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.
%I A341606 #25 Jul 24 2022 10:44:15
%S A341606 2,4,3,1,9,5,8,5,25,7,5,27,35,49,11,3,21,125,77,121,13,7,15,55,343,
%T A341606 143,169,17,16,11,175,13,1331,221,289,19,6,81,65,539,187,2197,323,361,
%U A341606 23,10,75,625,119,1573,247,4913,437,529,29,11,63,245,2401,209,2873,391,6859,667,841,31
%N A341606 Square array A(n,k) = A017666(A246278(n,k)), read by falling antidiagonals; denominator of abundancy index as applied onto prime shift array A246278.
%C A341606 See also comments and examples in A341605.
%H A341606 Antti Karttunen, <a href="/A341606/b341606.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals</a>
%H A341606 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A341606 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A341606 A(n, k) = A017666(A246278(n, k)).
%F A341606 A(n, k) = A246278(n, k) / A355925(n, k). - _Antti Karttunen_, Jul 22 2022
%e A341606 The top left corner of the array:
%e A341606 k= 1 2 3 4 5 6 7 8 9 10 11 12
%e A341606 2k= 2 4 6 8 10 12 14 16 18 20 22 24
%e A341606 |
%e A341606 ----+--------------------------------------------------------------------------
%e A341606 1 | 2, 4, 1, 8, 5, 3, 7, 16, 6, 10, 11, 2,
%e A341606 2 | 3, 9, 5, 27, 21, 15, 11, 81, 75, 63, 39, 9,
%e A341606 3 | 5, 25, 35, 125, 55, 175, 65, 625, 245, 275, 85, 875,
%e A341606 4 | 7, 49, 77, 343, 13, 539, 119, 2401, 121, 91, 133, 3773,
%e A341606 5 | 11, 121, 143, 1331, 187, 1573, 209, 14641, 1859, 2057, 253, 17303,
%e A341606 6 | 13, 169, 221, 2197, 247, 2873, 299, 28561, 3757, 3211, 377, 2197,
%e A341606 7 | 17, 289, 323, 4913, 391, 5491, 493, 83521, 6137, 6647, 527, 93347,
%e A341606 8 | 19, 361, 437, 6859, 551, 8303, 589, 130321, 10051, 10469, 37, 157757,
%e A341606 9 | 23, 529, 667, 12167, 713, 15341, 851, 279841, 19343, 16399, 943, 352843,
%e A341606 etc.
%e A341606 Arrays A341607 and A341608 give the largest prime factor (A006530) and the number of prime factors with multiplicity (A001222) of these terms. There are nonmonotonicities in both, for example, in columns 11, 12 and 14. This is illustrated below:
%e A341606 For column 11, with successive prime shifts of 22, we obtain:
%e A341606 n sigma(n) sigma(n)/n in lowest terms,
%e A341606 A017665(n)/A017666(n)
%e A341606 ---------------------------------------------------------------------------
%e A341606 22 36 = (2^2 * 3^2), 18/11 = (2 * 3^2)/11
%e A341606 39 56 = (2^3 * 7), 56/39 = (2^3 * 7)/(3 * 13)
%e A341606 85 108 = (2^2 * 3^3), 108/85 = (2^2 * 3^3)/(5 * 17)
%e A341606 133 160 = (2^5 * 5), 160/133 = (2^5 * 5)/(7 * 19)
%e A341606 253 288 = (2^5 * 3^2), 288/253 = (2^5 * 3^2)/(11 * 23)
%e A341606 377 420 = (2^2 * 3 * 5 * 7), 420/377 = (2^2 * 3 * 5 * 7)/(13 * 29)
%e A341606 527 576 = (2^6 * 3^2), 576/527 = (2^6 * 3^2)/(17 * 31)
%e A341606 703 760 = (2^3 * 5 * 19), 40/37 = (2^3 * 5)/37 <-- A001222 drops!
