures in A324213.

Links

Crossrefs

Cf. A228058, A324213, A325809.

Programs

  • PARI
    up_to = 25000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, May 29 2019
A324213(n) = { my(s=sigma(n)); sum(i=0, s, (1==gcd(n-i, n-(s-i)))); };
A325819(n) = A324213(A228058(n));

Formula

a(n) = A324213(A228058(n)).

A325822 Terms k of A228058 for which A325814(k) is a multiple of A034460(k).

Original entry on oeis.org

477, 3725, 29161, 107797, 166753, 205409, 500837, 535277, 780625, 1610389, 5649841, 6968125, 10292809, 10633429, 24231241, 32771201, 38322857, 40028661, 104861501, 170384117, 183593125, 277405641, 326081953, 488265625, 491716541, 704531953, 797338489, 836737393, 2053219321, 2359421369, 3012238153
Offset: 1

Views

Author

Antti Karttunen, May 23 2019

Keywords

Comments

Such terms A228058(n) that A325823(n) is a divisor of A325824(n).
If any odd perfect number exists, then it must occur in this sequence.
This is not a subsequence of A325376: 107797 is the first term that does not occur there.

Links

Crossrefs

Cf. A034460, A228058, A325376, A325812, A325814, A325823, A325824.

Programs

  • PARI
    A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
for(n=1,oo, if(isA228058(n) && !(A325814(n)%A034460(n)), print1(n, ", ")));

A325823 Sum of unitary proper divisors of A228058(n): a(n) = A034460(A228058(n)).

Original entry on oeis.org

15, 23, 27, 55, 39, 39, 47, 51, 87, 43, 63, 71, 127, 63, 83, 55, 99, 67, 175, 107, 111, 63, 119, 123, 67, 95, 147, 79, 159, 99, 167, 79, 295, 87, 183, 135, 191, 203, 207, 367, 87, 99, 91, 139, 239, 243, 251, 795, 115, 267, 111, 279, 123, 287, 127, 291, 103, 303, 535, 135, 323, 139, 327, 187, 715, 111, 119, 347, 359, 363, 123, 383
Offset: 1

Views

Author

Antti Karttunen, May 23 2019

Keywords

Links

Crossrefs

Cf. A034448, A034460, A228058, A325379, A325824.
Cf. also A325319, A325320.

Programs

  • PARI
    up_to = 10000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, May 23 2019
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A325823(n) = A034460(A228058(n));

Formula

a(n) = A034460(A228058(n)).
a(n) = A325824(n) - A325379(n).

A351574 Terms of A228058 missing from A347874.

Original entry on oeis.org

117, 245, 333, 425, 549, 605, 637, 657, 725, 833, 845, 873, 981, 1025, 1053, 1325, 1413, 1421, 1445, 1629, 1737, 1805, 1813, 2009, 2057, 2061, 2169, 2225, 2493, 2525, 2597, 2645, 2817, 2825, 2873, 2925, 2989, 2997, 3033, 3141, 3357, 3425, 3509, 3573, 3577, 3681, 3725, 3789, 3897, 4113, 4205, 4325, 4361, 4693, 4753
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Comments

Numbers that satisfy Euler's criterion for the odd perfect numbers (A228058), but do not satisfy the criterion specified in A347874.

Links

Crossrefs

Setwise difference A228058 \ A347874, also setwise difference A228058 \ A386429.
Cf. A000203, A003415.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));
isA351574(n) = (isA228058(n) && !isA347874(n));

A325377 a(n) = A001065(A228058(n)), where A001065(n) gives the sum of proper divisors of n.

Original entry on oeis.org

33, 65, 81, 97, 129, 109, 161, 177, 321, 133, 225, 257, 193, 161, 305, 205, 369, 193, 253, 401, 417, 253, 449, 465, 277, 641, 561, 349, 609, 801, 641, 289, 397, 397, 705, 289, 737, 785, 801, 481, 353, 469, 385, 337, 929, 945, 977, 2241, 565, 1041, 1281, 1089, 613, 1121, 637, 1137, 481, 1185, 673, 685, 1265, 709, 1281, 421, 2717, 545, 1601
Offset: 1

Views

Author

Antti Karttunen, Apr 22 2019

Keywords

Links

Crossrefs

Cf. A001065, A228058, A325320, A325380.

Programs

  • PARI
    up_to = 25000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n];
A001065(n) = (sigma(n)-n);
A325377(n) = A001065(A228058(n));

Formula

a(n) = A001065(A228058(n)).
a(n) > A325320(n) for all n.

A325380 Numbers k in A228058 such that also A001065(k) is in A228058.

Original entry on oeis.org

801, 1377, 1773, 2525, 3725, 4689, 4753, 6309, 6425, 7209, 7677, 8577, 8957, 9477, 11133, 11225, 11493, 11925, 12393, 12429, 12789, 13077, 15381, 15777, 18873, 19269, 19845, 20025, 20629, 21213, 24201, 26073, 26721, 28037, 28989, 29277, 29961, 30037, 30213, 31925, 32553, 33273, 34425, 34677, 36369, 36441, 38725, 39249, 40329
Offset: 1

Views

Author

Antti Karttunen, Apr 22 2019

Keywords

Comments

If any odd perfect number exists, then it must occur in this sequence.

Links

Crossrefs

Cf. A001065, A228058, A325377.

