A060990 -id:A060990 - cp's OEIS frontend

A262897 Nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent: a(n) = A259934(A262892(n)).

2, 12, 18, 30, 42, 54, 90, 94, 106, 121, 190, 194, 210, 236, 242, 254, 298, 302, 342, 346, 354, 366, 374, 390, 410, 426, 442, 466, 494, 530, 546, 558, 562, 566, 574, 606, 650, 658, 710, 716, 730, 746, 914, 942, 986, 1030, 1038, 1042, 1052, 1058, 1090, 1114, 1134, 1146, 1240, 1250, 1266, 1278, 1286, 1310, 1354, 1370, 1378, 1418, 1426, 1450, 1454, 1490, 1562, 1650, 1662, 1670, 1676, 1694, 1706

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

Equally, numbers n for which A060990(n)*A262693(n) = 1, thus an intersection of A262511 and A259934.

The next odd term after a(10) = 121 occurs at a(3372) = 113569.

Antti Karttunen, Table of n, a(n) for n = 1..4085

Cf. A000005, A049820, A060990, A259934, A262511, A262693, A262892, A262896.

Cf. A262510 (a subsequence).

a(n) = A259934(A262892(n)).

A262903 Numbers that are not leaves but all of whose children are leaves in the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

4, 5, 14, 16, 32, 41, 44, 77, 80, 92, 101, 110, 119, 128, 139, 148, 158, 161, 169, 176, 191, 192, 199, 215, 224, 227, 234, 238, 249, 262, 264, 277, 296, 311, 317, 327, 350, 351, 352, 360, 363, 382, 385, 389, 392, 395, 396, 411, 427, 430, 437, 448, 449, 461, 464, 483, 488, 518, 523, 531, 532, 542, 552, 561, 568, 570, 577, 579, 600, 601, 613, 619, 632, 634, 636, 645, 648, 659, 665, 666, 671, 682, 683, 696, 705, 723

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

Numbers n for which A060990(n) > 0 and A060990(n) = A262900(n).

Numbers n for which A262695(n) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A049820, A060990, A262695, A262900.
Subsequence of A262901 and A236562.
No common terms with A259934.
Cf. also A257512.

A263091 Primes p for which A049820(x) = p has no solution.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 103, 109, 113, 131, 163, 167, 193, 229, 241, 251, 257, 271, 293, 307, 313, 353, 359, 379, 383, 397, 401, 439, 463, 479, 487, 491, 499, 503, 509, 563, 571, 647, 653, 661, 673, 701, 739, 743, 757, 761, 773, 823, 859, 863, 883, 887, 911, 937, 941, 953, 967, 971, 977, 983, 1009, 1093, 1103, 1109, 1171, 1181, 1193, 1217, 1279, 1283, 1291, 1297, 1307, 1321, 1361

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Primes p that there is no such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).

Antti Karttunen, Table of n, a(n) for n = 1..15383

Complement among primes: A263090.
Intersection of A000040 and A045765.
Subsequence of A067774 (A049591).
Cf. A000005, A049820, A060990.

lim = 10000; s = Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], PrimeQ]; Take[s, 76] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem(123456789);
uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
v060990 = vector(uplim1);
for(n=3, uplim1, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
n=0; forprime(p=2, 524287, if((0 == A060990(p)), n++; write("b263091.txt", n, " ", p)));

;; With Antti Karttunen's IntSeq-library.
(define A263091 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (zero? (A060990 n))))))

A263093 Numbers whose squares are in A045765.

Original entry on oeis.org

5, 6, 7, 8, 10, 14, 16, 18, 20, 22, 26, 27, 28, 34, 35, 37, 46, 47, 50, 54, 56, 58, 59, 60, 62, 67, 73, 78, 82, 85, 89, 90, 94, 95, 98, 100, 103, 104, 106, 110, 114, 116, 118, 122, 124, 125, 126, 127, 128, 130, 135, 140, 141, 142, 148, 150, 155, 158, 161, 164, 170, 172, 174, 177, 178, 182, 184, 188, 190, 199, 202, 205, 207

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Numbers n such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Numbers n for which A060990(n^2) = A263087(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..12443
A. Karttunen, Ratio a(n)/A263092(n) drawn with the help of OEIS Plot2-utility

Cf. A000005, A000196, A045765, A049820, A060990, A263098.
Complement: A263092.
Positions of zeros in A263087 and positions of ones in A263088.
Cf. A263095 (the squares of these numbers).