%e A341606 943 1008 = (2^4 * 3^2 * 7), 1008/943 = (2^4 * 3^2 * 7)/(23 * 41)
%e A341606 -
%e A341606 On the second last row, the denominator of 760/703 (= 40/37) has only one prime factor (instead of two), namely 37, because sigma(703) has 19 as its divisor, which otherwise would be present in the denominator.
%e A341606 -
%e A341606 For column 12, with successive prime shifts of 24, we obtain:
%e A341606 n sigma(n) sigma(n)/n
%e A341606 ---------------------------------------------------------------------------
%e A341606 24 60 = (2^2 * 3 * 5), 5/2 = (5)/(2)
%e A341606 135 240 = (2^4 * 3 * 5), 16/9 = (2^4)/(3^2)
%e A341606 875 1248 = (2^5 * 3 * 13), 1248/875 = (2^5 * 3 * 13)/(5^3 * 7)
%e A341606 3773 4800 = (2^6 * 3 * 5^2), 4800/3773 = (2^6 * 3 * 5^2)/(7^3 * 11)
%e A341606 17303 20496 = (2^4 *3 *7 *61), 20496/17303 = (2^4 *3 *7 *61)/(11^3 * 13)
%e A341606 37349 42840 = (2^3 *3^2 *5 *7 *17), 2520/2197 = (2^3 * 3^2 *5 *7)/(13^3) !!
%e A341606 93347 104400 = (2^4 *3^2 *5^2 *29), 104400/93347 = (2^4 *3^2 *5^2 *29)/(17^3 *19)
%e A341606 -
%e A341606 On the second last row, the denominator of 42840/37349 (= 2520/2197) has no prime factor 17 (which would be otherwise present), because sigma(37349) has it as its divisor.
%e A341606 -
%e A341606 For column 14, with successive prime shifts of 28, we obtain:
%e A341606 n sigma(n) sigma(n)/n
%e A341606 ---------------------------------------------------------------------------
%e A341606 28 56 = (2^3 * 7), 2/1,
%e A341606 99 156 = (2^2 * 3 * 13), 52/33 = (2^2 * 13)/(3 * 11)
%e A341606 325 434 = (2 * 7 * 31), 434/325 = (2 * 7 * 31)/(5^2 * 13)
%e A341606 833 1026 = (2 * 3^3 * 19), 1026/833 = (2 * 3^3 * 19)/(7^2 * 17)
%e A341606 2299 2660 = (2^2 * 5 * 7 * 19), 140/121 = (2^2 * 5 * 7)/(11^2) <-- !!
%e A341606 3887 4392 = (2^3 * 3^2 * 61), 4392/3887 = (2^3 * 3^2 * 61)/(13^2 * 23)
%e A341606 On the second last row, the denominator of 2660/2299 (= 140/121) has no prime factor 19 (which would be otherwise present), because sigma(2299) has it as its divisor.
%e A341606 Note that if A006530 does not grow, then certainly A001222 drops.
%o A341606 (PARI)
%o A341606 up_to = 105;
%o A341606 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));
%o A341606 A017666(n) = denominator(sigma(n)/n);
%o A341606 A341606sq(row,col) = A017666(A246278sq(row,col));
%o A341606 A341606list(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] = A341606sq(col,(a-(col-1))))); (v); };
%o A341606 v341606 = A341606list(up_to);
%o A341606 A341606(n) = v341606[n];
%Y A341606 Cf. A017666, A246278.
%Y A341606 Cf. A341605 (numerators), A341626 (numerators of the columnwise first quotients of A341605/A341606), A341627 (and their denominators), A355925, A355927.
%Y A341606 Cf. A341607 (the largest prime factor in this array), A341608 (the number of prime factors, with multiplicity).
%Y A341606 Cf. also A007691, A341523, A341524.
%K A341606 nonn,frac,tabl,look
%O A341606 1,1
%A A341606 _Antti Karttunen_, Feb 16 2021