A341529 -id:A341529 - cp's OEIS frontend

A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

3, 7, 5, 5, 13, 7, 3, 7, 31, 11, 7, 5, 11, 11, 13, 7, 11, 13, 13, 19, 17, 11, 13, 13, 11, 17, 61, 19, 31, 13, 31, 17, 61, 19, 307, 23, 13, 11, 17, 13, 19, 17, 23, 127, 29, 7, 31, 71, 19, 19, 23, 29, 29, 79, 31, 13, 13, 11, 2801, 23, 61, 29, 181, 31, 67, 37, 5, 17, 31, 19, 3221, 29, 307, 31, 53, 37, 331, 41

Offset: 1

Antti Karttunen, Feb 16 2021

			The top left corner of the array:
   n=   1     2   3     4   5     6   7        8     9    10  11    12  13    14
  2n=   2     4   6     8  10    12  14       16    18    20  22    24  26    28
-----+---------------------------------------------------------------------------
   1 |  3,    7,  5,    3,  7,    7, 11,      31,   13,    7, 13,    5, 17,   11,
   2 |  5,   13,  7,    5, 11,   13, 13,      11,   31,   13, 17,    7, 19,   13,
   3 |  7,   31, 11,   13, 13,   31, 17,      71,   11,   31, 19,   13, 23,   31,
   4 | 11,   11, 13,   11, 17,   13, 19,    2801,   19,   17, 23,   13, 29,   19,
   5 | 13,   19, 17,   61, 19,   19, 23,    3221,   61,   19, 29,   61, 31,   23,
   6 | 17,   61, 19,   17, 23,   61, 29,   30941,  307,   61, 31,   19, 37,   61,
   7 | 19,  307, 23,   29, 29,  307, 31,   88741,  127,  307, 37,   29, 41,  307,
   8 | 23,  127, 29,  181, 31,  127, 37,     911,   79,  127, 41,  181, 43,  127,
   9 | 29,   79, 31,   53, 37,   79, 41,  292561,   67,   79, 43,   53, 47,   79,
  10 | 31,   67, 37,  421, 41,   67, 43,  732541,  331,   67, 47,  421, 53,   67,
  11 | 37,  331, 41,   37, 43,  331, 47,   17351,   67,  331, 53,   41, 59,  331,
  12 | 41,   67, 43,  137, 47,   67, 53,    4271, 1723,   67, 59,  137, 61,   67,
  13 | 43, 1723, 47,   43, 53, 1723, 59,  579281,  631, 1723, 61,   47, 67, 1723,
  14 | 47,  631, 53,   47, 59,  631, 61, 3500201,   61,  631, 67,   53, 71,  631,
  15 | 53,   61, 59,   53, 61,   61, 67,   14621,  409,   61, 71,   59, 73,   67,
  16 | 59,  409, 61,  281, 67,  409, 71,    5581, 3541,  409, 73,  281, 79,  409,
  17 | 61, 3541, 67, 1741, 71, 3541, 73,     181,   97, 3541, 79, 1741, 83, 3541,
  18 | 67,   97, 71, 1861, 73,   97, 79,   21491,   71,   97, 83, 1861, 89,   97,
  19 | 71,   71, 73,  449, 79,   73, 83,   26881, 5113,   79, 89,  449, 97,   83,

Cf. A006530, A246278, A341527, A341627.

Cf. also A341607.

up_to = 105;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341527(n) = denominator(A341528(n) / A341529(n));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A341628sq(row,col) = A006530(A341527(A246278sq(row,col)));
A341628list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341628sq(col,(a-(col-1))))); (v); };
v341628 = A341628list(up_to);
A341628(n) = v341628[n];

A(n,k) = A006530(A341627(n,k)) = A006530(A341527(A246278(n,k))).

A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).

Original entry on oeis.org

0, 1, 2, 10, 2, 33, 4, 74, 44, 55, 2, 278, 4, 115, 116, 490, 2, 613, 4, 498, 242, 169, 6, 1942, 92, 265, 742, 1046, 2, 1591, 6, 3086, 344, 355, 330, 4986, 4, 487, 542, 3570, 2, 3347, 4, 1638, 2326, 737, 6, 12542, 376, 2121, 716, 2546, 6, 9869, 388, 7510, 986, 943, 2, 12894, 6, 1225, 4872, 18970, 630, 5353, 4, 3498, 1492

Offset: 1

Antti Karttunen, Jul 13 2021

nonn

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346240, A347096, A347124, A347125.