\\ Compute A263093 and A263095 at the same time:
A060990(n) = { my(k = n + 1440, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit 1440 is good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 1; k = 0; while((n^2)<1102701600, if((0 == A060990(n*n)), k++; write("b263093.txt", k, " ", n); write("b263095.txt", k, " ", (n*n)); ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263093 (MATCHING-POS 1 1 (lambda (n) (zero? (A060990 (* n n))))))
(define A263093 (ZERO-POS 1 0 A263087))

a(n) = A000196(A263095(n)).

A263250 Even bisection of A263087; number of solutions to x - d(x) = 4(n^2), where d(x) is the number of divisors of x (A000005).

Original entry on oeis.org

2, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 0, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A060990, A263087, A263251.

Cf. also A263252 (partial sums).

A060990(n) = { my(k = n + 2400, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit A002183(77)=2400 good for at least up to A002182(77) = 10475665200.
A263087(n) = A060990(n^2);
A263250(n) = A263087(2*n);
p = 0; for(n=0, 10000, k = A263250(n); p += k; write("b263250.txt", n, " ", k); write("b263252.txt", n, " ", p)); \\ Compute A263250 and A263252 at the same time.

Scheme
```
(define (A263250 n) (A263087 (+ n n)))
```

a(n) = A263087(2*n).

A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 29, 30, 31, 32, 33, 36, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 51, 52, 53, 55, 57, 61, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 91, 92, 93, 96, 97, 99, 101, 102, 105, 107, 108, 109

Offset: 0

Antti Karttunen, Oct 11 2015

nonn

Starting offset is zero, because a(0)=0 is a special case in this sequence.
Numbers n for which A060990(n^2) = A263087(n) > 0.
Numbers n for which A049820(x) = n^2 has a solution.

Antti Karttunen, Table of n, a(n) for n = 0..20763

Complement: A263093.
Cf. A000005, A000196, A049820, A060990, A236562, A263087, A263098.
Cf. A263094 (the squares of these numbers).
Cf. A262515 (a subsequence).

\\ Compute A263092 and A263094 at the same time:
A060990(n) = { my(k = n + 1440, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit 1440 good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 0; k = 0; while((n^2)<1102701600, if((A060990(n*n) > 0), write("b263092.txt", k, " ", n); write("b263094.txt", k, " ", (n*n)); k++; ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263092 (MATCHING-POS 0 0 (lambda (n) (not (zero? (A060990 (* n n)))))))
(define A263092 (NONZERO-POS 0 0 A263087))

A263095 Squares in A045765; numbers n^2 such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

25, 36, 49, 64, 100, 196, 256, 324, 400, 484, 676, 729, 784, 1156, 1225, 1369, 2116, 2209, 2500, 2916, 3136, 3364, 3481, 3600, 3844, 4489, 5329, 6084, 6724, 7225, 7921, 8100, 8836, 9025, 9604, 10000, 10609, 10816, 11236, 12100, 12996, 13456, 13924, 14884, 15376, 15625, 15876, 16129, 16384, 16900, 18225, 19600, 19881, 20164, 21904, 22500, 24025, 24964, 25921, 26896

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

Some of the terms are shared with A262687, but none with A262514.

Antti Karttunen, Table of n, a(n) for n = 1..12443

Cf. A000005, A049820, A010052, A060990, A262514, A262687, A263098.
Cf. A263093 (gives the square roots).
Intersection of A000290 and A045765.
Cf. also A263091.

lim = 40000; Take[Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], IntegerQ@ Sqrt@ # &], 60] (* Michael De Vlieger, Oct 13 2015 *)

PARI
```
\\ See code in A263093.
```

;; With Antti Karttunen's IntSeq-library.
(define A263095 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010052 n)) (zero? (A060990 n))))))

A263251 Odd bisection of A263087; number of solutions to x - d(x) = (2n+1)^2, where d(x) is the number of divisors of x (A000005).