Programs

  • PARI
    A001065(n) = (sigma(n)-n);
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n)&&isA228058(A001065(n)), k++; print1(n,", ")));

A325809 Let k = A228058(n). a(n) is the number of ways to partition the divisors of k into complementary subsets x and y so that the (k-Sum(x)) and (k-Sum(y)) are coprime.

Original entry on oeis.org

8, 12, 8, 16, 8, 15, 16, 8, 113, 16, 8, 15, 16, 7, 14, 8, 8, 13, 16, 15, 8, 15, 14, 8, 15, 254, 8, 16, 8, 128, 16, 16, 16, 15, 8, 15, 16, 15, 8, 16, 13, 15, 7, 13, 16, 8, 16, 43008, 8, 8, 126, 8, 15, 15, 15, 8, 16, 8, 14, 8, 15, 16, 8, 16, 60672, 15, 256, 13, 16, 7, 103, 16, 16, 8, 16, 16, 16, 8, 2015, 16, 8, 15, 16, 39093, 16
Offset: 1

Views

Author

Antti Karttunen, May 25 2019

Keywords

Comments

The smallest value known so far occurs as a(449) = 6. A228058(449) = 23837 = 11^2 * 197.

Links

Crossrefs

Cf. A228058, A325807, A325819.

Programs

  • PARI
    up_to = 25000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n];
A325807(n) = { my(divs=divisors(n), s=sigma(n),r); sum(b=0,(2^(-1+length(divs)))-1,r=sumbybits(divs,2*b);(1==gcd(n-(s-r),n-r))); };
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };
A325809(n) = A325807(A228058(n));

Formula

a(n) = A325807(A228058(n)).

A387162 Numbers k satisfying Euler's criterion for odd perfect numbers (A228058), such that sigma(k)+k is also a multiple of 3, and sigma(k) preserves the 3-adic valuation of k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 325, 801, 925, 1525, 1573, 1773, 1825, 2097, 2205, 2425, 2725, 3757, 3825, 3925, 4041, 4477, 4525, 4689, 4825, 5013, 5725, 6025, 6877, 6925, 6957, 7381, 7605, 7825, 7929, 8125, 8425, 8577, 8725, 8833, 9325, 9549, 9873, 9925, 10225, 10525, 10693, 10825, 10933, 11425, 11493, 11737, 12789, 13189, 13437, 13525
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2025

Keywords

Links

Crossrefs

Intersection of A228058 and A349752.
Subsequence of A349755 from which this differs for the first time at n=109, with a(109) = 31225, while A349755(109) = 31213.
Probably the intersection of A349755 and A386429.

Programs

  • Mathematica
    nn=275;n=1;a228058={};While[Length[a228058 ] < nn,n=n+2;{p,e}=Transpose[FactorInteger[n]];od=Select[e,OddQ];If[Length[e]>1&&Length[od]==1&&Mod[od[[1]], 4]==1&&Mod[p[[Position[e, od[[1]]][[1,1]]]],4]==1,AppendTo[a228058,n]]];lim=a228058[[-1]];a349752=Select[Range[1,lim,2],Divisible[(s=DivisorSigma[1,#])+#,3] && IntegerExponent[s,3]==IntegerExponent[#,3]&];Intersection[a228058,a349752] (* James C. McMahon, Aug 27 2025 *)
  • PARI
    isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA349752(n) = if(!(n%2), 0, my(s=sigma(n)); (0==(s+n)%3) && valuation(s, 3)==valuation(n, 3));
isA387162(n) = (isA349752(n) && isA228058(n));

A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

117, 153, 333, 369, 425, 477, 549, 637, 657, 845, 873, 909, 925, 1017, 1053, 1233, 1325, 1377, 1413, 1421, 1445, 1525, 1557, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2097, 2169, 2225, 2313, 2493, 2525, 2529, 2597, 2637, 2725, 2817, 2825, 2853, 2989, 2997, 3033, 3177, 3321, 3357, 3425, 3509, 3573, 3577, 3609, 3725
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2025

Keywords

Comments

Terms k of A228058 for which A322361(k) = A342671(k), or equally, such that A319626(k) = A349164(k).

Links

Crossrefs

Intersection of A228058 and A349174.
Union of A387166 and A387167.
Differs from its subsequence A387167 for the first time at n=201, where a(201) = 14157, while A387167(201) = 14225.
Cf. A000203, A003961, A319626, A322361, A342671, A349164.
Cf. also A371082.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA349174(n) = if(!(n%2), 0, my(u=A003961(n)); gcd(u, sigma(n))==gcd(u, n));
isA387164(n) = (isA228058(n) && isA349174(n));

A387166 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, and that satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

14157, 33525, 101025, 118825, 129605, 281025, 300713, 301725, 335405, 348525, 358925, 438525, 573525, 618525, 686025, 688205, 696725, 742577, 776025, 838125, 909225, 911025, 978525, 1046025, 1079225, 1099805, 1226025, 1293525, 1316025, 1322893, 1428889, 1451025, 1529045, 1563525, 1698525, 1721025, 1788525, 1991025, 2036025
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2025

Keywords

Links

Crossrefs

Intersection of A228058 and A349176.
Intersection of A387164 and A104210, or equally, intersection of A387164 and A349166.
Setwise difference A387164 \ A387167.
Cf. A000203, A003961.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));
isA387166(n) = (isA228058(n) && isA349176(n));
Previous Showing 11-20 of 82 results. Next

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