Cf. also the sequences A001359, A029710, A031924 that give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));
A346239(n) = sumdiv(n,d,moebius(n/d)*A341512(d));

a(n) = Sum_{d|n} A008683(n/d) * A341512(d).
a(n) = A341512(n) - A346240(n).
a(n) = A347125(n) - A347124(n). - Antti Karttunen, Aug 25 2021

A342672 a(n) = lcm(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 20, 63, 42, 60, 88, 135, 325, 126, 156, 1260, 238, 264, 840, 2511, 342, 975, 460, 126, 1760, 468, 696, 540, 1519, 714, 1000, 5544, 930, 2520, 1184, 1701, 3120, 1026, 3696, 20475, 1558, 1380, 4760, 1890, 1806, 5280, 2068, 3276, 13650, 2088, 2544, 50220, 6897, 4557, 6840, 14994, 3186, 3000, 6552, 11880, 1840, 2790

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A341529, A342671.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342672(n) = lcm(sigma(n), A003961(n));

a(n) = lcm(A000203(n), A003961(n)).

a(n) = A341529(n) / A342671(n).

A346240 Difference between A341512 and its Möbius transform.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 11, 2, 3, 0, 46, 0, 5, 4, 85, 0, 80, 0, 68, 6, 3, 0, 398, 2, 5, 46, 130, 0, 209, 0, 575, 4, 3, 6, 981, 0, 5, 6, 640, 0, 397, 0, 182, 164, 7, 0, 2830, 4, 150, 4, 280, 0, 1435, 4, 1250, 6, 3, 0, 2586, 0, 7, 292, 3661, 6, 551, 0, 368, 8, 507, 0, 7983, 0, 5, 212, 502, 6, 847, 0, 4700, 788, 3, 0, 5078, 4, 5, 4, 1894

Offset: 1

Antti Karttunen, Jul 13 2021

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346239.

Cf. also A347097.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));
A346240(n) = -sumdiv(n,d,(dA341512(d));

a(n) = -Sum_{d|n, dA008683(n/d) * A341512(d).

a(n) = A341512(n) - A346239(n).

A351252 a(n) = sigma(n) * A276086(n), pointwise product of the sum of divisors function and the primorial base exp-function.

Original entry on oeis.org

2, 9, 24, 63, 108, 60, 80, 225, 390, 810, 1080, 700, 700, 1800, 3600, 6975, 8100, 4875, 5000, 15750, 24000, 40500, 54000, 37500, 38750, 78750, 150000, 315000, 337500, 504, 448, 1323, 2016, 3402, 6048, 3185, 2660, 6300, 11760, 28350, 26460, 16800, 15400, 44100, 81900, 113400, 151200, 108500, 99750, 244125, 378000

Offset: 1

Antti Karttunen, Feb 17 2022

nonn

Cf. A000203, A276086.

Cf. also A324580, A341529, A351458.

Array[Block[{i = 1, m = 1, n = #, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; DivisorSigma[1, #]*m] &, 51] (* Michael De Vlieger, Feb 17 2022, after Jean-François Alcover at A276086 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351252(n) = (sigma(n) * A276086(n));

a(n) = A000203(n) * A276086(n).

A378659 Positions of records for ratio A003961(n)/sigma(n), where A003961 is fully multiplicative with a(p) = nextprime(p) and sigma is the sum of the divisors function.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 16, 27, 32, 64, 128, 243, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

			  Term k       A003961(k)/A000203(k)    ratio
       1                1/1          =  1
       3                5/4          =  1.25
       4                9/7          =  1.2857143
       7               11/8          =  1.375
       8               27/15         =  1.8
       9               25/13         =  1.9230769
      16               81/31         =  2.6129032
      27              125/40         =  3.125
      32              243/63         =  3.8571429
      64              729/127        =  5.7401575
     128             2187/255        =  8.5764706
     243             3125/364        =  8.5851648
     256             6561/511        =  12.839530
     ...
33554432     847288609443/67108863   =  12625.584
67108864    2541865828329/134217727  =  18938.376

Cf. A000203, A003961.
Cf. A001359 (apparently gives the positions of successive minima of the ratio, for n > 2).
Cf. also A341529, A349627, A349628, A354827, A354828.

cp's OEIS Frontend

A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

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A346239 Möbius transform of A341512, sigma(n)*A003961(n) - n*sigma(A003961(n)).

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A342672 a(n) = lcm(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

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A346240 Difference between A341512 and its Möbius transform.

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A351252 a(n) = sigma(n) * A276086(n), pointwise product of the sum of divisors function and the primorial base exp-function.

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A378659 Positions of records for ratio A003961(n)/sigma(n), where A003961 is fully multiplicative with a(p) = nextprime(p) and sigma is the sum of the divisors function.

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A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).