Original entry on oeis.org

2, 1, 0, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 0, 0, 1, 2, 2, 0, 2, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 3, 2, 1, 0, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 2, 2, 0, 0, 1, 0, 1, 1, 1

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A060990, A263087, A263250.

Cf. also A263253 (partial sums).

A060990(n) = { my(k = n + 2400, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit A002183(77)=2400 good for at least up to A002182(77) = 10475665200.
A263087(n) = A060990(n^2);
A263251(n) = A263087((2*n)+1);
p = 0; for(n=0, 10000, k = A263251(n); p += k; write("b263251.txt", n, " ", k); write("b263253.txt", n, " ", p)); \\ Compute A263251 and A263253 at the same time.

(define (A263251 n) (A263087 (+ n n 1)))

a(n) = A263087(2*n + 1).

A264970 If A262686(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

12, 2, 11, 2, 1, 1, 10, 0, 0, 3, 2, 2, 9, 0, 1, 5, 1, 4, 8, 0, 0, 3, 7, 2, 0, 0, 2, 1, 0, 1, 6, 6, 1, 0, 5, 5, 0, 0, 6, 4, 0, 1, 4, 0, 1, 3, 3, 2, 5, 0, 0, 1, 0, 2, 2, 0, 0, 1, 1, 4, 4, 3, 3, 0, 0, 2, 0, 0, 0, 1, 2, 3, 3, 2, 0, 0, 2, 1, 4, 0, 1, 1, 3, 3, 2, 0, 2, 2, 0, 4, 3, 1, 1, 3, 2, 5, 1, 4, 0, 2, 0

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

a(n) = number of iterations of A262686 needed before zero is reached. In the context of tree (A263267) defined by edge-relation A049820(child) = parent, this is the number of hops we make before reaching one of the leaves (A045765), when we start from n and always select the largest child at each iteration.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A049820, A060990, A262686, A263267.
Cf. A045765 (positions of zeros).
One less than A264971.

If A060990(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A262686(n)).
Other identities. For all n >= 0:
a(n) = A264971(n) - 1.

A264971 If A262686(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

13, 3, 12, 3, 2, 2, 11, 1, 1, 4, 3, 3, 10, 1, 2, 6, 2, 5, 9, 1, 1, 4, 8, 3, 1, 1, 3, 2, 1, 2, 7, 7, 2, 1, 6, 6, 1, 1, 7, 5, 1, 2, 5, 1, 2, 4, 4, 3, 6, 1, 1, 2, 1, 3, 3, 1, 1, 2, 2, 5, 5, 4, 4, 1, 1, 3, 1, 1, 1, 2, 3, 4, 4, 3, 1, 1, 3, 2, 5, 1, 2, 2, 4, 4, 3, 1, 3, 3, 1, 5, 4, 2, 2, 4, 3, 6, 2, 5, 1, 3, 1

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

See comments at A264970.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A049820, A060990, A262686.
One more than A264970.
Number of significant terms on row n of A263271.

If A060990(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A262686(n)).
Other identities. For all n >= 0:
a(n) = 1 + A264970(n).

cp's OEIS Frontend

A262897 Nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent: a(n) = A259934(A262892(n)).

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A262903 Numbers that are not leaves but all of whose children are leaves in the tree generated by edge-relation A049820(child) = parent.

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A263091 Primes p for which A049820(x) = p has no solution.

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A263093 Numbers whose squares are in A045765.

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A263250 Even bisection of A263087; number of solutions to x - d(x) = 4(n^2), where d(x) is the number of divisors of x (A000005).

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Formula

A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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Scheme

A263095 Squares in A045765; numbers n^2 such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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Programs

Mathematica

PARI

Scheme

A263251 Odd bisection of A263087; number of solutions to x - d(x) = (2n+1)^2, where d(x) is the number of divisors of x (A000005).

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Programs

PARI

Scheme

Formula

A264970 If A262686(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).